Categorical Variables in Developmental Research: Methods of Analysis

Categorical Variables in

Developmental Research

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Categorical Variables in Developmental Research Methods of Analysis

Edited by

Alexander von Eye Michigan State University East Lansing, Michigan

Clifford C. Cloggm Pennsylvania State University University Park, Pennsylvania *Deceased

ACADEMIC PRESS

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London

Sydney Tokyo Toronto

This book is printed on acid-flee paper. ( ~ Copyright 9 1996 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. A c a d e m i c Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495

United Kingdom Edition publ&hed by Academic Press Limited 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Categorical variables in developmental research : methods of analysis / edited by Alexander von Eye, Clifford C. Clogg p. cm. Includes bibliographical references and index. ISBN 0-12-724965-6 1. Psychometrics. 2. Psychology--Statistical methods. 3. Categories (Mathematics) I. Eye, Alexander von. II. Clogg, Clifford C. BF39.C294 1995 155'.072--dc20 95-21950 PRINTED IN THE UNITED STATES OF AMERICA 95 96 97 98 99 00 QW 9 8 7 6 5

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Contents

Contributors xi Preface xiii Acknowledgments xvii In Memoriam xix

PART

1 Measurement and Repeated

Observations of Categorical Data 1. MeasurementCriteria for Choosing among Models with GradedResponses David Andrich

1. Introduction 3 2. Measurement Criteria for a Model for Graded Responses 4

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Contents

3. Models for Graded Responses 4. Examples 28 5. Summary and Discussion 32 References 34 11

Growth Modeling with Binary Responses

Bengt O. Muth~n

1. Introduction 37 2. Conventional Modeling and Estimation with Binary Longitudinal Data 39 3. More General Binary Growth Modeling 42 4. Analyses 46 5. Conclusions 52 References 52 11

Probit Models for the Analysis of Limited Dependent Panel Data Gerhard Arminger

1. 2. 3. 4.

Introduction 55 Model Specification 56 Estimation Method 61 Analysis of Production Output from German Business Test Data 68 5. Conclusion 72 References 73

PART 2 Catastrophe Theory 4. CatastropheAnalysis of Discontinuous Development

Han L. J. van der Maas and Peter C. M. Molenaar

1. 2. 3. 4. 5. 6.

Introduction 77 Catastrophe Theory 79 Issues in Conservation 80 The Cusp Model 84 Empirical Studies 89 Discussion 101 References 104

Contents B

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Catastrophe Theory of Stage Transitions in Metrical and Discrete Stochastic Systems

Peter C. M. Molenaar and Pascal Hartelman

1. Introduction 107 2. Elementary Catastrophe Theory 111 3. Catastrophe Theory for Metrical Stochastic Systems 115 4. Catastrophe Theory for Discrete Stochastic Systems 125 5. General Discussion and Conclusion 128 References 129

PART 3 Latent Class and Log-Linear Models

6. Some Practical Issues Related to the Estimation of Latent Class and Latent Transition Parameters

Linda M. Collins, Penny L. Fidler, and Stuart E. Wugalter

1. Introduction 133 2. Methods 137 3. Discussion 144 References 146

7. ContingencyTables and BetweenSubject Variability Thomas D. Wickens

1. 2. 3. 4. 5.

Introduction 147 Association Variability 148 The Simulation Procedure 150 Tests Based on Multinomial Variability 152 Tests Based on Between-Subject Variability 156 6. Procedures with Two Types of Variability 161 7. Discussion 163 References 167

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Contents

8. AssessingReliability of Categorical Measurements Using Latent Class Models Clifford C. Clogg and Wendy D. Manning

1. Introduction 169 2. The Latent Class Model: A Nonparametric Method of Assessing Reliability 171 3. Reliability of Dichotomous Measurements in a Prototypical Case 174 4. Assessment of Reliability by Group or by Time 178 5. Conclusion 181 References 182 W

Partitioning Chi-Square: Something Old, Something New, Something Borrowed, but Nothing BLUE (Just ML) David Rindskopf

1. 2. 3. 4. 5.

Introduction 183 Partitioning Independence Models 184 Analyzing Change and Stability 190 How to Partition Chi-Square 195 Discussion 199 References 201

10. Nonstandard Log-Linear Models for Measuring Change in Categorical Variables

Alexander von Eye and Christiane Spiel 1. 2. 3. 4.

Introduction 203 Bowker's Test 204 Log-Linear Models for Axial Symmetry 205 Axial Symmetry in Terms of a Nonstandard Log-Linear Model 206 5. Group Comparisons 209 6. Quasi-Symmetry 210 7. Discussion 213 References 214

Contents

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11. Application of the Multigraph

Representation of Hierarchical Log-Linear Models H. J. Khamis

Introduction 215 2. Notation and Review 216 3. The Generator Multigraph 217 4. Maximum Likelihood Estimation and Fundamental Conditional Independencies 5. Examples 222 6. Summary 228 References 229 ,

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PART 4 Applications 12. Correlation and Categorization under a Matching Hypothesis

Michael J. Rovine and Alexander von Eye

1. 2. 3. 4. 5. 6. 7. 8. .

10. 11.

Introduction 233 An Interesting Plot 234 The Binomial Effect Size Display 236 An Organizing Principle for Interval-Level Variables 237 Definition of the Matching Hypothesis 237 A Data Example 238 Correlation as a Count of Matches 241 Correlation as a Count of How Many Fall within a Set Range 243 Data Simulation 244 Building Uncertainties from Rounding Error into the Interpretation of a Correlation 246 Discussion 247 References 248

13. Residualized Categorical Phenotypes and Behavioral Genetic Modeling Scott L. Hershberger

1. The Problem 249 2. Weighted Least-Squares Estimation

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Contents

3. Proportional Effects Genotype-Environment Correlation Model 253 4. Method 256 5. Results 258 6. Conclusions 271 References 273 Index

275

Contributors

Numbers in parentheses indicate the pages on which the authors' contributions begin.

David Andrich (3) School of Education, Murdoch University, Murdoch,

Western Australia 6150, Australia Gerhard Arminger (55) Department of Economy, Bergische Universit~it, Wuppertal, Germany Clifford C. Clogg (169) Departments of Sociology and Statistics and Population Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802 Linda M. Collins (133) The Methodology Center and Department of Human Development and Family Studies, Pennsylvania State University, University Park, Pennsylvania 16802

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Contributors

Penny L. Fidler (133) J. P. Guilford Laboratory of Quantitative Psychology, University of Southern California, Los Angeles, California 90089 Pascal Hartelman (107) Faculty of Psychology, University of Amsterdam, The Netherlands Scott L. Hershberger (249) Department of Psychology, University of Kansas, Lawrence, Kansas 66045 H. J. Khamis (215) Statistical Consulting Center and Department of Community Health, School of Medicine, Wright State University, Dayton, Ohio 45435 Wendy D. Manning (169) Department of Sociology, Bowling Green State University, Bowling Green, Ohio 43403 Peter C. M. Molenaar (77, 107) Faculty of Psychology, University of Amsterdam, The Netherlands Bengt O. Muth~n (37) Graduate School of Education, University of California, Los Angeles, Los Angeles, California 90095 David Rindskopf (183) Educational Psychology, City University of New York Graduate School, Chestnut Ridge, New York 10977 Michael J. Rovine (233) Human Development and Family Studies, Pennsylvania State University, University Park, Pennsylvania 16802 Christiane Spiel (203) Department of Psychology, University of Vienna, Vienna, Austria Han L. J. van der Maas (77) Department of Developmental Psychology, University of Amsterdam, The Netherlands Alexander yon Eye (203, 233) Department of Psychology, Michigan State University, East Lansing, Michigan 48824 Thomas D. Wickens (147) Department of Psychology, University of California, Los Angeles, Los Angeles, California 90095 Stuart E. Wugalter (133) J. P. Guilford Laboratory of Quantitative Psychology, University of Southern California, Los Angeles, California 90089

Preface

Categorical variables come in many forms. Examples include classes that are naturally categorical such as gender or species; groups that have been formed by definition, such as classes of belief systems or classifications into users versus abusers of alcohol; groups that result from analysis of data, such as latent classes; classes that reflect grading of, for instance, performance or age; or categories that were formed for the purpose of discriminating as, for instance, nosological units. Developmental researchers have been reluctant to include categorical variables in their studies. The main reason for this reluctance is that it often is thought difficult to include categorical variables in plans for data analysis. Methods and computer programs for continuous variables seem to be more readily accessible. This volume presents methods for analysis of categorical data in developmental research. Thus, it fills the void perceived by many, by providing infor-

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mation in a very understandable fashion, about powerful methods for analysis of categorical data. These methods go beyond the more elementary tabulation and • methods still in widespread use. This volume covers a broad range of methods, concepts, and approaches. It is subdivided into the following four sections: 1. Measurement and Repeated Observations of Categorical Data 2. Catastrophe Theory 3. Latent Class and Log-Linear Models 4. Applications The first section, Measurement and Repeated Observations of Categorical Data, contains three chapters. The first of these, by David Andrich, discusses measurement criteria for choosing among models with graded responses. Specifically, this chapter is concerned with the important issues of criteria for measurement and choice of a model that satisfies these criteria. Proper selection of a measurement model ensures that measurement characteristics can be fully exploited for data analysis. The second chapter in the first section is by Bengt Muthrn. Titled Growth Modeling with Binary Responses, it describes methods for analysis of individual differences in growth or decline. It presents modeling with random intercepts and slopes when responses are binary. The importance of this paper lies in its demonstration of how to allow for restrictions on thresholds across time and differences across time in the variances of the underlying variables. The third chapter in this section, by Gerhard Arminger, is on Probit Models for the Analysis of Limited-Dependent Panel Data. It presents methods for analysis of equidistant panel data. Such data often involve metric and nonmetric variables that may be censored metric, dichotomously ordered, and unordered categorical. The chapter presents an extension of Heckman's (1981) models. The new methods allow researchers to analyze censored metric and ordered categorical variables and any blend of such variables using general threshold models. The second section of this volume contains two chapters on Catastrophe Theory. The first of these chapters, authored by Han van der Maas and Peter Molenaar, is Catastrophe Analysis of Discontinuous Development. This chapter presents results from application of catastrophe theory to the study of infants' acquisition of the skill of conservation. This skill, well-known since Piaget's investigations, is a very important step in a child's cognitive development. However, proper methodological analysis has been elusive. This paper presents methods and results at both the individual and the aggregate level. The second chapter in this section, by Peter Molenaar and Pascal Hartelman, addresses issues of Catastrophe Theory of Stage Transitions in Metrical and Discrete Stochastic Systems. This paper is conceptual in nature. It first gives an outline of elementary catastrophe theory and catastrophe theory for stochastic metrical systems. The chapter presents a principled approach to dealing with

Preface

XV

problems from Cobb's approach and introduces an approach for dealing with catastrophe theory of discrete stochastic systems. The importance of this chapter lies in the new course it charts for analysis of both continuous and categorical developmental change processes using catastrophe theory. The third section covers Latent Class and Log-Linear Models. Because of the widespread use of these models, they were allotted more space in this volume. The section contains six chapters. The first chapter, by Linda Collins, Penny Fidler, and Stuart Wugalter, addresses Some Practical Issues Related to Estimation of Latent Class and Latent Transition Parameters. Both of the issues addressed are of major practical relevance. The first concerns estimability of parameters for latent class and latent transition models when sample sizes are small. The second issue concerns the calculation of standard errors. The paper presents computer simulations that yielded very encouraging results. The second chapter in this section, by Thomas Wickens, covers issues of Contingency Tables and Between-Subject Variability. Specifically this chapter addresses problems that arise when between-subject variability prevents researchers from aggregating over subjects. Like the chapter before, this chapter presents simulation studies that show that various statistical tests differ in Type-I error rates and power when there is between-subject variability. The importance of this contribution lies in the presentation of recommendations for analysis of categorical data with between-subject variability. The third chapter in this section, by Clifford Clogg and Wendy Manning, is Assessing Reliability of Categorical Measurements Using Latent Class Models. This paper addresses the important topic of reliability assessment in categorical variables. The authors propose using the framework of the latent class model for assessing reliability of categorical variables. The model can be applied without assuming sufficiency of pairwise correlations and without assuming a special form for the underlying latent variable. The importance of this paper lies in that it provides a nonparametric approach to reliability assessment. The fourth chapter, by David Rindskopf, is Partitioning Chi-Square: Something Old, Something New, Something Borrowed, but Nothing BLUE (just ML). Partitioning chi-square is performed to identify reasons for departure from independence models. Rindskopf introduces methods for partitioning the likelihood chi-square. These methods avoid problems with the Lancaster methods of partitioning the Pearson chi-square. They are exact without needing complex adjustment formulas. Change-related hypotheses can be addressed using Rindskopf' s methods. The fifth chapter, by Alexander von Eye and Christiane Spiel, is Extending the Bowker Test for Symmetry Using Nonstandard Log-Linear Models for Measuring Change in Categorical Variables. The chapter presents three ways to formulate tests of axial symmetry in square cross-classifications: the Bowker test, standard log-linear models, and nonstandard log-linear models. The advan-

xvi

Preface

tage of the latter is that one can devise designs that allow one to simultaneously test symmetry and other developmental hypotheses. The sixth chapter in this section, by Harry Khamis, introduces readers into the Application of the Multigraph Representation of Hierarchical Log-Linear Models. The chapter focuses on hierarchical log-linear models. It introduces the generator multigraph and shows, using such graph-theoretic concepts as maximum spanning trees and edge cutsets, that the generator multigraph provides a useful tool for representing and interpreting hierarchical log-linear models. The fourth section of this volume contains application-oriented chapters. The first of these, written by Michael Rovine and Alexander von Eye, reinterprets Correlation and Categorization under a Matching Hypothesis. The authors show the relationship between the magnitude of a correlation coefficient and the number of times cases fall into certain segments of the range of values. The relationship of the correlation coefficient with the binomial effect size display is shown. The second chapter of this section, contributed by Scott Hershberger, discusses methods for Residualized Categorical Phenotypes and Behavioral Genetic Modeling. The paper presents methods for behavior genetic modeling of dichotomous variables that describe dichotomous phenotypes.

Alexander von Eye Clifford C. Clogg

Acknowledgments

We are indebted to many people who supported us during the production phase of this book. First there is the Dean of the Pennsylvania State College of Health and Human Development, Gerald McClearn. We thank him for his generous support. We thank Tina M. Meyers for help and support, particularly in phases of moves and transition. We thank a number of outside reviewers for investing time and effort to improve the quality of the chapters. Among them are Holger We[3els of the NIH, Ralph Levine of MSU, and Phil Wood of the University of Missouri who all read and commented on several chapters. We thank the authors of this volume for submitting excellent chapters and for responding to our requests for changes in a very professional fashion. We thank Academic Press, and especially Nikki Fine, for their interest in this project and for smooth and flexible collaboration during all phases of production. Most of all we thank our families, the ones we come from and the ones we live in. They make it possible for us to be what we are.

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In Memoriam

Clifford C. Clogg died on May 7, 1995. In Cliff we all lose a caring friend, a dedicated family man, a generous colleague, and a sociologist and statistician of the highest caliber. Cliff was a person with convictions, heart, and an incredibly sharp mind; he was a person to admire and to like. We needed him then and we still do now. The manuscripts for this book were submitted to the publisher two weeks before this tragedy. This includes the preface, which I left unchanged. Alexander von Eye

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..P A.R.T~ Measurement and

Repeated

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Measurement Criteria for Choosing among Models with Graded Responses 9

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David Andrich

Murdoch University Western Australia

1. INTRODUCTION It is generally accepted that measurement has been central to the advancement of empirical physical science. The prototype of measurement is the use o f an instrument to map the amount of a property on a real line divided into equal intervals by thresholds sufficiently fine that their own width can be ignored, and in its elementary form; this is understood readily by young school children. However, the function of measurement in science goes much deeper: it is central to the simultaneous definition of variables and the formulation of quantitative scientific theories and laws (Kuhn, 1961). Furthermore, when proper measurement has taken place, these laws take on a simple multiplicative structure (Ramsay, 1975). Although central to formulating physical laws, it is understood that measurement inevitably contains error. In expressions of deterministic theories, these errors are considered sufficiently small that they are ignored. In practice, the mean of independent repeated measurements, the variance of which is inversely Categorical Variables in Developmental Research: Methods of Analysis Copyright

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in any form

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David Andrich

proportional to the number of measurements, can be taken to increase precision to a point where errors can indeed be ignored. It is also understood that instruments have operating ranges; in principle, however, the measurement of an entity is not a function of the operating range of any one instrument, but of the size of the entity. Graded responses of one kind or another are used in social and other sciences when no measuring instrument is available, and these kinds of graded responses mirror the prototype of measurement in important ways. First, the property is envisaged to be continuous, such as an ability to perform in some domain, or an intensity of attitude, or the degree of a disease; second, the continuum is partitioned into ordered adjacent (contiguous) intervals, usually termed categories, that correspond to the units of length on the continuum. In elementary treatments of graded responses, the prototype of measurement is followed closely in that the successive categories are simply assigned successive integers, and these are then treated as measurements. In advanced treatments, a model with a random component is formalized for the response and classification processes, the sizes of the intervals are not presumed equal, and the number of categories is finite. This chapter is concerned with criteria, and the choice of a model that satisfies these criteria, so that the full force of measurement can be exploited with graded responses of this kind. The criteria are not applied to ordered variables where instruments for measurement already exist, such as age, height, income expressed in a given currency, and the like, which in a well-defined sense already meet the criteria. Although one new mathematical result is presented, this chapter is also not about statistical matters such as estimation, fit, and the like, which are already well established in the literature. Instead, it is about looking at a relatively familiar statistical situation from a relatively nonstandard perspective of measurement in science. Whether a variable is defined through levels of graded responses that characterize more or less of a property, or whether it is defined through the special case of measurement in which the accumulation of successive amounts of the property can be characterized in equal units, central to its definition is an understanding of what constitutes more or less of the property and what brings about changes in the property. It is in expressing this relationship between the definition and changes in terms of a model for measurement that generalizes to graded responses, and in articulating a perspective of empirical enquiry that backs up this expression, that this chapter contributes to the theme of the analysis of categorical variables in developmental research.

2. MEASUREMENTCRITERIA FOR A MODEL FOR GRADED RESPONSES In this section, three features of the relationship between measurement and theory are developed as criteria for models for measurement: first, the dominant direc-

1. Measurement Criteria for Choosing among Models with Graded Responses

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tion of the relationship between theory and measurement; second, the structure of the models that might be expected to apply when measurements have been used; and third, the invariance of the measurement under different partitions of the continuum. These are termed measurement criteria. It is stressed that in this argument, the criteria established are independent and a priori to any data to which they might apply. In effect, the model chosen is a formal rendition of the criteria; it expresses in mathematical terms the requirements to which the graded responses must conform if they are to be like measurements, and therefore the model itself must exhibit these criteria. Thus the model is not a description of any set of data, although it is expected that data sets composed of graded responses can be made to conform to the model, and that even some existing ones may do so. Moreover, the criteria are not assumptions about any set of data that might be analyzed by the model. If data collected in the form of graded responses do not accord with the model, then they do not meet the criteria embedded in the model, but this will not be evidence against the criteria or the model. Thus it is argued that graded responses, just like measurements, should subscribe to certain properties that can be expressed in mathematical terms, and also that the data should conform to the chosen model and not the other way around; that is, the model should not be chosen to summarize the data. This position may seem nonstandard, and because it is an aspect of a different perspective, it has been declared at the outset. It is not, however, novel, having been presented in one form or another by Thurstone (1928), Guttman (1950), and Rasch (1960/1980), and reinforced by Duncan (1984) and Wright (1984).

2.1. Theory Precedes Measurement Thomas Kuhn is well known for his theory of scientific revolutions (Kuhn, 1970). In this chapter, I will invoke a part of his case, apparently much less known than the revolutionary theory itself, concerning the function of measurement in science (Kuhn, 1961) in which he stands the relationship between measurement and theory as traditionally perceived on its head: In text books, the numbers that result from measurements usually appear as the archetypes of the "irreducible and stubborn facts" to which the scientist must, by struggle, make his theories conform. But scientific practice, as seen through the journal literature, the scientist often seems rather to be struggling with the facts, trying to force them to conformity with a theory he does not doubt. Quantitative facts cease to seem simply "the given." They must be fought for and with, and in this fight the theory with which they are to be compared proves the most potent weapon. Often scientists cannot get numbers that compare well with theory until they know what numbers they should be making nature yield. (Kuhn, 1961, p. 171)

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Kuhn (1961) elaborates the . . . "paper's most persistent thesis: The road from scientific law to scientific measurement can rarely be traveled in the reverse direction" (p. 219, emphasis in original). If this road can be seldom traveled in the physical sciences, then it is unlikely to be traveled in the social ' sciences. Yet, I suggest that social scientists attempt to travel this route most of the time by modeling available data, that is, by trying to find models that will account for the data as they appear. In relentlessly searching for statistical models that will account for the data as given, and finding them, the social scientist will eschew one of the main functions of measurement, the identification of anomalies: To the extent that measurement and quantitative technique play an especially significant role in scientific discovery, they do so precisely because, by displaying serious anomaly, they tell scientists when and where to look for a new qualitative phenomenon. To the nature of that phenomenon, they usually provide no clues. (Kuhn, 1961, p. 180) And this is because When measurement departs from theory, it is likely to yield mere numbers, and their very neutrality make them particularly sterile as a source of remedial suggestions. But numbers register the departure from theory with an authority and finesse that no qualitative technique can duplicate, and that departure often is enough to start a search. (Kuhn, 1961, p. 180) Although relevant in general, these remarks are specifically relevant to the role of measurement and therefore to the role that graded responses can have. Measurement formalizes quantitatively and efficiently a theoretical concept that can be summarized as a variable in terms of degree, similar in kind but greater or lesser in intensity, in terms of more or less, greater or smaller, stronger, or weaker, better or worse, and so on, that is to be studied empirically. If it is granted that the variable is an expression of a theory, that is, that it is an expression of what constitutes more or less, greater or smaller, and so on, according to the theory, then when studied empirically, it becomes important to invoke another principle of scientific enquiry, that of falsifiability (Popper, 1961). Although Popper and Kuhn have disagreed on significant aspects of the philosophy of science, the idea that any attempt at measurement arises from theory, and that measurement may provide, sooner or later, evidence against the theory, is common in both philosophies. The implication of this principle to the case of graded responses is that even the operational ordering of the categories, which defines the meaning of more or less of the variable, should be treated as a hypothesis. Clearly, when a researcher decides on the operational definition of the ordered categories, there is a strong conviction about their ordering, and any chosen model should reflect this. However, if the categories are to be treated as a hypothesis, then the model used to characterize the response process must itself

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also have the ordering as a hypothesis. Thus, the model should not provide an ordering irrespective of the data but should permit evidence to the contrary to arise: If the data refute the ordering, then an anomaly that must be explained is revealed. The explanation might involve no more than identifying a coding error, although most likely it will clarify the theory: however, whatever its source, the anomaly must be explained.

2.2. Fundamental Measurement and Laws in Physical Science Because of the strong analogy to be made between measurement in the physical sciences and the construction and operation of graded responses, any model chosen should be consistent with how measurements operate in the physical sciences. As already noted, graded responses differ from traditional measurements in that (a) they are finite in number, (b) the categories (units) are not equal in size, and (c) assignment of an entity to a category contains an uncertainty that must be formalized. Therefore, the model should specialize to the case in which the number of categories from an origin is, in principle, not finite, in which the sizes of the categories are equal (a constant unit), and in which measurement precision is increased to the point where the random component can be ignored. In general terms, the model should specialize to fundamental measurement. Fundamental measurement is an abstraction from the idea of concatenation of entities in which the successive addition of measurements corresponds to the concatenation (Wright, 1985, 1988). Measurements of mass exemplify fundamental measurement, in that if two entities are amalgamated, then the amalgamated entity behaves with respect to acceleration as a single body with a mass the sum of the masses of the original entities. It seems that because successive concatenations of a unit can be expressed as a multiplication of the unit, measurement leads to laws that have a multiplicative structure among the variables: "Throughout the gigantic range of physical knowledge, numerical laws assume a remarkably simple form provided fundamental measurement has taken place" (Ramsay, 1975, p. 262) and "Virtually all the laws of physics can be expressed numerically as multiplications or divisions of measurements" (p. 258). Thus it might be expected that any model that is to form a criterion for measurement would have this multiplicative structure, although this alone would not be sufficient to meet the criterion: it would still have to be demonstrated that when specialized to the case of physical measurement, the model leads to increased precision as the units are made smaller, and that the model reflects the process of concatenation.

2.3. Invariance of Location across an Arbitrary Partitioning of the Continuum Because an entity can be measured with different units, it is expected that the location of any entity on the continuum should be invariant when measured in

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different units, although with smaller units greater precision is expected. This difference in units corresponds to different partitions of the continuum. Therefore, this invariance of location across different partitions of the continuum should be met by graded responses (of which measurement is a special case).

2.4. Summary of Measurement Criteria In summary, the measurement criteria for a model for graded responses are 1. it should contain the ordering of the categories as a falsifiable hypothesis, 2. it should specialize to fundamental measurement, and 3. the location of an entity should be invariant under different partitions of the continuum. It should not come as a surprise that the preceding criteria are in some sense related, but they have been listed separately to focus on the salient features of measurement and its function. A model that satisfies these criteria provides a special opportunity for learning about variables in the form of graded responses. In the next section, these criteria are taken in reverse order, and the model that satisfies the first criterion by definition is presented first and then is shown to satisfy the other two criteria as well.

2.5. Statistical Criteria Partly because contrasts are helpful in any exposition, and partly because there is historically an alternate model for graded responses, this alternate model is outlined here. This model, too, has been presented as satisfying certain principles of scientific enquiry (McCullagh, 1980, 1985) for graded responses. These criteria, in contrast to what have been referred to as measurement criteria, will be referred to as statistical criteria: (i) If, as is usually the case, the response scale contains useful and scientifically important information such as order or factorial structure, the statistical model should take this information into account. (ii) If the response categories are on an arbitrary but ordinal scale, it is nearly always appropriate to consider models that are invariant under the grouping of adjacent categories (McCullagh, 1980). (iii) If sufficient care is taken in the design and execution of the experiment or survey, standard probability distributions such as the Poisson or multinomial may be applicable, but usually the data are over d i s p e r s e d . . . (McCullagh, 1985, p. 39). These statistical criteria seem compatible with the measurement criteria. However, it will be seen that the models chosen to meet each set of criteria are in

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fact incompatible and serve different situations. It is in part to help understand the basis of the choices and the change in perspective entailed in choosing the former model ahead of the latter for graded responses that the measurement criteria have been set in a wider context of the relationship between data, theory, and models. The distinction made here between measurement and statistical criteria is compatible with the same kind of distinction made in Duncan and Stenbeck (1988).

3. MODELS FOR GRADED RESPONSES The model that satisfies the criteria of invariance with graded responses arises from the Rasch (1961) class of models, and its properties have been elaborated by Andersen (1977) and Andrich (1978, 1985). The historically prior model is based on the work of Thurstone (Edwards & Thurstone, 1952) and has been further elaborated by Samejima (1969), Bock (1975), and McCullagh (1980, 1985).

3.1. Rasch Cumulative Threshold Model Rasch (1961) derived a class of models for measurement explicitly from the requirements of invariant comparisons. Because a measurement is the comparison of a number of units to a standard unit, and therefore the comparison of two measurements is indirectly a comparison of these comparisons with a standard, it is perhaps not surprising that the relationship can be taken in reverse, so that if conditions are imposed on comparisons, they may lead to measurement. Rasch's parsimonious criteria are the following. A. "The comparison between two stimuli should be independent of which particular individuals were instrumental for the comparison; and it should also be independent of which other stimuli within the considered class were or might also have been compared." B. "Symmetrically, a comparison between two individuals should be independent of which particular stimuli within the class considered were instrumental for comparison; and it should also be independent of which other individuals were also compared, on the same or on some other occasion (Rasch, 1961, p. 322)." The probabilistic models that satisfy these conditions have sufficient statistics for the parameters of the individuals and for the stimuli, which will be referred to in general now as entities and instruments, respectively. The existence of sufficient statistics means that the outcome space can be partitioned so that within the subspaces the distribution of responses depends only on either the

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parameters of the instrument or the entity, but not on both, thus providing the required invariance of comparison, either among instruments or among entities. The model for the probability Pr{xpi } of a response with respect to entity p and instrument i that satisfies these conditions of invariance can be expressed in different forms, one efficient form being Pr{xpi } = "rki + xpi[3p , ~1 exp 'Ypi k= 1

where ~pi = ~

exp - ~

x -- 0

-oO<~p
-rki + Xpi~p

-oc<'rxi<

oo, x =

1, m,

(1)

is a normalizing factor ensuring that the

k=l

probabilities sum to 1.0, J3p is the location of entity p, and -rxi, x = 1. . . . mi are m thresholds of instrument i which partition the continuum into m i + 1 categories, where "rxi > "rx(i_ 1), x = 2 . . . . m, and Xpi E {0,1,2 . . . . m} indicates the c o u n t of the number of thresholds exceeded: x therefore denotes the category of any response, just as in the prototype of measurement, and x = 0 denotes that no threshold has been exceeded. For completeness, although it does not exist, let "roi- 0. Because of the structure of the threshold parameters in Equation (1), the model is termed the c u m u l a t i v e t h r e s h o l d m o d e l (CTM), and because ~pi is always the sum of the numerators, it will not be redefined in subsequent expressions of the equation. Figure 1 shows its probability distribution. 3.1.1. Invariance under different partitions of the continuum The invariance property can be demonstrated readily, and has been (e.g., Andersen, 1977), but because a special point is to be emphasized, it will be demonstrated again briefly. Suppose there are two instruments i and j with m i and mj, m i 4= my, thresholds and m i + 1 and mj + 1 categories, respectively, that are supposed to elicit the same property in any entity p and the responses are supposed to conform to the model of Equation (1). Because the instruments have different numbers of thresholds, they must partition the continuum in two different ways. (They could of course partition the continuum in different ways even if they had the same number of thresholds.) Retaining the generality of different numbers of categories, Pr{(Xpi,Xpj) JrpiJ = Xpi + xpj} =

where

"Yrpij

=

1 exp - ? l "rki "~rp ij =

min(r, mi) ( Xpi XI~j = = E exp --Z~rki--Z~rkj max( O,r-- mi)

1

"rkj ,

(2)

=

. Equation (2) does not involve

1. Measurement Criteria for Choosing among Models with Graded Responses

11

FIGURE 1. Probability of response in each category as a function of the location of the entity and the thresholds according to the Rasch model.

the location parameter of the entity; therefore, when generalized, it permits the estimation of the threshold parameters of different instruments with different numbers of categories, and the thresholds so estimated reflect the different partitions o f the continuum independently o f the location parameters ~3p of the entities p, p = 1, E To illustrate Equation (2), suppose m i = 3, mj = 4 for instruments i and j, respectively. Then the outcome space for the response of entity p to each of these instruments is shown in the following matrix in which each element rpij is the sum of the outcomes; that is, rpij = Xpi -~ Xpj.

rpu Xpi

0

1

0 1 2 3

1 2 3 4

Xp~ 2 3 2 3 4 5

3 4 5 6

4 4 5 6 7

For each total score rpij, except 0 and m i -Jr-mj, there is more than one way in which the same total score can be obtained. Thus rpij partitions the outcome space into subspaces. For example, rpi~ = 2 can be obtained from {(2,0), (1,1), (0,2) }. The probability of each of these pairs of outcomes for entity p, taken to be independent, is given by

12

David Andrich

Pr{2,0} =

e x p ( - a ' l i - q'2i + 213p)

~pi Pr{ 1,1} =

9

1

,

"~pj

exp(-'rli + [3p) exp(-'rl9 + [3p)

~lpi 1 Pr{0,2} = - - .

"~/pj

exp(-'rl/-

'~pi

'r2j + 2~p)

'~pj

The probability of the sum rpi9 = 2, therefore, is given by Pr{ rpij = 2 } = Pr{ (2,0), (1,1), (0,2) } e x p ( - ' r l i - q'2i + 2 ~ p ) + e x p ( - ' r l i -

"rU + 2[~p) + exp(--'r~/

q'2j + 2[~p)

~pi~lpj Consider now the probability of any one of these outcomes, say (0, 2), given that the total is rpij - 2. This conditional probability is given by Pr{(O,2)lXpi + Xpj = 2} =

Pr{(0,2)} Pr{ (2,0), (1,1), (0,2) } e x p ( - ' r mj - "rzj + 2~p)[~lpi'Ypj

[ e x p ( - ' r l i - q ' 2 i - 2~p) + e x p ( - - ' r l i - "rU + 2~p) + exp(--'rl/-- "r2j + 2~p) exp(2[3p)eXp(--,rlj -- a-zj) exp(2[3p)[exp(-a-li- "r2i ) -Jr- e x p ( - - r ~ i - -rlj) + exp(-'r~j - a'2j)] exp(-'r 19 - a'29) exp(-'rli-

q'2i) + e x p ( - ' r l i -

"rlj) + exp(-'rlj - "r2j)'

which is independent of the location of the entity [3p and is the special case of Equation (2). The symmetrical case can also be readily demonstrated. Suppose two entities p and q are classified using instrument i with m i thresholds. Then, let S(xpi,Xqi ) = {(Xpi,Xqi),x = O,mi} be the outcome space of all possible pairs of responses which is partitioned into subspaces S*(xpi,Xqi ) -- {(Xpi,Xqi),(Xqi,Xpi)} of pairs for Xpi 4: Xqi, Xpi,Xqi E {0,1,2 . . . . mi}. Then, Pr{ (Xpi,Xqi) l S*(xpi,Xqi) } --

"~pqi

exp{Xpi~p i + Xqi6qi},

(3)

where ~lpqi--" exp{xpi~p + Xqi~q} + exp{Xqi~3p + Xpi~q} is independent of the thresholds of instrument i, that is, independent of the partition of the continuum. For example, consider an instrument i with m i -- 3 and outcomes Xpi = 1, Xqi "-- 2 for entities p and q, respectively. The outcome space for the response of entities p and q is given by the following matrix.

1. Measurement Criteria for Choosing among Models with Graded Responses

13

q

S(xpi,Xqi) p

Then for X p i - 1, Pr{ (lpi,2qi)[S* ( lpi,2qi) }

0 1 2 3

0

1

2

(0,0) (1,0) (2,0) (3,0)

(0,1) (1,1) (2,1) (3,1)

(0,2) (0,3) (1,2) (1,3) (2,2)(2,3) (3,2) (3,3)

Xqi = 2,

3

S*(lpi,2qi ) - {(lpi,2qi),

(2pi, lqi)} ,

and

exp(-'rli + [3p)exp(-'r~i- T2i-+- 2~q)/'~pfypj [exp(-'rli + [3p)exp(-'rli- q'z/ -+- 213p) + e x p ( - ' r l i - T2i-+- [3p)eXp(-'rli + ~q)]/'~pi'~pj e x p ( - ' r l i - "rli- "rzi)exp(l[3 p + 213q) e x p ( - ' r i i - - r l i - 'rzi)[exp(l[3 p + 213q) + exp(2[3p + l[3q)] exp([3p + 213q) exp([3p + 213q) + exp(2[3p + [3q) which is a special case of Equation (3). When generalized, Equation (3) could be used to estimate the parameters of many entities, although usually, alternate methods are used to estimate these parameters. The point is that Equation (3) itself is independent of the parameters of the instruments and how they partition the continuum. It is important, however, to appreciate that this invariance is of the location of the entities and the thresholds of the instruments, and not of the distributions of the outcomes themselves. In fact, the distributions themselves are not invariant in that if the data conform to the model, and two adjacent categories are collapsed, then the data will no longer conform to the model (Andersen, 1977" Andrich, 1978, 1992" Jansen & Roskam, 1986" Rasch, 1966). This may seem surprising, and it is central to the contrast between the measurement criteria of this chapter and the statistical criteria of McCullagh (1985) for a model for graded responses. It will be considered again when the alternate model is constructed. Although the preceding analysis demonstrates that the model does have the relevant invariance properties, as indicated earlier, this model is unique in having these properties for discrete data, resting as it does on the existence of sufficient statistics (Andersen, 1977).

3.1.2. Parametric structure and specialization to equal units from a natural origin Equation (1) has the required parametric structure of fundamental measurement expressed in the additive or logarithmic form. To see how the

14

David Andrich

model specializes to the case of physical measurement, it is instructive to use the multiplicative form, characteristic of equations in physical laws. Let log ~p = [3p and log o~x = %, then Equation (1) becomes

Pr{xpi} =

1

~p'"

-xpi- ,

'~pi H

m~i > O,

~p > O,

(4)

COki

k=O

where now the parameters must be greater than 0. Thus in the model, and in the multiplicative metric, a value of 0 provides the natural origin, as "roi -~ 0, tOoi ~ 1. Suppose that the first threshold in instrument i has the value o.)i as before, and then that each of the successive thresholds is an additional distance coi from the previous threshold. Thus the first threshold can be conceived of as one unit from the origin, with 03i being the unit of measurement, just as in the prototype of measurement. Then, the thresholds take on the value Xpi

COoi =

1

,

O.)xi = X(Di,

X = 1

,.

9

and

m,

.,

~I COki --- Xpilt'~xpi . w

!

9

k=O

Next, suppose that the number of categories (units) is not finite, but that in principle, it may be any count of the number of units" this being determined by the amount of the property and not by the finite range of a particular instruxpi

ment. Replacing m by ~ and I-I tOki by Xpi!to~," in Equation (4) gives k=0

Pr{xpi } -

However, with ~pi

~r x ~ O

I

(GI~,)~, '' ~ ,

"Ypi

X!

x - 0.....

(~#(.Di) xpi

~.

(5)

= exp(~p/eOi), Equation (5) is the 9

9

Poisson distribution, giving, in the conventional mode of expression,

Pr{xpi} = e -;~'/~~ (G/~

x!

(6)

Thus, repeated measurements with error specialize to the Poisson distribution as a function of the location of the entity and the size of the unit of the instrument. This result in itself, novel and perhaps surprising, has many implications. Lest they detract from the point of the chapter, only two will be considered here, the precision of measurement as a function of the size of the unit and the addition of measurements in relation to concatenation of entities. These are central to the specialization of graded responses to the physical measure-

1. Measurement Criteria for Choos&g among Models with Graded Responses

15

ment and can be studied by using only the well-known result that the mean and variance of a Poisson distribution are given by its parameter: E[Xpi] -- V[Xpi] ~- ~p/(D i.

Suppose that, as in physical measurement, a unit toI = 1 of instrument I is chosen as a standard. In this unit, ~p/tOI = ~p, giving E[Xpl ] = ~p, and just as in physical measurement, any observed count of units exceeding the origin is taken immediately as an unbiased estimate of the location of the entity. o)I Now, suppose that the units are reduced to o~n = m in some instrument n, n

and that the count in these units is denoted by the random variable Xpn. Then in these units

E[Xp n]

_

o~1/n

__

?/~p.

Thus reducing the units to (1/n)th of an original unit gives n times the value in the new units. Let the values of the variable Xp, be denoted by ~tF, n(1) - when expressed in standard units of instrument I: then, if xp,, E {0,1,2,3,4 . . . . } in units .(I){ 01234 } of to,,, ~p,, E ........ in units of toI. (For example, if the original

nnnnn

units are in centimeters and the new ones are in millimeters, then measurements in millimeters when expressed in centimeters will be 1/lOth their millimeter values.) Therefore, expressed in terms of the original unit ~oI,

n

n

n

as required to show the invariance of location of entity p when measured with different instruments when the measurement is carried out in different units (but expressed in the same unit). In some sense this result is in fact rather trivial, but it is important because it highlights the need to distinguish between the effect of changing the unit in relation to the original unit in making the actual measurements and the effect of reexpressing the measurement in the new unit in terms of the original unit. In the former, the changed unit actually changes the parameter of the distribution, and therefore the probabilities, whereas in the latter, the parameter and the probabilities do not change, only the values of the measurements change into the original units. The same logic then can be used to establish the variance, and this is not trivial. Thus v[.~'21

=

v

~

n

:

-n~

v[x,j

=

n2

=-,

n

16

DavidAndrich

indicating that when the unit is (1/n)th of the original unit, the variance is also 1/nth of the original variance when expressed in the original metric and therefore increases the precision in the location of ~p. Although beyond the scope of this chapter, it can be readily shown that if the measurements are Poisson distributed according to Equation (6), then the distribution of the m e a n of n independent measurements in a particular unit is identical to the distribution of the measurements in a unit (1/n)th the size of that unit. Thus taking n measurements and finding the mean to estimate the location has an identical effect as taking one measurement in a unit (1/n)th the size of the original unit. This result itself is rather impressive, but here the only point emphasized is that it is consistent with what is expected in measurements, namely, that a measurement in smaller units should be more precise than a measurement in larger units. Figure 2 shows the distribution of measurements of Equation (5) for an entity with ~p = 1 in standard units, and when the units are halved successively up to units of to//8. The figure clearly shows that the distribution also tends to the normal, and this arises because as the value of its parameter increases, the Poisson distribution tends to the normal. This is all as expected in physical measurement: When expressed in the original metric, the expected value remains constant, while the precision becomes greater as the unit is reduced and the distribution of measurements is expected to become normal. If the unit is made

0.6

"

Pr{x} o.5 0 0.4

Unit =1

9 Unit = 1/2 9 Unit = 1/4 9 Unit = 1/8

( 0.3

0.2

0.1

0.0 0.0

0.5

1.0

= 1

1.5

2.0

2.5

3.0

3.5

Measurement x

FIGURE2. Distributions of measurements of entity of size 1 and units starting with 1.0 then halved for each new set of measurements.

1. Measurement Criteria for Choosing among Models with Graded Responses

17

sufficiently small relative to the size of the entity measured, then it might be small enough to be ignored in a deterministic theory. In addition to behaving as measurements with respect to their location, dispersion, and distribution in physical science, the model also incorporates the characteristic of concatenation. As is well known, the sum of two Poisson distributions is Poisson again, with the new parameter equal to the sum of parameters of the individual distributions. Suppose entities p and q with location values ~p and ~q, respectively, are concatenated and then measured with instrument i with unit coi. Then, from Equation (6) and some straightforward algebra, Pr{xp+q = Xp + Xq} = e

with E[xp+q]

_ ~,,+~q 0,, [(~p + ~q)/O~i]~,'+',/Xp+q!

+ eq =

~

, 1.oi

(7)

indicating that the value of the new concatenated entity is the sum of the values of the original entities and that an estimate of this value is the sum of the measurements of the original entities.

3.1.3. Falsification of the ordering of the categories Rasch (1961) originally specialized a m u l t i d i m e n s i o n a l version of a response model that satisfied criteria A and B and expressed it in the form equivalent to Pr{xpi} =

1

eXp{Kxi

@

+xi~p}'

(8)

"~pi

where the coefficient Kxi and scoring function +xi characterized each category x. Andersen (1977) showed that the model had sufficient statistics only if +(x+ 1)i - +xi = +xi - +(~-l)i and that categories x and x = 1 could be combined only if +(~+1)i = +~i. Andrich (1978) constructed the K'S and + ' s in terms of the prototype of measurement. The steps in the construction, reviewed here with an emphasis on falsification of threshold order, are 1. assume in the first instance independent dichotomous decisions at each threshold x, x = 1, m; 2. characterize the dichotomous decision by the model Pr{y~p i = 1 } = e x p O[xi(~ p -"rxi)/T]xpi , Pr{yxp i = 0 } = 1]Ylxpi, where Yxpi is a Bernoulli random variable in which Yxpi = 1 denotes threshold x is exceeded, Yxpi = 0 denotes threshold x is not exceeded, %i is the discrimination at threshold x, Tlxpi -= 1 + exp O[xi(~ p -- Txi); and 3. restrict and renormalize the original outcome space 1~ = { ( 0 , 0 , 0 . . . 0), (1,0,0 . . . . 0) . . . . (1,1,1 . . . . 1) } of all possible 2 m outcomes to the subspace 1~' of those outcomes which conform to the Guttman pattern consistent with

18

David Andrich

the threshold order according to (0,0,0 . . . . O) corresponding to xpi = 0 , ( 1 , 0 , 0 . . . O) corresponding to xpi = 1, (1,1,0 . . . . O) corresponding to xpi = 2 , and (1,1,1 . . . . 1) corresponding to xpi = m. This restriction makes the final outcome at each threshold dependent on the location of all thresholds (Andrich, 1992). For example, consider the case of a continuum partitioned into two thresholds. Then t} = {(0,0), Pr{ (0,0) } = Pr{(1,O)} = Pr{ (0,1) } = Pr{(1,1)} =

(1,0), (0,1), (1,1) } and ( 1)( 1 )/"qlpi'q2pi, [exp oLli([31, - Tli)](1)/"qlpi'q2pi , (1)[exp c~2i([3p - T2i)]/TllpiTl2pi , [exp OLli([3 p -- "rli) exp oL2i([3 p -- 'T2i ) ]/'lq l l, iTI21, i.

However, the outcome must be in one of only three categories, and this is given by the subspace 1 1 ' = {(0,0), (1,0), (1,1)}, in which case a response in the first category x~,i = 0 implies failure at both thresholds, xpi = 1 implies success at the first threshold and failure at the second, and xpi = 2 implies success at both thresholds. The element (0,1) is excluded because it implies exceeding the second more difficult threshold and failing the first, easier threshold, which violates the ordering of the thresholds. If the element (1,0) had been excluded and the element (0,1) included, then the intended ordering of the thresholds and direction of the continuum would have been reversed. The choice of elements which are legitimate, and therefore others which are inadmissible, is emphasized because it is a key step in defining the direction of the continuum and the interpretation of the application of the model. The probability of the outcomes conditional on the subspace t}', and after some simplification, is given by Pr{(O,O) ID'} = Pr{x~,i = 0 } =

(1)(1)

'~pi 1 Pr{(1,O) lt)'} = Pr{xpi = 1} = -- {exp(-otli'rli "~pi 1

+ otii[3p) }

Pr{(1,O) lO'} = Pr{x~; = 2} = - - { e x p ( - o t j i - r l i - ot2i'rzi + (otli + o~zi)[3p)}. "~i,i

Defining the successive elements of the Guttman pattern {(0,0), (1,0), (1,1)} by the integer random variable X, x E {0,1,2}, and generalizing, gives

1. Measurement Criteria for Choosing among Models with Graded Responses

Pr{xpi-

19

1 0} = ~ exp{Kx/ + d~xi[3p}, "Ypi

which is Equation (8) in which +xi and K~i are identified as +oi---0 +xi

--

and

OLli nt- OL2i -[-

9 9 9 -[- OLxi

K0/------0 Kxi

x-

--

--OLliTil

1. . . .

--

OL2i'r2i

9 9 9 --c~ .viT r~i ,

m,

for -oo < T x i ~ oo, X - - 1 . . . . m; "rxi > "r(~_ ~)i, x - 2 . . . . m. Thus the scoring functions +~i are the sums of the successive discriminations at the thresholds, and if %; > 0, x - 1. . . . m, as would normally be required, the successive scoring functions increase, qb(x+1); > 4'~/x - 1. . . . m. If the discriminations are constrained to be equal, O L l i - - O L 2 i 0~3i = . . . OLxi = . . . c~,,,i = oti > 0, then the scoring functions and the category coefficients become +o = 0 +xi

and

- - XOLi

Ko = 0 Kxi

--

--OL(TIi

-[- T 2 i

-q- " " " -'[- T x i ) ,

X --

1. . . .

m.

It can now be seen that the successive scoring functions have the property ( b ~ + l - (b~ = ( b ~ - (bx-l = a specified by Andersen (1977). Thus the mathematical requirement of sufficient statistics, which arises from the requirement of invariant comparisons, leads to the specialization of e q u a l d i s c r i m i n a t i o n s at the t h r e s h o l d s , exactly as in the prototype of measurement. Furthermore, if at threshold x, % i - 0, then (b(~+~)~- (bxi, which explains why categories can be combined only if threshold x does not discriminate, that is, if it is in any case artificial. At the same time, if a threshold does not discriminate, then the categories should not be combined. This analysis of the construction of the CTM provides an explanation of the known result that the collapsing of categories with such a model has nontrivial implications for interpreting relationships among variables (Clogg and Shihadeh, 1994). The parameter oL can be absorbed into the other parameters without loss of generality, giving the model of Equation (1), in which the scoring functions now are simply the successive counts x E {0,1,2 . . . . m} of the number of thresholds exceeded, as in the prototype of measurement. The category coefficients then become simply the opposite of the sums of the thresholds up to the category of the coefficient x, giving in summary +oi-0 ~)xi -- Xi

and

K0/--0 Kxi

-'- - - ( T I i

-[- T 2 i

-[-

9 9 9 -nt- T x i ) ,

X --

1. . . .

m,

and the resultant model is the CTM. It is important to note that the scoring of successive categories with successive integers d o e s n o t r e q u i r e a n y a s s u m p t i o n t h a t the d i s t a n c e s b e t w e e n t h r e s h o l d s are e q u a l - - t h e s e differences are estimated through the estimates of the thresholds.

20

David Andrich

The requirement of equal discriminations at the thresholds, and the strict ordering of the thresholds, is therefore central to the construction of the model. However, it turns out that there is nothing in the structure of the parameters or in the way the summary statistics appear in any of the estimation equations (irrespective of method of estimation) that constrains the threshold estimates to be ordered; and in the estimates, the parameters may appear in an order different from that intended. If they appear in the incorrect order, it is taken that the thresholds are not operating as intended, and that the hypothesis of threshold order must be rejected. This reversal of order in the estimates will happen readily if the discriminations at the thresholds are not equal in the data and the data are analyzed using the CTM which requires equal discriminations. A common reaction to this evidence that the model permits the estimates to be disordered is that the disordered estimates should simply be accepted--either there is something wrong with the model or the parameters should be interpreted as showing something other than that the data fail to satisfy the ordering requirement. This perspective is supported by the feature that the usual statistical tests of fit based on the chi-square distribution can be constructed and that the data can be shown to fit the model according to such a test of fit. However, such reasoning does not take into account that none of these tests of fit, which operate with different powers, are necessary and sufficient to conclude that the data fit the model. One hypothesis is that the thresholds are ordered correctly. Thus the fit can be satisfied with a global test of fit or with respect to some specific hypothesis about the data, even though some other specific hypothesis may have to be rejected. In addition, such tests of fit involve the data in the estimates of parameters, and degrees of freedom are lost to the test of fit: the test of fit checks whether, given these estimates and the model, the detail of these very same data can be recovered. In the CTM, in which the threshold estimates can show any order, the test of fit involves those estimates and so it may not reveal any misfit, and, in particular, it cannot in itself reveal anything special about the reversed estimates of the thresholds. It is a test of fit of the internal consistency of the data given the parameter estimates, and features of the parameter estimates such as order have to be tested separately. Thus, evidence from this kind of statistical test of fit is incomplete and does not obviate the need to study the estimates in relation to other evidence and other criteria related to the construction of the model. This argument is so central to the application of the model that it is now elaborated from an alternative perspective. This perspective involves the log-odds of successive categories: log (Pr{(x)i}/Pr{(x

-

1)i}) - [([3p

If "r(x+ 1)i > "r(x)i, then from Equation (9),

--

~i)

--

"rxi].

(9)

1. Measurement Criteria for Choos&g among Models with Graded Responses

21

log (Pr{(x)i}/Pr{(x - 1)i}) - log (Pr{(x + 1)i}/Pr{(x)i}) =

[(~p

--

T(x+

-

a;)

1)i - -

-

-r~;] -

[([3p -

a/)

-

(Pr{xpi} )2 1 )pi})(Pr {(x

+

%+~;]

Txi ~ O,

it follows that

(Pr {(x -

1

)pi } )

>1,

which ensures a unimodal distribution of Xpi. Specifically, no matter how close two successive thresholds are, if "~x)i < "r~x+~); and if "rex); < [3p < "r~x+~)i, then the probability of the outcome x is greater than the probability of an outcome in any other category. Figure 3 illustrates this feature where thresholds 'r3i and "r4i are close together. When successive thresholds are in an identical location, or reversed, unimodality no longer holds. The unimodal shape when the thresholds are ordered is consistent with interpreting the values Xpi to parallel measurements, because the distribution of Ypi is then simply a distribution of random error conditional on the location of the entity and the thresholds, and upon replication, regression effects would produce a unimodal distribution. The Poisson distribution which has been shown to be the special case when thresholds are equidistant in the multiplicative metric, exemplifies this feature. Other random error distributions, such as the binomial and negative binomial, as well as the normal

FIGURE 3.

Probability of response in seven categories with thresholds 3 and 4 close together.

22

David Andrich

and other continuous distributions all are unimodal. It is stressed, however, that unimodality is a property of the model, and that there is no guarantee that in any set of data, this unimodal property would hold--that is an empirical issue-and if it does not hold, then reversed threshold estimates will be obtained. When one tests for the normal, binomial, or Poisson distributions in usual circumstances, one can readily find that the data do not fit the model because they are bimodal. To return to the idea that the threshold order is a hypothesis about the data, there must be a myriad of ways in empirical work in which data may be collected in graded responses where the discriminations at the thresholds will not be equal, or where some thresholds do not discriminate at all, or where some may even discriminate negatively so that a nonunimodal distribution, with reversed estimates of thresholds, is obtained. If the estimates show threshold disorder, then it is taken that the intended ordering has been refuted by the data and that an anomaly has been disclosed, an anomaly that needs substantive investigation. The CTM permits the data to reveal whether or not they are ordered. 3.1.4. S u m m a r y of C T M and measurement criteria The CTM therefore satisfies the criteria specified, and also therefore characterizes the responses in ordered response categories in a way that reflects measurement. The only difference is that the categories in which the continuum is partitioned is not in equal units and the number of categories is finite. This means that an estimate of the location is not available explicitly but must be obtained by solving an equation iteratively; otherwise, it has the same features. How each of the criteria, and how the model's characterization of these criteria, can be exploited in data is reflected in the examples provided in the next section, after the competing model is presented.

3.2. Thurstone Cumulative Probability Model The model historically prior to the CTM for ordered response categories is also based on the partitioning of a continuum into categories. The derivation of the model is based on the plausible assumption of a single continuous response process so that the entity is classified into a category depending on the realization of this process. If "rli, "rzi, "r3i . . . . . "rxi . . . . 'rmi are again m-ordered thresholds dividing the continuum into m + 1 categories, then Figure 4 shows the construction and the continuous process that Thurstone originally assumed to be normal. However, because it is much more tractable, and because with a scaling constant it is virtually indistinguishable from the normal (Bock, 1975), the process is now often assumed to be the double exponential density function f ( y ) = (exp(y))/(1 + exp(y)) 2. In addition, various parameterizations have been

1. Measurement Criteria for Choosing among Models with Graded Responses

FIGURE 4.

23

The double exponential density function with four thresholds.

used but the differences among these is not relevant to the chapter; therefore only the one that is the simplest and closest to the parameterization of the CTM, the one with only the location and thresholds parameters, is examined. Thus if Ypi is a random continuous process on the continuum about [3p, and if successive categories are denoted by successive integers Xpi, then an outcome "rxi <- Ypi <- "r(x+ 1~i leads to the outcome Xpi, with Xpi = 0 if Ypi <- "rli, and xpi = m if Y p i ~ "rmi" According to the double exponential with only the location parameter [3p, Pr{Ypi > "rxi }

~:(

C Y/,i -- [~p

e [~/' -- Tvi

J (1 + e~,--'Y% ~dy~i : 1 + e~,'-~.'~

ff'x i

Let Pr*{Xpi} = Pr{xpi} + Pr{(x + 1)pi} Jr- Pr{(x + 2)pi } + . . .

+ Pr{mpi }, then

ei3p--Tki Pr{Ypi > "rki} = Pr*{Xpi} = 1 + e ~,'-'~ki"

(10)

Thus in contrast to the CTM, in which the successive thresholds are accumulated in the relevant expression, in this model it is the successive probabilities that are accumulated: hence the designation cumulative probability model (CPM). This model may now be assessed according to the measurement criteria specified in section 2.

3.2.1. Invariance under different partitions of the continuum probability for each value of the outcomes Xpi is given by Pr{xpi } = Pr* {Xpi} that is,

--

Pr* {(x +

1)pi}, X

=

1 ....

m -

1,

First, the

24

David Andrich e~,,-r

ef3,,-r

Pr{xpi} = 1 + e ~,'-~-';

-

+ , ,i

1 + e ~,'-~'.'+'" . x . = . 1 . . .

m-

1

(11)

and Pr {Opi }

"-

1-

e[Bp--Tli 1 + e 6p-'rli'

Pr { mpi } :

e6p-%,i 1 + e 6p-%'i"

In this model, no sufficient statistics exist. Therefore, the kind of invariance through the separation of parameters and the conditioning out of one set of parameters while the other set is estimated, as established in the CTM, cannot be demonstrated. However, and in terms of the statistical criteria presented in McCullagh (1985) and previously quoted, another kind of invariance does exist. Because there is only the one random process characterized by the d i s t r i b u t i o n Ypi, this distribution itself remains invariant under different partitions of the continuum; that is, it remains invariant with different numbers of thresholds with different locations. Even so, this does not ensure that the estimate of [3p will be the same under these different partitions. It may turn out that in empirical or simulated data there is not a great change in the estimates under different partitions, but in the model itself, there is no property from which it follows that the location of the entity would be invariant. 3.2.2. Structure a n d s p e c i a l i z a t i o n Although it contains, in part, an additive structure within each of the two terms of the probability P r { x p i }, because the two terms are related by a difference, the structure of the model is not multiplicative (or additive in the logarithmic metric), as found when fundamental measurement has taken place, and this also precludes the possibility of sufficient statistics. Next, consider the effect of increasing the number of thresholds. Figure 5 shows the distribution of the response process Ypi a s in Figure 4, but with another threshold placed between the original thresholds. It is evident that the distribution has neither narrowed nor changed in location as a result of increasing the number of thresholds. In principle, this means that the estimate of the location [3p will not be improved, that is, the precision of the estimate will not be improved by the addition of thresholds and the reduction in the size of the unit. Thus although the location of the observed value Xpi is itself now within a narrower region, it has the same probability of being beyond any specified value from [3p in both sets of thresholds. Therefore, it does not have the effect that as the number of thresholds increases, the precision of the location of [3p also increases. Interestingly, it is this very feature that might make the model applicable to another kind of situation, and it is perhaps the metaphor of this situation that has led it to be used with graded responses. The single distribution of the process Ypi with a location and a variance that

1. Measurement Criteria for Choosing among Models with Graded Responses

FIGURE 5.

25

The double exponential density function with seven thresholds.

does not change with different partitions of the continuum, implies that the variance is a fixed property of the distribution and not a property of the process of measurement or construction of the categories. An example might be helpful. Suppose persons are measured for their height and then classified into class intervals. On the assumption of no errors of measurement, this distribution has both a mean and a variance, and if the data were partitioned into different sets of class intervals, then an invariance of distribution might be expected. In this case, the variance is not a function of errors of measurement, but is a variation that describes the differences among the persons when the construction of different class intervals occurs post hoc, that is, after the data are collected with a particular unit of measurement. In this case, the parameter to be identified is not simply the location of an entity, in which case the smaller the unit the narrower the expected error distribution and the greater the precision of measurement, but two parameters, one for the location and one for the variance of entities. It seems that CPM is constructed from this kind of situation and that it might therefore be appropriate to apply it to such situations. For example, if the probability of the classification of members of a population is to be characterized as a function of some explanatory variable, say income as a function of age, then the CPM may be appropriate. This case is different in principle from that of graded responses which are in the process of being constructed to characterize the meaning of more or less of a variable, and the model is used to locate each single entity as precisely as possible in parallel to measurement in physical science. 3.2.3. Falsification The strict ordering of threshold estimates is ensured in the CPM. This is because the estimates of the thresholds are a function of the

26

David Andrich

cumulative probabilities in the model, specifically of the log-odds of the cumulative probabilities relative to their complement, which can be seen from log (Pr* {Xpi } )/( 1 -- Pr* {Xpi } ) = ~p -- 'Txi.

(12)

Because the P r * {Xpi } are estimated by the cumulative frequencies beyond "rxi, they are strictly decreasing, and so the threshold estimates will never be disordered. (Two successive thresholds may have the same value if the frequency in a category is 0.) Because the model has built into it this strict ordering, when viewed from the perspective of the statistical criteria this is considered a strength. However, from the point of view of measurement criteria, it means that no matter how badly or incorrectly the data have been collected in relation to the theory from which they arise, or how incorrect the theory, the data will always show a strict ordering of thresholds. Then, in part because of the model and in part because of the perspective that leads to the model, no anomalies in ordering will ever appear. This means that the ordering is strictly a property of the model and not of the data, and in fact is independent of the data. Because the statistical tests of fit based on the chi-square distribution are again not necessary and sufficient, it is possible to obtain tests of fit that show that the data fit the model even when from inspection of the data it is evident that a threshold is not discriminating. One of the upcoming examples reveals this.

3.2.4. Summary of the CPM in relation to the measurement criteria It is evident that the CPM satisfies none of the measurement criteria: it does not provide invariance of location, it does not specialize to the case of measurement when precision is to be increased by increasing the number of thresholds, and it does not permit the ordering of the response categories to be falsified. The reason it has none of these features is that it arises from a situation in which the distribution is not that of measurement error, but of a property already measured in which the entities are described by a mean and a variance.

3.3. Combining and Juxtaposing the CTM and the CPM It is possible to consider some combination of ideas associated with the two principles for deriving models for ordered categories when one has groups of entities to be compared, when the comparison is essentially with respect to the location parameter of the group even though it is known that each entity within each group g has its own location. In the presence of error or measurement of individual entities, it is possible to superimpose the distribution of the entities, as defined in the CPM, on the probability of the classification of each entity in a standard way:

1. Measurement Criteria for Choosing among Models with Graded Responses

Pr{xgi} - ~ Pr{xpil~p ~ g} 13,,

Pr{[3p}

27

orPr{xgi} - f Pr{x, il6p E g}f([3p)d[3,, 13,,

for a discrete or continuous distribution of [3, respectively. However, it would still be necessary to assume a shape for the distribution of [3. If the distribution for the entities themselves is assumed to be unimodal (say, based on the normal or double exponential or gamma distributions), then the reversal of the thresholds may imply either a violation of the operation of the graded responses or a violation of the unimodal assumption of the distribution of the entities. Alternatively, and more simply, it is possible to apply the CTM directly to the location of the groups. Then, if the groups are assumed to have homogeneous unimodal distributions differing only in their locations, the variation among entities within a group would be absorbed into the locations of the thresholds. In this case, the reversal of thresholds again would reflect either on the operation of the graded responses or on the assumption of a unimodal distribution of the entities. It should be stressed, however, that in the case in which estimates of individual entities are characterized by two sets of graded responses, when the parameters of the entity are conditioned out, then the reversal of threshold estimates reflects only on the graded responses. Estimation and other features of the CTM and the CPM are well established in the literature for various design structures and therefore these will not be discussed here. The CTM may be identified as a member of the power series distributions (Leunbach, 1976; Noack, 1950), and can also be identified as a loglinear model (Andrich, 1979; Duncan, 1986; Goodman, 1981). This means that the statistical properties that are relevant for estimation and fit are known. However, it should be stressed that the CTM is not chosen for graded responses because it can be identified in this way statistically, but because it turns out to be the case that the CTM, chosen because it meets the criteria for measurement, can be so identified. The CPM is established as a standard link function in the analysis of general linear models (McCullagh, 1980). If the task were simply to model a particular set of data, that is, to describe the data with a model that fits according to some general statistical criterion, then very little difference would be found between the CTM and the CPM. In addition, if the criterion of threshold order is not taken as a hypothesis but as given, and if the distribution of outcomes is defined by a location and dispersion that are to be taken as invariant across different partitions of the continuum, then the application of the CPM becomes attractive. However, if the graded responses are used to locate an entity as accurately as possible, as in measurement so that the dispersion is only a function of the accuracy of the instrument, and if the ordering of the thresholds is taken as a hypothesis about the variable, then the application of the CTM is required.

28

David Andrich

4. EXAMPLES 4.1. Example 1: Boys' Dreams The first example follows up an analysis of the distribution of dreams among 223 boys aged 5 to 15 by McCullagh (1980) using the CPM. These data, taken from Maxwell (1961) and reproduced in Table 1 in a rearranged format, have also been analyzed by Nelder and Wedderburn (1972) using a log-linear model with a linear scoring function for location but without concern for any threshold parameters. For convenience of interpretation here, the order of the table has been reversed so that the most severe disturbance of dreams has the highest rating. Because it concerns the development through age of the severity of dreams along a variable defined through a set of graded responses, the example is directly relevant to the theme of this book. The data are analyzed here according to the CTM but with only one instrument i and the location of an entity characterizes the location ~.Lgof each group g. The model then takes the form Pr{xgi}

=

1 exp -

"rk +

XglXg

, -~

< tx < oc, - c o

< .r k < oc, x = 1 . . . .

m.

k=l

"~gi

As an aside, note that l o g [ P r { x g i } / P r { ( x - 1)gi}] = I.Lg -- Txi , which permits a linear regression-type analysis between the ordered thresholds "rxi, x = 1. . . . m of the dependent variable and iXg, g -- 1,6 of the explanatory variable. This approach to the analysis of ordered dependent variables, but without specific con-

TABLE 1 Frequencyof Disturbed Dreams Among Boys Aged 5 to 15 Degree of suffering from disturbed dreams Not severe Age

x:

5-7 8-9 10-11 12-13 14-15

1

2

3

~g

7 10 23 28 32

4 15 9 9 5

3 ll 11 12 4

7 13 7 10 3

.281 .337 -.034 -.013 -.573

0.481

q-, •

Very severe

0

= 9.631, p < .292

-.261

-.220

13, = - . 3 2 0

29

1. Measurement Criteria for Choosing among Models with Graded Responses

cern for the empirical ordering of the thresholds, is described in Goodman (1981) and now exists in some statistical packages featuring log-linear analyses. Table 1 also shows the estimates of the group effects ~l.g and of the thresholds § and a chi-square test of fit of the recovery of the frequencies according to the model. In describing the data, the model is adequate: X 2 = 9.631 on 8 df p < .292. Before making a study of the thresholds, it is also evident that the age groups 5 to 7 and 8 to 9 have very similar location values, and likewise age groups 10 to 11 and 12 to 13, and that when taking this grouping into account, the basic shift is for the dreams to decrease in severity as a function of age. To highlight this feature, Table 2 shows a reanalysis according to the CTM but with ages 5 to 9 combined and with ages 10 to 13 combined. Again, the data accord well with the model - X 2 = 6.738 on 4 df p < .150. This grouping, which is rather straightforward in that it involves no more than making class intervals larger than originally set, has a major implication for understanding the rate of change along the variable as a function of age: changes are noticeable in the range of 5 to 13 years in intervals of 4 years, with a rather sharper change in the age group of 14 to 15 years. Returning to the thresholds, in both analyses, the estimates show reversals, with the second and third thresholds having lesser values than the first threshold, and the second and third thresholds being close in value: an anomaly has been disclosed. To study this anomaly, recall that in the construction of the CTM, equal discriminations at the thresholds are required, and that if a category does not discriminate, but the model assumes it does, then it is likely that reversed threshold estimates will be obtained. Moreover, if a threshold does not discriminate, then the two categories it separates could and should be

TABLE 2 Frequency of Disturbed Dreams Among Boys Aged 5 to 15 with Age Groups 5 to 9, 10 to 13, and 14 to 15 Degree of suffering from disturbed dreams Not severe Age

x:

5-9 10-13 14-15

Very servere

0

1

2

3

~Lg

17 51 32

19 18 5

14 23 4

20 17 3

0.412 0.069 -0.482 m

;r.,.

0.482 X4z = 6.738, p < .150

-0.261

-0.221

12 = -.411

30

David Andrich

collapsed. Directed by this perspective, it can be seen just by looking at the data that the middle threshold does not discriminatemfrequencies in the categories on either side of it are about equal irrespective of age. That is, even though the overall grading of the severity of dreams changes as a function of age in the extreme two categories, the relative distribution in the middle two categories does not change. A number of points regarding the relationship between a theory and the construction of graded responses can be illustrated using this example. First, there is a need for a theory that relates changes in the minds of boys as they become older to the definition of degrees of severity of dreams. Thus, because it is not age as such, but the changes that take place through aging that are relevant, it is not sufficient to observe that severity of dreams is related to age and that a statistical model can be found that accounts for the data. Ideally, the theory comes before the data are collected, but the need for a theory arises even if the relation is observed after the data are collected. In this case, the theory should bear on the observation that changes in the 5 to 13 year range are noticeable only in class intervals of 4 years and are more sharply noticeable in the same direction in the age group of 14 to 15 years, and on how this relates to the severity of dreams. Second, then, is that the changes in dreaming that occur through age and the intensity of the variable defined through the graded responses are defined simultaneously: the locations of the thresholds are an integral part of the definition of the intensity of the variable--they define what it means to have more or less severe dreams. Third, although there is a trend in the change in severity as a function of what happens through a change in age, there is no change in severity as characterized by the frequencies in the middle two categories: the change occurs only in relation to the extreme categories, and this, it should be stressed, is a characteristic of the data (which the model of analysis exposed). If the middle two categories operated consistently with the extreme categories, with the third being more severe than the second, then they too should show the same trend, even if it is not as strong as with the extreme categories. If the two extreme categories were not present, but only the middle two, then no trend would be observed. Although the emphasis here is theoretical, to help make the point, it is noted that there could be undesirable practical consequences in ignoring the evidence: for example, suppose treatments on the basis of the classification were recommended, then different treatments might be recommended for classifications in the middle two categories, which from this empirical evidence are no better than random. If the classifications in the middle two categories are combined, which is justified under the specific hypothesis that the middle two categories do not discriminate, then the estimates of the parameters and the general test of fit is as shown in Table 3. The data again fit the model, but in addition, the two thresh-

31

1. Measurement Criteria for Choosing among Models with Graded Responses

TABLE3 Frequencyof Disturbed Dreams among Boys Aged 5 to 15 with Age Groups 5 to 9, 10 to 13, and 14 to 15, and the Original Middle Two Categories Degree of suffering from disturbed dreams Not severe

Age

x:

5-9 10-13 14-15

Very severe

0

1

2

~g

17 51 32

33 41 9

20 17 3

.708 .068 -.777 B

~,

-.414

.414

t2 = - . 6 3 7

X2 = 2.632, p < .268

olds are now in the correct order. However, the task is not simply to model the data, but to understand their meaning. Therefore, the implication of this analysis is not that the model accounts for the data and therefore that closure on the analysis of the data has been reached. Quite the contrary: the implication is that the definition of what constitutes the middle two categories of severity, how these might be related to changes in boys' dreams with age, and how the variable of severity has been operationalized all have to be examined from the point of view of the relative substantive theory. In addition, the assumption of some kind of homogeneity of the boys' location in each age group should be examined for any possible splitting into two or more subpopulations, especially on the basis of some background factor that might be relevant to severity of dreams, perhaps recent family or personal trauma or the like. In the words of Kuhn quoted earlier, an anomaly has been disclosed with quantitative finesse, although as to the source of the anomaly there is no clue: a search for an explanation is required. Unfortunately, Maxwell (1961) provides no additional substantive information regarding the source of the data, and indeed from a statistical fit perspective no further action is required--the model accounts for the data.

4.2. Example 2: Serological Readings The second example involves 12 serological readings classified into five levels of reaction of 12 blood cells analyzed by Fisher (1958, p. 290). If the readings from the sera in the five levels of reaction are scored by successive integers beginning with 0, then the data may be arranged as in Table 4. Fisher described a method of analysis in which he assigned the score of 0 to the first level, a score of 1 to the last, and then estimated the values for the remaining categories

32 TABLE 4

David Andrich

Serological Readings of 12 Cells Frequency of level of reaction of 12 sera/ceU

Cell

x:

1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

~g

0 0 0 1 0 0 0 0 0 0 0 0

1 4 0 1 1 1 0 0 0 4 1 0

8 8 12 10 7 8 3 5 6 8 10 4

3 0 0 0 4 3 7 6 6 0 1 6

0 0 0 0 0 4 2 1 4 0 0 2

-.031 -2.184 -0.794 - 1.903 .273 1.224 1.770 1.307 1.695 -2.184 -0.794 1.620

;r,

-3.543 X23

=

-2.095

2.357

3.282

12 = - . 9 0 9

25.58, p < .818

by maximizing the sum of squares of the main effects relative to the total sum of squares. This is exactly the same number of constraints as required in the analysis using the CTM, and as in the CTM, no ordering is forced on the estimates o f the weights. Table 4 also shows the parameter estimates and the test of fit according to the CTM, and shows both that the data accord with the model and that the threshold estimates are in the correct order. One purpose in reproducing this simple example is to appreciate Fisher's simple and powerful remark, consistent with the perspective in this chapter on the evidence that the estimates of the values in his analysis (which correspond to the thresholds in the CTM) were in the correct order: "It will be observed that the numerical v a l u e s . . , l i e . . , in the proper order for increasing reaction. This is not a consequence of the procedure by which they have been obtained, but a property of the data examined" (Fisher, 1958, p. 294).

5. SUMMARY AND DISCUSSION This chapter discusses the criteria for choosing among models for analyzing graded responses that are ubiquitous in the social, biological, and other sciences. The observation on which these criteria and choice of model was pursued is

1. Measurement Criteria for Choosing among Models with Graded Responses

33

that graded responses are used in those situations in which measuring instruments of the kind found in the physical sciences are not available for dependent variables, and that the graded responses have, therefore, as important a role in empirical enquiry as do measurements, especially in developmental research. Graded responses are similar to measurements in that, just as units do in measurement, they partition a latent unidimensional continuum into adjacent intervals. They both define, therefore, what it means to have more or less of the property in question, which is considered central to the definition of the variable and to the operationalization of the theory from which the variable arises, and to an understanding of development and change along the variable. Graded responses differ from measurements in that they are finite in number, they are not of equal length, and generally the origin is arbitrary. Therefore, it was considered appropriate to set criteria for a model for graded responses in the wider context of the role played by measurements in the physical sciences, in particular, that the model is required to 1. locate the entity on the continuum in a way that is invariant under different partitions of the continuum, that is, invariant across a choice of different units; 2. specialize to the case of measurements where the graded response categories are equal in size, where there is a natural origin, where the number of categories is not finite, and where a decrease in the size of categories increases the precision of the estimate of the location of the entity; and 3. permit the data collected to falsify the theory of ordering of the graded responses. The Rasch model for graded responses was shown to satisfy these criteria. For purposes of exposition, the model based on the work of Thurstone, which has also been advocated for use with graded responses, was contrasted with the Rasch model and was shown not to meet the same criteria. Instead, it was shown that the Thurstone model provides invariant distributions under different partitions of the continuum (rather than invariant locations), greater precision of location of each observation (rather than of the location of the entity), and ensures that the graded responses are ordered irrespective of the data, that is, that the ordering is a property of the model. The differences in the approaches of the two models goes beyond the way they describe the data, because if they are used descriptively, that is, to summarize the data, then in general if one model is satisfactory the other will also be satisfactory. A constant theme of this chapter is that because the models reflected different understandings of the relationship between theory, measurement, and data, to appreciate the difference between the two models a different perspective is required--in short, a different understanding of scientific modeling when measurement or graded responses are involved. In other words, the two

34

David Andrich

models are not simply alternative descriptions of the data from within the same scientific framework, but reflect different scientific frameworks.

ACKNOWLEDGMENTS This research was supported by the Australian Research Council. Clifford C. Clogg and Tzuwei Cheng read the manuscript and made constructive comments.

REFERENCES Andersen, E. B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42, 69-81. Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 357-374. Andrich, D. (1979). A model for contingency tables having an ordered response classification. Biometrics, 35, 403-415. Andrich, D. (1985). An elaboration of Guttman scaling with Rasch models for measurement. In N. B. Tuma (Ed.), Sociological methology (Chapter 2, pp. 33-80). San Francisco: Jossey-Bass. Andrich, D. (1992, June 22-26). On the function of fundamental measurement in the social sciences: The objective measurement of subjective meaning. Invited Keynote Address, International Conference, Social Science Methodology, Research Committee on Logic and Methodology of the International Sociological Association, Trento, Italy. Bock, R. D. (1975). Multivariate statistical methods in behavioral research. New York: McGrawHill. Clogg, C. C., & Shihadeh, E. S. (1994). Statistical models for ordinal variables. Thousand Oaks: SAGE Publications. Duncan, O. D. (1984). The latent trait approach in survey research. In C. E Turner & E. Martin (Eds.), Surveying subjective phenomena (Vol. 1, pp. 210-229). New York: Russell Sage Foundation. Duncan, O. D. (1986). Probability, disposition, and the inconsistency of attitudes and behavior. Synthese, 68, 65-98. Duncan, O. D., & Stenbeck, M. (1988). Panels and cohorts: Design and model in the study of voting turnout. In C. C. Clogg (Ed.), Sociological methodology (pp. 1-35). Washington, DC: American Sociological Association. Edwards, A. L., & Thurstone, L. L. (1952). An internal consistency check for scale values determined by the method of successive integers. Psychometrika, 17, 169-180. Fisher, R. A. (1958). Statistical methods for research workers (13th ed.). New York: Hafner. Goodman, L. A. (1981). Three elementary views of log linear models for the analysis of crossclassifications having ordered categories. In S. Leinhardt (Ed.), Sociological methodology, pp. 193-239. San Francisco: Jossey-Bass. Guttman, L. (1950). The basis for scalogram analysis. In S. A. Stouffer (Ed.), Measurement and prediction (pp. 60-90). New York: Wiley. Jansen E G. W., & Roskam, E. E. (1986). Latent trait models and dichotomization of graded responses. Psychometrika, 51(1), 69-91. Kuhn, T. S. (1961). The function of measurement in modern physical science. Isis, 52(Pt. 2), 161-193.

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Kuhn, T. S. (1970). The structure of scientific revolutions (2nd enlarged ed.) Chicago: University of Chicago Press. Leunbach, G. (1976). A probabilistic measurement model for assessing whether tests measure the same personal factor. Copenhagen, Denmark: Danish Institute for Educational Research. Maxwell, A. E. (1961). Analysis of qualitative data. London: Methuen. McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society, Series B. 42(2), 109-142. McCullagh, E (1985). Statistical and scientific aspects of models for qualitative data. In P. Nijkamp, H. Lielner, and N. Wrigley (Eds.), Measuring the unmeasurable, pp. 39-49. Dordrecht, The Netherlands: Martinus Nijhoff. Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society, Series A, 135, 370-384. Noack, A. (1950). A class of random variables with discrete distributions. Annals of Mathematical Statistics, 21, 127-132. Popper, K. (1961). The logic of scientific discovery. New York: Science Editions. Ramsay, J. O. (1975). Review of the book Foundations of measurement, Vol. 1, by D. H. Krantz, R. D. Luce, E Suppes, and A. Tverskey. Psychometrika, 40, 257-262. Rasch, G. (1960/1980). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research. (Expanded edition with foreword and afterword by B. D. Wright. Chicago: University of Chicago Press, 1980). Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In J. Neyman (Ed.), Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (Vol. 4, pp. 321-334). Berkleley: University of California Press. Rash, G. (1966). An individualistic approach to item analysis. In E F. Lazarsfeld & N. W. Henry (Eds.), Readings in mathematical social science (pp. 89-108). Chicago: Science Research Associates. Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometric Monographs, 34(2, No. 17). Thurstone, L. L. (1928). Attitudes can be measured. American Journal of Sociology, 33, 529-554. Wright, B. D. (1984). Despair and hope for educational measurement. Contemporary Education Review, 3, 281-288. Wright, B. D. (1985). Additivity in psychological measurement. In E. E. Roskam (Ed.), Measurement and personality assessment (Selected papers). 23rd International Congress of Psychology, 8, 101-111. Wright, B. D. (1988). Campbell concatenation for mental testing. Special Interest Group: Rasch Measurement, 2(1), 3-4.

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Growth Modeling with Binary Responses Bengt O. Muth~n

University of California, Los Angeles Los Angeles, Cafifornia 90024

1. INTRODUCTION In developmental research it is natural to place an emphasis on the study of individual differences in change, growth, and decline over time. It is useful to formulate a longitudinal model for these processes to assess the amount of individual variation and to relate the individual variation to background information on the individuals. Random coefficient growth modeling (see, e.g., Laird & Ware, 1982; Rutter & Elashoff, 1994) is suitable for such longitudinal analysis. It goes beyond conventional structural equation modeling of longitudinal data with its focus on autoregressive models (see, e.g., Jfreskog & Sfrbom, 1977; Wheaton, Muthfn, Alwin, & Summers, 1977) in that it describes individual differences in the longitudinal processes. It is instructive to consider first some examples of longitudinal studies with binary responses in which random coefficient modeling has been used.

1.1. Example 1: Decline in Depression Gibbons and Bock (1987) reported on the longitudinal analyses of data on Danish psychiatric patients. A total of 100 clinically depressed patients were given Categorical Variables in Developmental Research: Methods of Analysis Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

37

38

Bengt O. Muth~n

one of two drugs and were observed five times weekly. The response variable was recorded as recovered (scored 0) or still depressed (scored 1). Information was collected on time-invariant covariates such as severity of illness and age. Time-varying covariates included the plasma level of the drug. The patients were divided into two groups. One group received the drug imipramine and the other received chlorimipramine. The object of the study was to find out which of the two drugs resulted in the sharpest decline in the probability that the patient still felt depressed.

1.2. Example 2: Change/Stability of Neuroticism Muth6n (1983) studied the data described in Henderson, Byrne, and DuncanJones (1981) for 231 Canberra adults interviewed four times at 4-month intervals regarding aspects of "neurotic illness." In a short form of a general health questionnaire, the four questions asked were "In the last month have you suffered from any of the following? Anxiety. Depression. Irritability. Nervousness." A yes response was denoted 1 and no was denoted 0. Time-invariant covariates included gender and a measure of long-term susceptibility to neurosis (the N scale from the Eysenck Personality Inventory). Time-varying covariates included life events in the four months prior to the interview. The object of the study was to assess the stability over time of the level of neuroticism of this population of individuals.

1.3. Example 3: Correlated Observations on Asthma Attacks Stiratelli, Laird, and Ware (1984) studied data on daily observations of 64 asthmatics living in Garden Grove, California. The response variable was the presence or the absence of an asthma attack recorded over a period of about 7 months. Time-invariant covariates included gender, age, and history of hay fever. Timevarying covariates included air pollution and weather conditions. The object of the study was to assess the relative importance of various risk factors for increased probability of asthma attacks.

1.4. Contrasting the Examples It is interesting to contrast these three examples. Example 3 illustrates the fact that often the longitudinal structure of the data is only a nuisance. Here, the interest is the same as in regression analysis. However, observations over time for the same individual are correlated so that the usual assumption of independent observations does not hold. The focus is on how to do the regression analysis while properly taking into account the nonindependence. In Example 2, the longitudinal structure of the data is not a nuisance but is essential to the analy-

2. Growth Modeling with Binary Responses

39

sis. The longitudinal process is one in which no particular trend over time is expected; the interest is in assessing how much responses vary over time for a typical individual. Observations fluctuate up and down over time for each individual and it is the amount of fluctuation that is the focus of the study. Example 1 involves a further elaboration of the longitudinal study. Here, observations do not only fluctuate over time for a given individual but also follow a decreasing trend over time. This chapter focuses on situations illustrated by Examples 1 and 2. The aim of this chapter is to discuss conventional random effects modeling for binary longitudinal response and to compare that with a generalized random effects model for longitudinal data which draws on techniques used in latent variable modeling. In section 2, the conventional modeling and estimation is presented. Section 3 critiques this approach and gives a more general formulation. Section 4 presents a small Monte Carlo study in which the more general approach is studied and analyses of real data is also presented.

2. CONVENTIONALMODELING AND ESTIMATION WITH BINARY LONGITUDINAL DATA Consider a binary variable y and a corresponding continuous latent response variable y* for which "r is a threshold parameter determining the y outcomes: y = 1 when y* > -r and y = 0 otherwise. Here, the progress over time of the latent response variable y* is described as Yik = cti + [3itk + ~l~Vik + ~ik,

(1)

where i denotes an individual, tk denotes a time-related variable with tk = k (e.g., k = 0, 1, 2 . . . . . K - 1), oLi is a random intercept at t = 0, [3i is a random slope, ~/k are fixed slopes, vik is a time-varying covariate, and ~ik is a residual, --- In(0, ~ ) . Furthermore, { O~i -'- ~Lor + "IToLW i "+" ~ ori ~ i -- ~ f3 + 3T [3W i -+" ~ [3i'

(e)

where IX~, tx~, rr~,, and 7r~ are parameters, w i is a time-invariant covariate, and ~ , ~ are residuals assumed to have a bivariate normal distribution,

With tk = k and a linear function of time, for example, k = 0, 1, 2 . . . . . K - 1, the variables a and [3 can be interpreted as the initial status level and the rate of growth/decline, respectively.

40

Bengt O. Muth~n

The residuals of ~ik a r e commonly assumed to be uncorrelated across time 9 In line with Gibbons and Bock (1987), however, a first-order autoregressive structure over time for these residuals is presented. Letting x = (w, v)', the model implies multivariate normality for y* conditional on x with

Yi.o !X c7

I

~l~176

YillX 9

(

~

~]lvil

= T tx~ + qTe~Wi jr_ ~ ~ -Jr-"rr~3W i ]

"~2Vi2

k Y i K - 1 IX

(4)

~K-lViK-I

and 1 p p2

V(y* Ix) = Ttlr~/f~T ' + t~;

9 pK-1

p

13 2

1 p

p 1

o pK-2

.

.

.

... ...

. pK--3

. . .

pK-- 1 pK-2 pK--3

(5) 1

where with linear growth or decline

r ___

1 1 1

0 1 2

.

(6)

.

1

K

1

and ~ , / ~ is the 2 x 2 covariance matrix in Eq. (3). For given tk, w, and v, the model expresses the probability of a certain observed response Yi~: as a function of the random coefficients oL and [3,

P(Yi~

=

I I OLi, ~i' X) = P(Yik > a'[ oti, =

[~i' X)

f: tp(slz, qJ~)ds

(7) where q~ is a (univariate) normal density, ~; denotes the standard deviation of ~, and

Z -- OLi At- ~itk -t- ~lkVik.

(8)

To identify the model, the standardization a" = 0, ~ -- 1 can be used as in conventional probit regression (see, e.g., Gibbons & Bock, 1987).

2. Growth Modeling with Binary Responses

41

The probability of a certain response may be expressed as P(Yo . . . . .

YK- 1 IX) --

.,-~f+~j -~ P(Yo, Y l . . . . .

YK-1 [OL, ~3, X) q~(oL, ~31x)doL d~3,

(9)

where P(Yo, Yl . . . .

fc - ( Yi o)

YK-iI e~, [3, X) = 9

9 9fc

- ( YiK- 1)

q~(Yo, Yl*. . . . .

YK-1 * I OL, ~,

x)dYio..,

*

dYiK_

1,

(10)

where c - ( Y i k ) denotes the integration domain for Yik given that the kth variable takes on the value Yik. Here, the integration domain is either ( - ~ , "r) or ('r, +~). In the special case of uncorrelated residuals, that is, p = 0 in Equation (5), the y* variables are independent when conditioning on a, [3, and x so that P(Yo, Yl . . . . . YK-iI a, [3, x) simplifies considerably, K--1

P(Yo,

Yl . . . . .

YK-1 Io~, 13, x) - k=oI-[ -(y.~,q~(YikloL,

fc

*

[3, x)dYik. *

(11)

In this case, only univariate normal distribution functions are involved so that the essential computations of Equation (9) involve the two-dimensional integral over ~ and [3. Perhaps because of the computational simplifications, the special case of p = 0 appears to be the standard model used in growth analysis with binary response 9 This model was used in Gibbons and Bock (1987; see also Gibbons & Hedeker, 1993). The analogous model with logit link was studied in Stiratelli et al. (1984) and in Zeger and Karim (1991). Gibbons and Bock (1987) considered maximum likelihood estimation using Fisher scoring and EM procedures developed for binary factor analysis in Bock and Lieberman (1970) and Bock and Aitkin (1981). Stiratelli et al. (1984) considered restricted maximum likelihood using the EM algorithm. Gibbons and Bock (1987) used a computational simplification obtained by orthogonalizing the bivariate normal variables oL and [3 using a Cholesky factor so that the bivariate normal density is written as a product of two univariate normal densities. They used numerical integration by Gauss-Hermite quadrature, with the weights being the product of the onedimensional weights. For the case of p 4: 0, Gibbons and Bock (1987) used the Clark algorithm to approximate the probabilities of the multivariate normal distribution for y* in Equation (10). Even when p = 0 the computations are heavy when there is a large number of distinct x values in the sample. Zeger and Karim (1991) employed a Bayesian approach using the Gibbs sampler algorithm. For recent overviews, see Fitzmaurice, Laird, and Rotnitzky (1993); Longford (1993); Diggle, Liang, and Zeger (1994); and Rutter and Elashoff (1994).

42

Bengt O. Muth~,n

3. MORE GENERAL BINARY GROWTH MODELING 3.1. Critique of Conventional Approaches In this section, weaknesses in conventional growth modeling with binary data are presented, along with a more general model and its estimation. The maximum likelihood approach to binary growth modeling leads to heavy computations when p 4: 0. This seems to have caused a tendency to restrict modeling to an assumption of p = 0. Experience with continuous response variables, that is, when y =- y*, indicates that p = 0 is not always a realistic assumption. The assumption of a single p parameter that is different from zero, as in the Gibbons-Bock first-order autoregressive model, also may not be realistic in some cases. Instead, it appears necessary to include a separate parameter for at least the correlations among residuals that are adjacent in time. Furthermore, the conventional model specification of a- = 0, qJ~ = 1 has no effect when, as in standard probit regression, there is only a single equation that is being estimated. It is important to note, however, that this is not the case in longitudinal analysis. The longitudinal analysis can be characterized as a multivariate probit regression in which the multivariate response consists of the same response variable at different time points. This has the following consequences. First, the standardization of "r to zero at all time points needs clarification. In the binary case, this does not lead to incorrect results but does not show the generalization to the case of ordered categorical response or to the case of multiple indicators. The threshold a" is a parameter describing a measurement characteristic of the variable y, namely, the level (proportion) of y with zero values on x. Because the same y variable is measured at all time points, equality of this measurement characteristic over time is the natural model specification. In the binary case, however, the equality of the level of y across time points is accomplished by tx~, in Equation (2) affecting y equally over time as a result of the unit coefficient of oLi in Equation (1), which is not explicitly shown. Setting a- = 0 is therefore correct, although an equivalent specification would take "r as a parameter held equal over time points while fixing tx~, at zero. In the ordered categorical case, however, there are several "r parameters involved for a y variable and equality over time of such "r's is called for. In this case, tx~, cannot be separately estimated but may be fixed at zero. The multiple indicator case will be discussed in the next section. Second, ~ is the standard deviation of the residual variation of the latent response variable y*, and fixing it at unity implicitly assumes that the residual variation has the same value over time. This is not realistic because over time different sources of variation not accounted for by the time-varying variable vi~, are likely to be introduced. Again, experiences with continuous response variables indicate that the residual variance often changes over time.

2. Growth Modeling with Binary Responses

43

In presentations using the logit version of the model, the parameters of a" and ~ are usually not mentioned (see, e.g., Diggle et al., 1994). This is probably because the threshold formulation, often used in the probit case, is seldom used in the logit case. This has inadvertently lead to an unnecessarily restrictive logit formulation in growth modeling.

3.2. The Approach of Muth6n An important methodological consideration is whether computational difficulties should lead to a simplified model, such as using p = 0 or t~ = 1, or whether it is better to maintain a general model and instead use a simpler estimator. Here, I describe the latter approach, building on the model of Equations (1) and (2) to consider a more general model and a limited-information estimator. First, the ~ik variables of Equation (1) are allowed to be correlated among themselves and are allowed to have different variances over time. Second, multiple indicators Yikj, J = 1, 2 . . . . . p are allowed at each time point, (12)

Yil,j = kjxlik + eikj,

where kj is a measurement (slope) parameter for indicator j, eikj is a measurement error residual for variable j at time k, and Yikj 1 if Yikj > "rj. The multiple indicator case is illustrated by Example 2 in which four measurements of a single construct "neurotic illness" ('q) were considered. Given that the a"s and the k's are measurement parameters, a natural model specification would impose equality over time for each of these parameters. Using normality assumptions for all three types of residuals, ~, ~, and e, again leads to a multivariate probit regression model. This generalized binary growth model is a special case of the structural equation model of Muth6n (1983, 1984). The longitudinal modeling issues just discussed were also brought up in Muth6n (1983), where a random intercept model like Equations (1), (2), and (12) was fitted to the Example 2 data. The problems with standardization issues related to ~- and t~ have also been emphasized by Arminger (see, e.g., ch. 3, this volume) and Muth6n and Christofferson (1981) in the context of structural equation modeling. In the approach of Muth6n (1983, 1984), conditional mean and covariance matrix expressions corresponding to Equations (4) and (5) are considered. This is sufficient given the conditional normality assumptions. Muth6n (1983, 1984) introduced a diagonal scaling matrix A containing the inverse of the conditional standard deviations of the latent response variable at each time point, =

A = d i a g [ V ( y * Ix)] -1/2.

(13)

Muth6n (1983, 1984) describes three model parts. Using the single-indicator growth model example of Equations (4) and (5),

44

Bengt O. MuthOn 0.1 = ATIx 0" 2 -- A[T-rr ~/] 0"3 = 1 ) = A ( T ~ , ~ / ~ T ' + ~ g g ) A ,

(14) (15) (16)

where ~ is the K • K covariance matrix of ~ (cf. Eq. 5). The three parts correspond to the intercepts, slopes, and residual correlation matrix of a multivariate probit regression. Note that 1) is a correlation matrix. The general model of Muth6n (1983, 1984), including multiple indicators as well as multiple factors at each time point, can be expressed as follows. Consider a set of measurement relations for a p-dimensional vector y*, y* = Axl + e,

(17)

and a set of structural relations for an m-dimensional vector of latent variable constructs -q, n =a+

13n + F x +

~,

(18)

where A, a, B, F are parameter arrays, e is a residual (measurement error) vector with mean zero and covariance matrix 19, and ~ is a residual vector with mean zero and covariance matrix ~ . The scaling matrix ZX is also included in this general framework, as is the ability to analyze independent samples from multiple populations simultaneously. Muthrn (1983, 1984, 1987) used a least-squares estimator where with 0. (if'l, 0.;, 0.;)t, F = (s - 0.)' W - I ( s

--

0.),

(19)

where the s elements are arranged in line with 0. and are maximum likelihood estimates of the intercepts, slopes, and residual correlations. Here, s I and s 2 are estimates from probit regressions of each y variable on all the x variables, whereas each s 3 element is a residual correlation from a bivariate probit regression of a pair of y variables regressed on all x variables. A generalized leastsquares estimator is obtained when the weight matrix W is a consistent estimate of the asymptotic covariance matrix of s. In this case, a chi-square test of model fit is obtained as n. F, where n is the sample size and F refers to the minimum value of the function in Equation (19). (For additional technical details on the asymptotic theory behind this approach, see Muthrn & Satorra, 1995.) Muthrn (1984, 1987) presented a general computer program LISCOMP which carries out these calculations.

3.3. Model Identification Given that the response variables are categorical, the general binary growth model needs to be studied carefully in terms of parameter identification. Under

2. Growth Modeling with Binary Responses

45

the normality assumptions of the model, the number of distinct elements of cr represents the total number of parameters that can be identified, therefore the number of growth model parameters can be no larger than this. The growth model parameters are identified if and only if they are identified in terms of the elements of er. It is instructive to consider first the conditional y* variance in some detail for the case of binary growth modeling. Let [A]~ 2 denote the conditional variance of Ykk given x. For simplicity, the focus is on the case with linear growth. With four time points, the conditional variances of y* can be expressed in model parameter terms as [A]o2 = + ~ + tb~o~o [z~]H2 = +~,~ + 2tb~ + tb~ + tb~l~l [A]222 = tb~,~, + 4tb~,~ + 4tb~ + ~2~2 [A]332 = ~,,~ + 6~,~ + 9qs~ + ~3~3"

(20) (21) (22) (23)

Note that the A elements are different because of across-time differences in contributions from ~,~/~ as well as ~ . In Equation (5), using the Gibbons-Beck standardization of +~ - 1, there are no free ~ parameters to be estimated. Contrary to this conventional approach, there are four different ~ parameters in Equations (18) through (21). Because the y* variables are not directly observed, not all of these parameters are identifiable. Instead of assuming ~ = 1 for all time points, as in Gibbons-Beck, the first diagonal element of A can be fixed to unity, corresponding to the first time point. For the remaining time points, the A elements in Equations (19) through (21) are the unrestricted parameters instead of the residual y* variances of ~. The residual variances are not taken as free parameters to be estimated, but can be obtained from the other parameters using Equations (18) through (21). Allowing the A parameters to be different across time allows the residual variances to be different across time. With four time points, this adds three parameters to the model relative to the conventional model. The (co)variance-related parameters of the model are in this case the three free/X elements (not the the's) and the three elements of ~,~/~. With covariates, added parameters are lX~, tx~, -rr~, -rr~, and ~/o. . . . . ~/K-1. It can be shown that with four time points, covariances between pairs of residuals at adjacent time points can also be identified in this model (see Muthdn & Liu, 1994). As opposed to the case of a single response variable, multiple indicator models allow for separate identification of the residual variances and the measurement errors of each indicator. In this case, there is the additional advantage that the residual variances for the latent variable constructs xl are identified at all time points. Multiple-indicator models would assume equality of the measurement parameters (the -r' s and the )t' s) for the same response variable across time. In this case, the p~,~intercept is fixed at zero. The A matrix scaling is general-

46

Bengt O. Muth~n

ized as follows. The scaling factors of A are fixed at the first time point for all of the indicators to eliminate the indeterminacy of scale for each different y* variable (corresponding to each indicator) at this time point. The scaling factors of A are free for all indicators at later time points so that the measurement error variances are not restricted to be equal across time points for the same response variable.

3.4. Implementation in Latent Variable Modeling Software In the case of continuous response variables, Meredith and Tisak (1984, 1990) have shown that the random coefficient model of the previous section can be formulated as a latent variable model. For applications in psychology, see McArdle and Epstein (1987); for applications in education, see Muth6n (1993) and Willett and Sayer (1993); and for applications in mental health, see Muth6n (1983, 1991). For a pedagogical introduction to the continuous case, see Muth6n, Khoo, and Nelson Goff (1994) and Willett and Sayer (1993). Muth6n (1983, 1993) pointed out that this idea could be carried over to the binary and ordered categorical case. The basic idea is easy to describe. In Equation 1, c~i is unobserved and varies randomly across individuals. Hence, it is a latent variable. Furthermore, in the product t e r m ~itk, ~i is a latent variable multiplied by a term tk which is constant over individuals and can therefore be treated as a parameter. The tks may be fixed as in Equation (6), but with three or more time points they may be estimated for the third and later time points to represent nonlinear growth. More than one growth factor may also be used.

4. ANALYSES Simulated and real data will now be used to illustrate analyses using the general growth model with binary data.

4.1. A Monte Carlo Study A limited Monte Carlo study was carried out to demonstrate the sampling behavior of the generalized least-squares estimator in the binary case. The model chosen for the study has a single binary response variable observed at four time points. There is one time-invariant covariate and one time-varying covariate (one for each time point). The simulated data can be thought of as being in line with the Example 1 situation in which the probability of a problem behavior declines over time. Linear decline is specified with T as in Equation (6). Both the random intercept (et) and the random slope ([3) show individual variation as represented both by their common dependence on the time-invariant covariate (w) and their residual variation (represented by ~ / ~ ) . The e~ variable regression has a

2. Growth Modeling with Binary Responses

47

positive intercept (Ix~) and a positive slope (Try,) for w, whereas the [3 variable regression has a negative intercept (tx~) and a negative slope ('rr~) for w. In this way, the time-invariant covariate can be seen as a risk factor, which with increasing value increases o~, that is, it increases the initial probability of the problem and decreases [3, making the rate of decline larger (this latter means that the higher the risk factor value, the more likely the improvement is in the problem behavior over time). The regression coefficient for the response variable on the time-varying covariates (the vk's) is positive and the same at all time points. The residual variances ( ~ ) are changing over time and there is a nonzero residual covariance between adjacent pairs of residuals that is assumed to be equal. The time-varying covariates are correlated 0.5 and are each correlated 0.25 with the time-invariant covariate. All covariates have means of 0 and variances of 1. The population values of the model parameters are given in Table 1.

TABLE 1 Monte Carlo Study: 500 Replications Using LISCOMP GLS for Binary Response Variables Parameter

True value

n = 1000

n = 250

~ ~

0.50 -0.50

0.50 (0.05, 0.05) -0.51 (0.04, 0.04)

0.50 (0.10, 0.09) -0.50 (0.10, 0.09)

"rr~ "rr~ ~/1 ~/2 ~/3 ~/4

0.50 -0.50 0.70 0.70 0.70 0.70

0.50 -0.51 0.70 0.72 0.71 0.71

0.50 -0.50 0.70 0.76 0.71 0.71

qJe~ot ~[3ot ~[313

0.50 -0.10 0.10

0.49 (0.13, 0.13) -0.09 (0.06, 0.05) 0.10 (0.04, 0.03)

0.48 (0.29, 0.26) -0.08 (0.13, 0.11) 0.10 (0.09, 0.08)

~k+ 1, k

0.20

0.22 (0.09, 0.08)

0.26 (0.31, 0.22)

All

1.00 1.00 1.00

0.99 (0.13, 0.13) 1.00 (0.11, 0.11) 1.00 (0.11, 0.11)

1.01 (0.29, 0.25) 1.02 (0.25, 0.22) 1.03 (0.25, 0.23)

14.75 5.53 5.2 1.0

15.51 5.61 5.4 2.0

A22 A33 X 2 Average (df= 15) SD 5% Reject proportion 1% Reject proportion

(0.05, (0.04, (0.06, (0.11, (0.09, (0.08,

0.05) 0.04) 0.05) 0.11) 0.09) 0.08)

(0.10, 0.10) (0.09, (0.11, (0.40, (0.19, (0.18,

0.09) 0.11) 0.29) 0.18) 0.17)

Note. In parentheses are empirical standard deviations, standard errors, df degrees of freedom; SD, standard deviation.

48

Bengt O. Muth#n

Two sample sizes were used, a larger sample size of n = 1000 and a smaller sample of n = 250. Multivariate normal data were generated for y* and x, and the y* variables were dichotomized at zero. The generalized least-squares estimator was used. The parameter values chosen (see Table 1) imply that the proportion of y = 1 at the four time points is .64, .50, .34, and .25. The parameters estimated were IX~, IX[3, "rr,~, 7r[3, ~/o, ~tl, '5/2, ~/3, q/oto~, q/[3o~,11/[313,q/~l,+z~k(a single parameter), [A] ll, [A]22, and [A]33. The threshold parameter -r was fixed at zero and the scaling factor [A]lz was fixed at one. As discussed in section 3.3, the four residual variances of qJ;~ are not free parameters to be estimated but they are still allowed to differ freely across time (their population values are .5, .6, .5, and .2) because the A parameters are free. The degrees of freedom for the chi-square model test is 15. A total of 500 replications were used for both sample sizes in Table 1. Table 1 gives the parameter estimates, the empirical standard deviation of the estimates across the 500 replications, the mean of the estimated standard errors for the 500 replications, and a summary of the chi-square test of model fit for the 500 replications. As seen in Table 1, the estimates for the n = 1000 case show almost no parameter bias, the empirical variation is very close to the mean of the standard errors, and the chi-square test behaves correctly. As expected, the empirical standard deviation is cut in half when reducing the sample size to a quarter, from 1000 to 250. Exceptions to this, however, are the regression slope for the second time point "Y2and the residual covariance ~+,~k" The cause for these anomalies needs further research. In these cases, the standard errors are also strongly underestimated. In the remaining cases, the standard errors agree rather well with the empirical variation, with perhaps a minor tendency to underestimate the standard errors for the (co)variance-related parameters of ~ and A. At n = 250, the variation in the regression intercept and slope parameter (IX and 7r) estimates is low enough for the hypotheses of zero values to be rejected at the 5% level. For the (co)variance-related parameters of +, however, this is not the case and the A parameters also have relatively large variation. The chi-square test behavior at n = 250 is quite good.

4.2. Analysis of Example 2 Data The model used for the preceding simulation study will now be applied to the Example 2 data of neurotic illness as described previously (for more details, see Henderson et al., 1981). Each of the four response variables will be modeled separately. They can also be analyzed together as multiple indicators of neurotic illness, but this will not be done here. Previous longitudinal analyses of these data were done in Muth6n (1983, 1991). Summaries of the data are given in Table 2. As is shown in Table 2, there is a certain drop from the first to the remaining occasions in the proportion of

2. Growth Modeling with Binary Responses

49

TABLE2 Descriptive Statistics for Example 2 Data Response variables Percentage yes: Anxiety Depression Irritability Nervousness

Time 1

Time 2

Time 3

Time 4

26.4 25.5 40.3 24.2

16.5 14.7 31.2 19.0

15.6 17.7 28.1 16.0

16.0 13.9 29.0 15.6

Covariates

N L1 L2 L3 L4

Means

Variances

9.31 3.86 3.17 2.58 2.42

20.66 6.54 5.89 4.90 5.27 Correlations

N L1 L2 L3 L4

1.00 0.22 0.16 0.18 0.21

1.00 0.54 0.50 0.53 .

.

.

1.00 0.49 0.49

1.00 0.51

1.00

.

people answering yes to the neuroticism items. There is also a corresponding drop in the mean of the life event score. Because the latter is used as a timevarying covariate, this means that the data could be fit by a model that does not include a factor for a decline in the response variable, but which instead uses only a random intercept factor model. Given previous analysis results, gender is dropped as a time-invariant covariate. Only the N score, the long-term susceptibility to neurosis, will be used to predict the variation in the random intercept factor. Two types of models will be fit to the data. First, the general binary growth model will be fit, allowing for across-time variation in the latent response variable residual variance and nonzero covariances between pairs of residuals (the residual covariances are restricted to being equal). Second, the conventional binary growth model, in which these features are not allowed for, will be fit as a comparison. In both cases, the same quantities as in Table 1 will be studied, along with two types of summary statistics. One summary statistic is R 2, that is, the proportion of variation is the e~ factor accounted for by N. A second statistic is the proportion that the e~ factor variation makes up of the total variation in the latent response variable y*, calculated at all time points. The oL factor

50

Bengt O. Muth6n

represents individual variation in a neurotic illness trait, variation that is present at all time points. In addition to this variation, the y* variation is also influenced by time-specific variation caused by time-varying, measured covariates (Ls) and time-specific unmeasured residuals (~s). In this way, the proportion is a time-varying measure of how predominant the trait variation is in the responses. Table 3 shows the results for the general binary growth model. The model fits each of the four response variables very well. As expected, the N score has a significantly positive influence (4r~) on the random intercept factor and the L scores have significantly positive influences (C/s) on the probability of yes answers for the response variables. For none of these four response variables is the residual covariance significantly different from zero. Note, however, from the simulation study at n = 250 that the variation in this estimate is quite large and that a large sample size is required to reject zero covariance. As shown in the simulation study, the point estimate of the covariance may be of reasonable magnitude. The model with

TABLE3 Analysis of Example 2 Data Using the General Growth Model ,,

,

Anxiety

Depression

Irritability

Nervousness

ixoL

-1.46 (0.21)

-2.56 (0.34)

-1.21 (0.20)

-2.43 (0.27)

"rre~ yl y2 ~3 ~4

0.06 (0.01) 0.08 (0.03) 0.05 (0.02) 0.06 (0.03) 0.03 (0.02)

0.13 (0.02) 0.14 (0.04) 0.02 (0.03) 0.12 (0.03) 0.04 (0.05)

0.06 (0.01) 0.09 (0.03) 0.08 (0.02) 0.07 (0.02) 0.06 (0.02)

0.14 (0.02) 0.08 (0.03) 0.03 (0.02) 0.05 (0.03) 0.01 (0.03)

~o~o~

0.31 (0.08)

0.39 (0.10)

0.27 (0.07)

0.69 (0.09)

-0.01 (0.05)

-0.12 (0.10)

-0.04 (0.04)

-0.11 (0.06)

1.37 (0.19) 1.48 (0.21) 1.20 (0.18)

0.88 (0.15) 1.08 (0.19) 0.88 (0.14)

1.41 (0.22) 1.43 (0.24) 1.32 (0.22)

1.01 (0.11) 1.14 (0.11) 1.00 (0.11)

~k, k+l All A22 A33 X2(19) p-value R2ot P1 P2 P3 P4

16.49 .624 0.19 0.34 0.31 0.31 0.30

21.73 .298 0.47 0.50 0.41 0.42 0.41

26.02 .130 0.22 0.30 0.28 0.29 0.28

23.30 .225 0.37 0.75 0.52 0.54 0.52

2. Growth Modeling with Binary Responses

51

nonzero residual covariance is therefore maintained. In this particular application, the estimate is small. The scaling factors of A are not significantly different from unity at the 5% level for all but one of the cases. Because in this model there is no [3 factor, A is a function of the oL factor residual variance + , ~ and the residual variance thcc (cf. Eqs. 18-21). Unit values for the A scaling factors would therefore indicate that the residual variances are constant over time in this application. Note, however, from the Table 1 simulation results that the sampling variation in the A estimates is quite large at n = 250 which makes it difficult to reject equality of residual variances over time. The Table 1 results also indicate that the point estimates for A are good. Table 4 shows the results for the conventional binary growth model. This model cannot be rejected at the 5% level in these applications. The parameter estimates are, in most cases, similar to those for the generalized model of Table 3. Differences do, however, show up in the values for the trait variance proportions, labeled P~ through P4 in Tables 3 and 4. Relative to the more general model, the conventional model overestimates these proportions for three out of the four response variables. For example, the conventional model indicates that there is a considerable dominance of trait variation in the response variable Nervousness, with a proportion of .81 for the last three time points (see Table 4).

TABLE 4

Analysis of Example 2 Data Using the Conventional Growth Model Anxiety

Depression

Irritability

Nervousness

~OL

-1.79 (0.17)

-2.40 (0.16)

-1.56 (0.16)

-2.56 (0.23)

9roL ~1 ~/2 "y3 ~/4

0.07 (0.01) 0.12 (0.02) 0.06 (0.03) 0.05 (0.04) 0.04 (0.03)

0.12 (0.01) 0.13 (0.02) 0.03 (0.03) 0.10 (0.03) 0.05 (0.03)

0.08 (0.01) 0.12 (0.02) 0.10 (0.02) 0.09 (0.03) 0.07 (0.03)

0.15 (0.02) 0.09 (0.03) 0.03 (0.02) 0.03 (0.03) 0.01 (0.03)

~c~c~

0.50 (0.51)

0.31 (0.06)

0.42 (0.06)

0.72 (0.05)

X2(23) p-value RZoL P1 P2 P3 P4

26.12 .295 0.17 0.50 0.54 0.54 0.54

26.29 .287 0.49 0.43 0.47 0.45 0.46

33.97 .066 0.24 0.45 0.46 0.47 0.48

29.02 .180 0.39 0.78 0.81 0.81 0.81

52

Bengt O. Muth~n

The more general model of Table 3 points to a much lower range of values for the last three time points, .52 to .54.

5. CONCLUSIONS This chapter has discussed a general framework for longitudinal analysis with binary response variables. As compared with conventional random effects modeling with binary response, this general approach allows for residuals that are correlated over time and variances that vary over time. It also allows for multiple indicators of latent variable constructs, in which case it is possible to identify separately residual variation and measurement error variation. The more general model can be estimated by a limited-information generalized least-squares estimator. The general approach fits into an existing latent variable modeling framework for which software has been developed. A Monte Carlo study showed that the limited-information generalized leastsquares estimator performed well with sample sizes at least as low as n - 250. At this sample size, the sampling variability is not unduly large for the regression parameters of the model, but it is rather high for the (co)variance-related parameters of the model. Analyses of a real data set indicated that the differences in key estimates obtained by the conventional model are not always markedly different from those obtained by the more general model, but can lead to quite different conclusions about certain aspects of the phenomenon that is being modeled. The general approach should be of value for developmental studies in which variables are often binary and in which the variables are often very skewed and essentially binary. The general model allows for a flexible analysis which has so far been used very little with binary responses. A multiple-cohort analysis of this type is carried out in Muthdn and Muthdn (1995), which describes the development of heavy drinking over age for young adults.

ACKNOWLEDGMENTS This research was supported by Grant AA 08651-01 from NIAAA for the project "Psychometric Advances for Alcohol and Depression Studies" and Grant 40859 from the National Institute of Mental Health.

REFERENCES Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443-459.

2. Growth Modeling with Binary Responses

53

Bock, R. D., & Lieberman (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179-197. Diggle, P. J., Liang, K. Y., & Zeger, S. (1994). Analysis of longitudinal data. Oxford: Oxford University Press. Fitzmaurice, G. M., Laird, N. M., & Rotnitzky, A. G. (1993). Regression models for discrete longitudinal data. Statistical Science, 8, 284-309. Gibbons, R. D., & Bock, R. D. (1987). Trend in correlated proportions. Psychometrika, 52, 113-124. Gibbons, R. D., & Hedeker, D. R. (1993). Application of random-effects probit regression models (Technical report). Chicago: University of Illinois at Chicago, UIC Biometric Laboratory. Henderson, A. S., Byrne, D. G., & Duncan-Jones, E (1981). Neurosis and the social environment. Sydney: Academic Press. J6reskog, K. G., & S6rbom, D. (1977). Statistical models and methods for analysis of longitudinal data. In D. J. Aigner & A. S. Goldberger (Eds.), Latent variables in socio-economic models (pp. 285-325). Amsterdam: North-Holland. Laird, N. M., & Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38, 963-974. Longford, N. T. (1993). Random coefficient models. Oxford: Oxford University Press. McArdle, J. J., & Epstein, D. (1987). Latent growth curves within developmental structural equation models. Child Development, 58, 110-133. Meredith, W., & Tisak, J. (1984). "Tuckerizing" curves. Paper presented at the annual meetings of the Psychometric Society, Santa Barbara, CA. Meredith, W., & Tisak, J. (1990). Latent curve analysis. Psychometrika, 55, 107-122. Muth6n, B. (1983). Latent variable structural equation modeling with categorical data. Journal of Econometrics, 22, 43-65. Muth6n, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115-132. Muth6n, B. (1987). LISCOMP. Analysis of linear structural equations with a comprehensive measurement model. Theoretical integration and user's guide. Mooresville, IN: Scientific Software. Muth6n, B. (1991). Analysis of longitudinal data using latent variable models with varying parameters. In L. Collins & J. Horn (Eds.), Best methods for the analysis of change. Recent advances, unanswered questions, future directions (pp. 1-17). Washington, DC: American Psychological Association. Muth6n, B. (1993). Latent variable modeling of growth with missing data and multilevel data. In C. R. Rao & C. M. Cuadras (Eds.), Multivariate analysis: Future directions 2 (pp. 199-210). Amsterdam: North-Holland. Muth6n, B., & Christofferson, A. (1981). Simultaneous factor analysis of dichotomous variables in several groups. Psychometrika, 46, 407-419. Muth6n, B., Khoo, S. T., & Nelson Goff, G. (1994). Longitudinal studies of achievement growth using latent variable modeling (Technical report). Los Angeles: UCLA, Graduate School of Education. Muth6n, B., & Liu, G. (1994). Identification issues related to binary growth modeling (Technical report). Manuscript in preparation. Muth6n, B., & Muth6n, L. (1995). Longitudinal modeling of non-normal data with latent variable techniques: Applications to developmental curves of heavy drinking among young adults (Technical report). Manuscript in preparation. Muth6n, B., & Satorra, A. (1995). Technical aspects of Muth6n's LISCOMP approach to estimation of latent variable relations with a comprehensive measurement model. Psychometrika. Rutter, C. M., & Elashoff, R. M. (1994). Analysis of longitudinal data: Random coefficient regression modeling. Statistics in Medicine, 13, 1211-1231.

54

Bengt O. Muth~n

Stiratelli, R., Laird, N., & Ware, J. H. (1984). Random-effects models for serial observations with binary response. Biometrics, 40, 961-971. Wheaton, B., Muth6n, B., Alwin, D., & Summers, G. (1977). Assessing reliability and stability in panel models. In D. R. Heise (Ed.), Sociological methodology 1977 (pp. 84-136). San Francisco: Jossey-Bass. Willett, J. B., & Sayer, A. G. (1993). Using covariance structure analysis to detect correlates and predictors of individual change over time. Psychological Bulletin. Zeger, S. L., & Karim, M. R. (1991). Generalized linear models with random effects; a Gibbs sampling approach. Journal of the American Statistical Association, 86, 79-86.

Probit Models for the Analysis of Limited Dependent Panel Data GerhardArminger

Bergische Universit~t Wuppertal, Germany

1. INTRODUCTION Panel data are observations from a random sample of n elements from a population collected over a fixed number T of time points. Usually, n is fairly large (e.g., approximately 5000 households in the German Socio-Economic Panel [GSOEP; Wagner, Schupp, & Rendtel, 1991] or approximately 4000 firms in the 1993 panel of the German bureau of labor) and T is fairly small (T = 10 panel waves in the GSOEP). Usually, the time points t = 1. . . . . T are equidistant. In psychology, panel data are often referred to as repeated measurements; in epidemiology they are referred to as cohort data. In analyzing panel data using regression models, one often finds that the dependent variables of interest are nonmetric, that is, dichotomous, censored metric, and ordered or unordered categorical. Typical examples are found in the following areas: 9 The analysis of individual behavior in the labor market. Flaig, Licht, and Steiner (1993) model and test hypotheses about whether a person is unemployed Categorical Variables in Developmental Research: Methods of Analysis Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

55

56

GerhardArminger

or not at time t depending on variables such as unemployment at t - l, length of former unemployment, education, professional experience, and other personal characteristics using data from the GSOEE Here, the dependent variable is dichotomous with the categories "employed" versus "unemployed." 9 The analysis of the behavior of individual firms. Arminger and Ronning (1991) analyze the simultaneous dependence of changes in output, prices, and stock using data from the German business test conducted quarterly by IFO Institute in Munich. Here, the dependent variables of change in output (less, equal, and more than at time t - 1) and change in price are ordered trichotomous and the variable stock is metric as it is measured in production months. The special problem in specifying a model for these data is that trichotomous variables appear on the left a n d on the right side of regression equations. 9 The analysis of marketing data. Marketing data come often in the form of preferences measured on a Likert scale with five or seven ordered categorical outcomes such as "1 = I like product A very much" to "5 = I don't like product A at all." Again, the variables are ordered categorical and often a great number of variables must be reduced to a smaller set of variables by using factor analytic models for ordered categorical data. Ordered categorical variables may again appear on the left and on the right side of regression equations. In the following section the construction principles of Heckman (1981a, 1981b) on the specification and estimation of dichotomous outcomes in panel data are extended to include ordered categorical and censored dependent variables as well as simultaneous equation systems of nonmetric dependent variables. The parameters of models in which strict exogeneity of error terms holds can be estimated by assuming multivariate normality of the error terms in threshold models and by using conditional polychoric and polyserial covariance coefficients in the framework of mean and covariance structure models for nonmetric dependent variables. These models and estimation techniques have been introduced by Muth6n (1984) and extended by Ktisters (1987) and Schepers and Arminger (1992). The estimation of special models in which strict exogeneity of error terms does not hold is also briefly discussed. Special attention is given to the problem of initial states. As an example, the trichotomous output variable of a four-wave panel of 656 firms from the German business test conducted by the IFO Institute is analyzed.

2. MODEL SPECIFICATION 2.1. Heckman's Model for Dichotomous Variables Heckman (1981a, chap. 3.3) considers the following model for an unobserved variable Yit, i = 1 . . . . . n, t = 1 ..... T , where i denotes the individual and t denotes a sequence of equispaced time points:

3. Probit Model and Limited Dependent Panel Data

Yit -- ~it + 6-it"

57

(1)

The term txi, is the expected value of Y~t, which itself may be written as a function of the exogenous variables x;,, lagged observed variables Yi,t-j, and lagged unobserved variables Yi,,-j: vc

I'Lit - Xit~ + E

vc

3"t--j, tYi,t--j + E

j=l

j--1

j

K

~kJ,t--J ]--I Yi,t--I + E l--1

k=l

~

9

kYi,t-k"

(2)

The error terms 6-it are collected in a vector and are assumed to be normally distributed: 6-i "-" (6-il

6-iT)" 6-i "~ t (0, Y,).

.....

(3)

A threshold model is used to model the relation between the unobserved variable Yit and the observed variable Yit: if Yzt > 0 if Yit <- O.

Yit --

(4)

The unobserved variable Y;t is considered to be a disposition or utility that is connected to the observed dependent variable Y;t through a dichotomous threshold model with threshold 0. The values Y~t, Yit and the 1 • P vectors xit are collected in T x 1 vectors Yi, Yi and the T • P matrix X i. The random variables {y;, X~} are identically and independently distributed, which corresponds to a simple random sample from a population. The model specification consists of determining the structures of the systematic part ~Lit and the stochastic part % of the model. A discussion of the different parts of the model specification for Y~t is found in Heckman (1981a) and in Hamerle and Ronning (1995). Here, only the most important elements of the specification are repeated. The first component of ix,.t in Equation (2) is the variation induced by possible time-varying explanatory variables xit. The P x 1 parameter vector [3 is time-constant in this model, but can be changed to P x 1 parameter vectors [3t, t - 1. . . . . T that vary over time. The second component of txit captures the influence of former states of the observed dependent variable Yi,t-9, J > 1, which is called the true state dependence. If 3't-l,, = 3'J 4 : 0 and 3"t--j,t 0 for all j > 1 and t - 1. . . . . T, we have a simple Markov model. Note that the inclusion of former states requires the knowledge of initial states Y~o, Y i , - 1 . . . . . depending on the specification of the parameters 3't-i,t. If the initial states are known and are nonstochastic, they can be included in the vectors xit as additional explanatory variables. If the initial states are themselves outcomes of the process that generates Yit, the distribution of the initial states must be taken into account, as discussed by Heckman (1981 b) and in Section 3 of this chapter. Note that the effects of the former states Y;,t-~ may change for each time point. This is captured by 3't-j,t. In most applications, 3't-j,, is set to 3',_j and almost all of the parameters are set to 0. =

50

GerhardArminger

The third component of [JLi, models the dependence of Yit on the duration of the state Yi,t = 1. Again, the effects of duration may be different for each time length. These different effects are parameterized in ha,,_j. If hj,,_j = h, this is the simple case of a linear duration effect. The fourth component of ].Lit models the dependence of Yit on former values Yi,t-j o f the unobserved endogenous variables. Structures that incorporate this component are called models with habit formation or habit persistence. The idea behind such models is that Yit is not dependent on the actual former state of the observed variable, but rather on the former disposition or habit identified with Yi,t-j instead of Yi,t-j. If the initial dispositions Yi,o, Yi,-1 . . . . . Yi,-(K-I) are known and nonstochastic, they may be included in the list of explanatory variables. Otherwise, assumptions about the distribution of the initial variables Yi,o, Yi,-1 . . . . . Y i , - ( K - 1 ) m u s t be made. Now turn to the specification of the error term Eit in Equation (3). The error term is often decomposed in the form (5)

~'it-- oti at- ~-it'

where oti denotes the error term which varies across individuals but not over time and which may be considered to be an unobserved heterogeneity just as in the metric case (cf. Hsiao, 1986). The values of oti may be considered as fixed effects for each i or may be considered as random effects such that oti "" I "(0, o-2). In the first case, oti is an individual specific parameter. If Yi, is modeled by Yit = xitf3 + oti Jr- E.it and is actually observed, as in the metric case, then oti may be eliminated by taking the first differences Y i , - Y i , t - ~ - ( x i , - x/,t-1)[3 + ei.t - %,-1. This technique does not work for nonmetric models such as the probit model. In the case of a dichotomous logit model, the otis may be eliminated by conditioning on a sufficient statistic, as shown in Hsiao (1986) and Hamerle and Ronning (1995). If oti is a random variable, then it is assumed to be uncorrelated with xit and ~-it. If ~-it has constant variance %2 and is serially uncorrelated, then Ei has the typical covariance structure 2+ orot

2 oroL

2 ore

2 + 2 O'o~ ore

211' + or2I,

v( *.] "2 orc~

"2 orc~

9 9 9

2 _1_ 2 O"OL ore

where 1 is a T X 1 vector of ones and I is the T • T identity matrix 9 More generally, a serial or a factor analytic structure may be assumed for V(e i). (Details are found in Heckman, 1981 a, or Arminger, 1992.) Discussion of the estimation of the parameters of this model under various assumption is deferred to section 3.

3. Probit Model and Limited Dependent Panel Data

59

2.2. Extension to General Threshold Models We now extend Heckman's (198 la) dichotomous models in a systematic way to censored, metrically classified and ordered categorical dependent variables and simultaneous equation systems that allow as dependent variables any mixture of metric and/or limited dependent variables. Only random effect models are considered. The T x 1 vector Yi of utilities Yit is formulated as a multivariate linear regression model

Y i -- ~

-+- 1-[x i "~ e-i,

where x i is a R • 1 vector of observed explanatory variables, ~/is a T • 1 vector of regression constants, and II is a T X R matrix of regression coefficients. The T X 1 vector of error terms follows a normal distribution L I ~ Note that there is a slight change in notation compared with section 2.2. The R X 1 vector x; may be interpreted as the vectorized form of X i in section 2.1 and may additionally include dummy variables denoting the lagged values Y~t-~, Y~t-2 . . . . and duration of observed states. The Heckman model of the form P~i,- x;,[3, t = 1. . . . . T, which in matrix form is written as p~; = XJ3, is then written as ~J'~i =

I-Ixi

with

x i,

!' H

0

...

0 )

[3' 0

... ...

0 [3'

Xi2

,

)Ci ~

\XiT

If the regression parameters are not serially constant, [3' in the first row is ret placed by [3 l, in the second row by [3'2, and so forth. Together with the specification of E through a model for unobserved heterogeneity and for serial correlation, the preceding specification of Yi-~l + Hxi + Ei yields a conditional mean and covariance structure in the latent variable vector Yi, with Yi " ~ t~(~/+ IIxi, E). The model is now extended by allowing not only the dichotomous threshold model of Equation (4), but any one of the following threshold models that maps Yit onto the observed variable y;, (cf. Schepers, Arminger, & Kiisters, 1991). For convenience, the case index i = 1. . . . . n is omitted. 9 Variable Yt is metric (identity relation). Examples are variables such as monthly income and psychological test scores. y, = y,.

(6)

60

GerhardArminger

9 Variable Yt is ordered categorical with unknown thresholds n-t,~ < -rt,2 < . . . < "rt,K, and categories y, = 1. . . . . K t + 1 (ordinal probit relation; McKelvey & Zavoina, 1975). Examples are dichotomous variables such as e m p l o y m e n t (employed vs. unemployed) and five-point Likert scales with categories ranging from "I like very much" (1) to "I don't like at all" (5). yt - k r

yt

E[Tt,k_I, Tt,k)

with ['r,,o, n,, l) - (-oc, ,r,,l) and "r,,K,+l -- +oo.

(7)

Note that for reasons of identification, the threshold "rt,, is set to 0 and the variance of the reduced form error term o52 is set to 1 The parameters in [3, are only identified up to scale. If one considers simultaneous equation models or analyzes two or more panel waves simultaneously, only hypotheses of proportionality of regression coefficients across equations can be tested in general. Hypotheses of equality of regression coefficients across equations can only be tested under additional, and sometimes nontestable, assumptions (Sobel & Arminger, 1992). 9 Classified metric variables may be treated analogously to the orginal probit case with the difference that the class limits are now used as known thresholds (Stewart, 1983). No identification restrictions are necessary. An important example is grouped income with known class boundaries. 9 Variable Yt is one-sided censored with a threshold value q't, 1 known a priori (tobit relation; Tobin, 1958). The classical example is household expenditure for durable goods, such as cars, when only a subset of the households in a sample acquires a car during a given time interval. Yt Y' = ['rt, ,

if Yt > "rt,1 ifyt <- Tt, l"

(8)

9 Variable Yt is double-sided censored with threshold values a-t,, < q't,2 known a priori (two-limit probit relation; Rosett & Nelson, 1975).

Yt-

I]Yt-r~l t "r,,2

if y; -< "rt,~ if r,, 1 < y,. < Tt,2 ifT t2 <-y,.

(9)

In the case of general threshold models, several modifications of the dichotomous models for panel data have to be considered. First, the state dependence must be modified for nondichotomous dependent variables. In the case of a censored dependent variable, there should be a dummy variable di,t_ j that takes on the value of 1 if the variable Yi,t-j has been observed and a value of 0 if Yi,t-j takes on the threshold value. For metrically classified variables and ordered categorical variables, a K • 1 vector of d u m m y variables ~,*t)_j must be defined. The d u m m y variable ~.*]_j equals 1 if Yi,,-j falls in category k, k = 2 . . . . . K + 1 and is 0 otherwise.

3. Probit Model and Limited Dependent Panel Data

61

Second, the identification restrictions for the variances (r 2t, t = 1. . . . . T of q-it and for the thresholds collected in the vector "r~t) for each panel wave t must be chosen with great care. In my opinion, the threshold vectors "r~t) should be set equal across panel waves, otherwise the meaning of the categories of an ordered categorical variable is assumed to vary across time. This restriction immediately implies that, except for the first wave, the variances o',2 need not be restricted but can vary across the panel waves. For censored dependent vari2 ables, no restrictions for o-t are necessary. A further extension concerns the specification of models for a system of variables. Instead of considering only one variable over T waves, one may consider a vector Yit of H dependent variables over time. Each component of Yit is denoted by Yith, h - 1 . . . . . H. The vector of dependent variables Yi of the ith observation is then a H . T • 1 vector of dependent variables observed at T time points. Each latent variable Yith at each time point is then mapped onto the observation Yith through a threshold model of the form given previously (different thresholds may be used for each component). The covariance matrix E of e; contains in this case not only the conditional serial covariance structure for each variable Yth but also the conditional covariance structure between the variables across all time points. An example for such a model is found in Arminger and Ronning (1991). e.

e.

3. ESTIMATION METHOD The estimation of the parameters of the models discussed in section 2 is usually performed with full information or sequential marginal limited-information maximum likelihood (ML) estimation methods. To set up the likelihood equation, an important distinction must be made concerning whether the error terms % are strictly exogenous or not (cf. Keane & Runkle, 1992). q-it is strictly exogenous if Eit 1s assumed to be independent of all present, past, and future values of explanatory variables. Examples of strictly exogenous error terms are included in the following models in which xit is assumed to be scalar and the error terms or/ and % are assumed to be independent of xit and of each other. o

Y i t - [3o + {3jxit + e~i + % Yit = [3o + [3 lxi, + ~lYit- 1 + %.

(10) (11)

The first model is a model without state dependence Yit-1 but with unobserved heterogeneity o~i. The second model includes state dependence but not unobserved heterogeneity. However, if both models are combined, then the error term Eit = Ol.i + q'it is not strictly exogenous" Yit = [3o + [31xit + ~lYit-1 + oLi + %.

(12)

62

GerhardArminger

Here, ~-it is correlated with the regressor Yit-1 through the unobserved heterogeneity term oti. Hence, the likelihood set up in subsection 3.2 must be used for ML estimation in which oti is integrated out. However, if state dependence is replaced by habit persistence, then the error term % is again strictly exogenous. Let the initial value Yio be given by (13)

Yio = No + [3oXio + r

and the following waves by (14)

Yit : ~t + ~lXit -Jr" OLi + Eit,

where ~io, ai, and % are independent of each other and all values Xio, Xil . . . . . XiT. Successive substitution of Yio in the equation for Yil and of Yil in the equation for Yi2, and so forth, yields the result that the error terms r of the reduced form are serially correlated but are independent of all values of Xio, xi~ . . . . . XiT.

3.1. Sequential Marginal ML Estimation Under Strict Exogeneity In this section, strict exogeneity of errors is assumed. For general mean and covariance structures, it is assumed that a P x 1 vector Yi of latent dependent variables follows a multivariate normal distribution with conditional mean and covariance: E(yT[xi) = ~/(O) + 1-I(O)xi,

V(y*lxi) = ~,(O).

(15)

In the analysis of panel data, P equals T if a univariate dependent variable is analyzed or P equals H . T if a multivariate dependent variable is analyzed. ~/(O) Is a P x 1 vector of regression constants and 1-I(O) is a P X R matrix of reduced form regression coefficients, x i Is a R x 1 vector of explanatory variables, 2~(O) is the P x P covariance matrix of the errors of the reduced form, and O is the q x 1 vector of structural parameters to be estimated. The reduced form parameters ~/(O), II(O), and ~ ( 0 ) are continously differentiable functions of a common vector O. Typical examples are simultaneous equation systems Yi = Byi + Fxi + ~i

with

~i "~ N(0, f~),

(16)

with the reduced form parameters II(O) = ( I -

B)-~F

and

~(O) = (I - B)- 111(1 - B)-

I',

(17)

and confirmatory factor analysis Yi = Axli + ~-i

with

rli "" N(0, ~ )

and

ei ~" N(0, |

(18)

and the reduced form parameters I I ( O ) = 0,

E(O) = A ~ A ' + |

(19)

3. Probit Model and Limited Dependent Panel Data

63

In the first example, O consists of the structural parameter matrices B, F, and 11. In the second example, 0 consists of A, ~, | The estimation of the structural parameter vector and the threshold parameters collected in 0 from the observed data vector y; proceeds in three stages. This section is based on Schepers et al. (1991). Computation of the estimates with the MECOSA program is described in Schepers and Arminger (1992). 1. In the first stage, the threshold parameters "r, the reduced form coefficients y and II of the regression equation, and the reduced form error variance 00,2 of the tth equation are estimated using marginal ML. Note that this first stage is the estimation of the mean structure without restrictions as in Equation (15). The parameters to be estimated in the tth equation are the thresholds denoted by the vector % the regression constant denoted by y,, the regression coefficients, 2 The that is, the tth row of H denoted by Hr., and the variance denoted by 00,. marginal estimation is performed using ordinary univariate regression, tobit, and ordinal probit regression. 2. In the second stage, the problem is to estimate the covariances of the error terms in the reduced form equations. Note that in this stage the covariances are estimated without parametric restrictions. Because the errors are assumed to be normally distributed and strongly consistent estimators of the reduced form coefficients have already been obtained in the first stage, the estimation problem reduces to maximizing the loglikelihood function

10(000) : k In P(Yit, Yij ]Xi' "r,, "~t, fit . ,^2~,, 2rj, yj, ^ flj., 6.2, % 1 , ^

(20)

i=1

in which P(Yi,, Yijlxi, "r,, ~/,, I],., or,^2,?j, %, (Ij., 6"2, 000)is the bivariate probability of Yit and Yiy given x i and the reduced form coefficients. A typical example of this bivariate probability is the case when Yt and yj are both ordinal. Then the probability that Yit = k and yij = 1 is given by P(Yit --- k, Yij = l[xi) -- f'~,ll,') s

~ ( Y t ' * Yj* ] ~-Lit' 00t' "2 ~Lij' 002 , 00o)dyj*d y , *,

(21)

^ 2 p~a, ~ 000)is in which ~i, = ~/, + II,.xi, O~ij = ~/j + (-lj.xi and 0(y,,* yj.* I tx,, 00,, the bivariate normal density function. Note that in the ordinal case 6.2 = 6.2 = 1. Hence, 000 is a correlation coefficient called the polychoric correlation coefficient. The loglikelihood function lo(00tj ) has to be modified accordingly if variables with other measurement levels are used. Note that the covariances 000 are the covariances of the error terms in the equations for y,, t = 1. . . . P conditional on x i. The estimated thresholds ?,, the reduced form coefficients ~/, and lit., the vailances 00,, ^ 2 and the covariances 6.o from all equations are then collected in a vec-

64

GerhardArminger

tor ~,, which depends on the sample size n. For the final estimation stage, a strongly consistent estimate of the asymptotic covariance matrix W of ~,, is computed. This estimate is denoted by I,V,,. The asymptotic covariance matrix W is difficult to derive because the estimates of 6-u of the second stage depend on the estimated coefficients 'Tf, ~ f = t, j of the first stage. The various elements of the asymptotic covariance matrix W are given in Ktisters (1987). The estimate ff'n is computed in MECOSA by using analytical first order and numerical second order derivatives of the first and second stage loglikelihood function. 3. In the third stage, the vector of K of thresholds, the reduced form regression coefficients, and the reduced form covariance matrix is written as a function of the structural parameters of interest, collected in the parameter vector O. The parameter vector 0 is then estimated by minimizing the quadratic form

1-~If.,O'~,

Q,,(O) - ( ~ , , -

K(O))tW,-~I(I~n-K(O)),

(22)

which is a minim distance approach based on the asymptotic normality of the estimators of the reduced form coefficients. The vector ~,, is asymptotically normal with expected value K(O) and covariance matrix W. Because W, is a strongly consistent estimate of W, the quadratic form Qn(O) is centrally xZ-distributed with p - q degrees of freedom if the model is specified correctly and the sample size is sufficiently large. The number p indicates the number of elements in ~:,,, and q is the number of elements in O. The computation of Wn is quite cumbersome for models with many parameters. The function Q,,(a3) is minimized using the Davidon-Fletcher-Powell algorithm with numerical first derivatives. The program MECOSAfollows these three estimation stages. In the third stage, the facilities of GAUSS are fully exploited to estimate parameters under arbitrary restrictions. The parameter vector K(O) may be defined using the matrix language and the procedure facility of GAUSS. Consequently, arbitrary restrictions can be placed on -r(O), ~/(~), II(O), and ~(O), including the restrictions placed on V(%* = ~(0) in the analysis of panel data. The MECOSAprogram provides estimates of the reduced form parameters II(O) in the first stage by regressing all dependent variables in Yi = (Y/~. . . . . Y/T)' on all regressors collected in xi without the usual restriction that the effect of xi, o n Yi,t-j, J >- 1 is 0. This restriction has to be put into the program in the third stage. If the model includes lagged dependent variables, it may be necessary to restrict the parameter estimates already in the first stage. In this case, one can trick the program into restricted estimation in the first stage by running the first stage of MECOSA T times separately for each Yit which is then regressed on xit. In the second stage, the standard batch input of MECOSA has to be corrected to deal with the T estimation results from the first stage. The polychoric covariances and/or correlation coefficients of the reduced form are then computed under the restrictions of the first stage. :,r:.

3. Probit Model and Limited Dependent Panel Data

65

3.2. ML Estimation for a Model with State Dependence and Unobserved Heterogeneity A model with state dependence and unobserved heterogeneity % --~ N(0, 0~ for a dichotomous outcome y , ~ {0, 1} will now be considered. Such a model has been found to be useful in labor market research in which Yir denotes employment which may depend on former employment status, on observed exogenous variables, and on unobserved variables collected in % that vary across individuals but not across time. The initial value Yio is assumed to be fixed. The model is written as

Yir

[3 Xir + "lYit-I + % + e-it"

(23)

The % are assumed to be serially independent where % --~ N(0, 002) and are assumed to be independent of or;. If the variance 00,2 is set to 1, then the conditional individual likelihood function for [3 and 3, given ot may be formulated as Li([3, "y I'rl) = T

I-I [di)(~3'xit + "Yyit-i + 00aTI)]''" [1 -- Cb(f3'xit + "YYit-1 + O'e~TI)]i-y;',

(24)

t=l

where ~ ( . ) in the standard normal acumulative distribution function and rl "-" N(0, 1) such that oL = 00,~,rI. Hence, L;(6;, "YIxI) is the product of T conditional probit likelihood functions. To estimate [3, "y, and 00., the likelihood function is set up by integrating over rl and computing the product over all individuals i = 1, 9

.

.

,F/:

(25) i=1

zc

Here, ~ ( x I) is the density function of the standard normal distribution. The integral has to be evaluated numerically using Gauss-Hermite numerical integration. This estimation method may be extended to ordered categorical data as shown in Arminger (1992). Unfortunately, no standard software is available yet to deal with the general model.

3.3. The Problem of Initial States For the estimation of the general Heckman model, it was assumed that the initial states for a model with state dependence or with habit persistence are known and nonstochastic or that information about the initial states could be used in the list of explanatory variables. If this is not the case, one must assume that either a new process begins at the first wave of the panel or that the error terms

66

GerhardArminger

of the process for the initial states are uncorrelated with the error terms of the T panel waves (cf. Heckman, 1981 a, 198 lb). Note that the first assumption holds for the attrition process in a panel where the dependent variable is the participation of a person in panel wave t given that this person has participated in wave 1. At the first wave, all persons participate. Hence, the initial states are known and a new process starts at time 1. Heckman (1981b) considers the following special case of the initial states problem: 6"it-- 6 -+- "YYi t--~ -+- O~i -1- ~-it

Yit =

1 if ' y i . t > 0 0 if Yir <- 0.

with

(26)

The random effects are oLi "~ i~(0, (7a), 2 E(oLi~-it ) = O, ~-it --~ I~(0, o'~) with o'~2 = 1. In this model, the only initial value is Y;o. What happens if Y/o is not known and nonstochastic but is determined by the same process as Yit for t = 1. . . . . T? In this case, Yio is dependent on oti and the sample conditional likelihood function for Yit, t - 1 . . . . . T given Yio is given by integration over the unobserved variable a which may be written as a -- o'~xl with ~1 ~"- I~(0, 1):

L([3o, ~l, cry) =

i= 1

~

zc t =

(Op(~3 q- ~lYi,t-1

+ O'aT]) y`'

(1 -- Oi~(~ + "YYi,t-1 -k- O'oLTI))l-yi' )<

P(Yio [cr~xl)OP(rl)dxl,

(27)

where cI)(.) is the standard normal density. The term P(Yiol oO is the probability that the random variable Yio is either > 0 or-< 0. Possible specifications of P(Yiola) by introducing restrictions on the process generating the exogenous variables xit and application of the estimation method of Kiefer and Wolfowitz (1956) are discussed by Heckman (1981b). However, for all practical purposes this estimation method is too cumbersome. As a first simple solution for this estimation problem, Heckman proposes the estimation of a i as a parameter by ML methods. This method, however, yields only consistent estimates of the structural parameters 13 and "y if T ~ w (cf. Andersen, 1973; Neyman & Scott, 1948), which cannot be assumed for panel data. The limited Monte Carlo evidence given by Heckman (1981b) for T = 8 panel waves in the model of Equation (27) is certainly not sufficient to recommend the ML estimation of cr i in conjunction with 13 and "y as a general solution. The second simple solution proposed by Heckman (1981b) is the substitution of P(Yiola) by F(xio~) ), where Xio is a vector of explanatory variables for Yio and F(.) is the distribution function of the random variable Yio = Xio ~ -at- ~-iO"

(28)

3. Probit Model and Limited Dependent Panel Data

67

This ad hoc solution is simply the attempt to replace the unobserved heterogeneity in Y;o by observed heterogeneity in X~o. The error term is allowed to be correlated with OLi and % for t = 1. . . . . T without restrictions to capture the serial correlation between eio and % = o~; + Eit induced by o~;. In practice, this procedure amounts to the inclusion of Yio in the vector of dependent variables and additional parameters in the augmented covariance matrix for E i = ( E i 0 , 9 If %9 = oLi + % with V(c~i) = O"2 and V(ei, ) = O"G, 2 then the augell9. . . . . eiT)'. mented covariance matrix is given by 2 O'G0

2 0"2 + O'e

t3"10

v(~2) =

~2o

~

9 0"70

"2 O-oL

~

2+

"2 O-oL

~

2

.

9 9 9

(29)

2 q_ 2 O-c~ O"e

The limited Monte Carlo simulation of Heckman (1981b) shows that this approach works better than the simultaneous ML method to yield consistent estimates of 13 and y. This solution for the initial states problem depends crucially on the specification of a model for the initial states. The key idea is to specify a model for the initial states that gives as good predictions as possible. The model for the initial states need not be the same model as for Yit, t - 1 . . . . . T. The errors eio, ei_ j, ei_ K are allowed to correlate freely with %. If the model for the initial states is misspecified, the simple solution will work poorly 9 A problem that is easier to handle occurs if habit persistence instead of state dependence is taken into account. Assume that the following model is formu2 g ( E i t ) _. o e2, and E(Eitq.is ) "-- 0 for t 4: s: l a t e d w i t h Eit* __ ot i -k- q-it, W ( ~ = O-c~, Yit = xit[3 + ~/Yi,t-I + eit,

t = 1. . . .

T.

(30)

Furthermore, we assume like Bhargava and Sargan (1983) for the metric case, that the start value Yio has been generated by the same process a s Yi,t, that is, Yio = Xio~ -k- ~/Yi,- 1 + Eio.

(31)

This implies that y*go is determined by :xz

Yio = ~

~/k(Xi,-k[3 + e/,-k) = Yio q- ~i,-k,

(32)

k=O .._~

k *

where ~i.-k k=O ~/ e/,-k" Assume that the optimal predictor of Yio given all observed values x,,, t - 1. . . . . T of the exogenous variables is given by Yio -- X i ~ '

(33)

68

GerhardArminger

where 8 is the vector of regression coefficients for the vector Then Y;o may be written as

x i -- (Xio,

Xi

I,

9

9

9

,

x;r).

Yio =

(34)

Yio -nt- RiO'

where Uio ~ I~ o"2) and Uio is independent of oti and %. The model for the augmented vector Yi - (Y;o, Y T ' ) ' can then be written in the form of a simultaneous equation system: Yi = B y i

+ Fx;

(35)

+ e. i ,

with the (T + 1) X (T + 1) parameter matrix B and the (T + l) X q parameter matrix F" 0 ~ B=

0 o

. . . ...

0 o

0 o

y

...

0

0

,

F=

8 0

0 13

0 0

0

0

13 . . .

9

,

o o . . . y o

oo

. . . ...

0 0 0

(36)

.

o...

j3

The vector of explanatory variables )~i is "~i = (X/o, Xil ..... X i T ) ' . The covariance matrix V({~) of the augmented vector e i = (eio, e i) is given by the elements (cf. Bhargava & Sargan, 1983): 2 O'er -- ~ 2

v~176 =1 VOt - -

2 F O"u

2 O'cx

1-y 2+

V tt = O'oL V,s -

t

2 O'e 1 - - ,,~2

2

er a ,

2

O'e ,

t =

1, . . . , T

t 4= s 9

The reduced form parameters estimated in the first two stages of MECOSA are given by H=(I-B)-~F

and

~=(I-B)-~V(~)(I-B)

-~'.

(37)

4. ANALYSIS OF PRODUCTION OUTPUT FROM GERMAN BUSINESS TEST DATA To illustrate the model in section 2 and the estimation methods of section 3.1, I have analyzed business test data from the IFO Institute 9Four waves obtained in August and November 1987 and February and May 1988 from 656 German firms are presented. The variables and codes used in the analysis are given in Table 1.

3. Probit Model and Limited Dependent Panel Data

69

TABLE 1 Questions and Variables from the IFO Business Test

Question Our domestic production activity concerning XY has been with regard to the month before brisker (3), unchanged (2), weaker

Measurement level

Variable in model

Variable name

Ordinal

Ay, = y, - y,_ l

Output O

(1). At the moment, our stock of unsold finished products of XY corresponds to: no stock, less than 0.5/4, 1/4 . . . . . 6/4, more than 6/4, o r . . . months of production.

Metric

At the moment, our stock of raw materials for the production of XY corresponds to: no stock, less than 0.5/4, 1/4 . . . . . 6/4, more than 6/4, o r . . . months of production.

Metric

At the moment, we feel that our stock of orders of XY is relatively big (e.g., extended delivery period) (3), stayed the same (2), be rather unfavorable (I).

Ordinal

a,-

Under elimination of mere seasonal fluctuations, our business situation for XY during the coming 6 months will be rather favorable (3), stay more or less the same (2), be rather unfavorable (1).

Ordinal

d,,,+, - d,

Business expectation GL

With regard to the month before, for us the demand situation (at home and abroad) has improved (3), stayed the same (2), decreased ( 1).

Ordinal

Ad,=d,-d,_l

Demand a t t D

Number of employes in the business (grouped).

Metric

In k

LNE

Production activity at time t - 1 with 1 if Ay,_ ~ = 1 and 0 otherwise.

Dummy

u,

Output OL1

Production activity at time t - 1 with 1 if Ay,_ ~ = 3 and 0 otherwise.

Dummy

v,

Output OL2

ft-i

Stock of finished products LFP

rt rt-l

Stock of raw material LRP

f,

In

a~'

Order stock AB

70

GerhardArminger

The variables are indexed by the number of the panel wave; hence O o, LFP o, LRP o. . . . refer to the variables at time 0, and O~, LFP~, LRP~ . . . . refer to the variables at time 1. The variables O (output), AB (order stack), GL (business expectation), and D (demand) are only measured as ordered categorical variables, the variables LFP, LRP, and LNE are metric, and OL 1 and OL2 are d u m m y variables. The model that is specified now is set out in Arminger and Ronning (1991). The process described by the model is conditional on wave 0, hence the process is not modeled for this wave. AYt = P~t + ~3tl(at-1 -- a t - l )

~rt-1

+ [3,2(dt,t+ 1 - dr) + ~,3st + ~/tl( In K)

(

+~/,3 In ~

+

'~t4blt

+

"~t5Vt

+%t=

1, 2, 3,

(38)

Where e, --- In(0, cr2) is assumed to be serially uncorrelated. This model describes the change in output at time t as a function of the stock of orders at time t - 1, business expectation at time t, the number of employees, the relative change of stock in raw material from t - 1 to t, the relative change in the stock of finished products from t - 1 to t, and the states of the output variable at time t - 1. The variable In K, that is, the number of employees, does not change over time. Unobserved heterogeneity is not incorporated. The variable s t is defined as s t = [(dt - dr_ ~) - (d t_ ~,t - d , _ ~)],

(39)

which may be interpreted as a shock or surprise effect. If (dt - d t _ ~ ) > ( d t - ~ . , - d,_,), then the shock is positive, that is, the demand is higher than the past expectation; otherwise, the shock is 0 or negative. The effect of s t may be estimated by setting the parameter of the demand (dt - dt-~) to [3,3 and the parameter of the past business expectation (dr.,-~ - d r - ~ ) to -[3t3. Note that the variables Ay,, (a,_ 1 - a,_ 1), (d,.,_ l - dr), and ( d , - dr_ 1) are only observed at an ordinal scale where the following observation rule is supposed to hold:

Ot =

i

if if if

Ayt_<.r] 1)

(40)

"r I < A y t <-- "r~2l) A y t > "r~21).

The observation rules for AB t, GL t, and D t are analogous. For identification, the first threshold is set to 0. Because AB t, GL t, and Dt are only observed at an ordinal scale, it is assumed that ( a t - l - at*-l), (dt,t-~ - dr), and ( d r - d r - ~ ) are endogenous variables with means conditioned by the exogenous variables In K,

\

In ~

rt-I/

,

In

:,-I

, u t,

71

3. Probit Model and Limited Dependent Panel Data

and v, and a multivariate normal error vector that is uncorrelated with % The model for all endogenous variables in the first wave is therefore given by a0 - a0 d21 - dl d I - do

IX1 [&2 -

d l , o - do Yl - Yo

q-

~4

~5 Yll

-q-

~1,3

Y12

0 0

0 0

0 0

0 0

0

0

0

0

0 [~51

0 ~52

0 [~53 -InK

Y14

Y,5~

/ln

"~31

"~32

'~34

'~/42

~44

Y35 ] Y45 ]

~

'~/41 Y51

Y52

Y54

Y55 /

Ik

\

0 0

a0 - a0 d2, l - d l

0

d I - do

0 0

dl, o - d o Yl - Yo

ra

In

f~

+

.

(41)

u1 P1

The models for wave 2 and wave 3 are constructed in similar ways. Note that the variable GL occurs two times in the first wave as GL o and GL~. In the second wave, GL~ must be taken from the first wave. Hence the whole model consists of 13 equations. The focus here, however, is only on Equations 5, 9, and 13, that is, y~ - Y o , Y 2 - Y ~ , and Y3 -- Y2. The first model does not take into account restrictions by proportionality of coefficients for each wave and unobserved heterogeneity. The parameter estimates obtained from MECOSA are shown in Table 2. The pseudo R2s of McKelvey and Zavoina (1975) show that only a small portion of the variance of the output is explained. The output increases in the second wave in comparison to the first and third waves. Judged by the z-values, the variables stock order (AB) and surprise effect are more important than business expectation (GL). The firms react primarily to shocks of the recent past. If the shock has been positive, more output is produced. The variables stock of raw materials and stock of unsold finished products are of lesser importance than the dependence on the state of the period before. Here, however, only the decrease of output in the past period matters. The covariances between the errors are rather small, indicating that the assumption of uncorrelated errors over time given the former states is correct. In Table 3, the results of the restricted parameter estimation under the hypothesis of proportionality of the regression coefficients except for the constants and the effect of the number of employees are shown. The hypothesis of proportionality is not rejected at the 0.05 test level. The error variance of the second wave is greater in absolute terms than the error variances of the first and second waves as judged from the inverse of the proportionality coefficient ~.

72

GerhardArminger

Unrestricted Parameter Estimates for IFO Output Model

TABLE 2

Explanatory variables

"r~ a"2 IX (a,_~ - a~;~) d,+ ~., - d, s, In K In

Wave 1

2.175 0.604 0.439 0.053 0.413 --0.049

r,

0 (28.989) (1.613) (5.685) (1.046) (10.993) (-- 1.433)

Wave 2

2.175 1.147 0.305 0.089 0.312 --0.023

0 (28.989) (3.281) (4.161) (2.319) (9.844) (--0.707)

0.097 (0.679)

--0.077 (0.481)

Wave 3

2.175 0.599 0.293 0.125 0.385 0.015

0 (28.989) (2.793) (6.236) (2.822) (13.532) (0.468)

0.086(0.632)

Ft-- 1

f, ft-~ u, v,

0.238 (2.973)

-0.031 (-0.369)

-0.058 (-0.796)

-0.543 (-3.040) 0.129 (1.039)

-0.834 (-5.916) -0.204 (-1.697)

-0.227 (-2.386) 0.429 (3.217)

R 2MZ

0.089

0.184

0.116

0.514 O. 127 0.045

0.715 -0.010

0.616

In

Covariances Wave l Wave 2 Wave 3 Note: z-values are in parentheses.

5. CONCLUSION The models discussed in this chapter serve as an introduction to the wide range of models that have been and may be formulated for the analysis of nonmetric longitudinal data with few time points observed in many independent sample elements. Other models include extensions of legit models and of loglinear models for count data (cf. Hamerle & Ronning, 1995). The estimation methods given here have been proven to be useful. However, new approaches for estimation are emerging. Beck and Gibbons (1994) exploit new techniques for highdimensional numerical integration. Muth6n and Arminger (1994) report first results for using the Gibbs sampler to describe the a posteriori distribution of parameters in MECOSA-type models.

ACKNOWLEDGMENTS I am grateful to the IFO Institute Munich for providing the business test data and to Professor G. Ronning of the University of Ttibingen and to R. Jung of the University of Konstanz for preparing

3. Probit Model and Limited Dependent Panel Data

TABLE3

Restricted Parameter Estimates for IFO Output Model

Explanatory variables

-rI "r2 tx (a,_~ - a~"__~) d,+~.,- d, s, In K In

73

Wave 1

2.133 0.850 0.393 0.097 0.413 --0.073

0 (30.454) (2.514) (9.130) (3.3169) (13.255) (--2.292)

Wave 2

2.133 1.212 0.393 0.097 0.413 --0.036

0 (30.454) (3.051) (9.130) (3.3169) (13.255) (--0.881)

Wave 3

2.133 0.731 0.393 0.097 0.413 0.006

0 (30.454) (3.072) (9.130) (3.3169) (13.255) (0.172)

r, rt-l

0.125 (1.505)

0.125 (1.505)

0.125 (1.505)

In f' fr-J u, v,

0.036 (0.774)

0.036 (0.774)

0.036 (0.774)

--0.501 (--7.466) 0.014 (0.224)

--0.501 (--7.466) 0.014 (0.224)

--0.501 (--7.466) 0.014 (0.224)

0.753

0.908

df

15

X2 Statistic

5.781

Note. z-values are in parentheses.

the data. Helpful comments on an earlier version of the chapter have been provided by an unkown reviewer. Comments should be sent to Gerhard Arminger, Bergische Universit~it Wuppertal, Department of Economics (FB 6), D-42097 Wuppertal, Germany.

REFERENCES Andersen, E. B. (1973). Conditional inference and models for measuring. Copenhagen: Mentalhygiejnisk Forsknings Institut. Arminger, G. (1992). Analyzing panel dam with non-metric dependent variables: Probit models, generalized estimating equations, missing data and absorbing states, (Discuss. Pap. No. 59). Berlin: Deutsches Institut for Wirtschaftsforschung. Arminger, G., & Ronning, G. (1991). Ein Strukturmodell ftir Preis-, Produktions- und Lagerhaltungsentscheidungen von Firmen, IFO-STUDIEN. Zeischrift fiir Empirische Wirtschaftsforschung, 37, 229-254. Bhargava, A. and Sargan, G. D. (1983). Estimating Dynamic Random Effect Models from Panel Data Covering Short Time Periods, Econometrica, 51, 1635-1359. Bock, R. D., & Gibbons, R. D. (1994). High-dimensional multivariate probit analysis. Unpublished manuscript. University of Chicago, Department of Psychology. Flaig, G., Licht, G., & Steiner, V. (1993). Testing for state dependence effects in a dynamic model of male unemployment behavior (Discuss. Pap. No. 93-07). Mannheim: Zentrum ftir Europ~iische Wirtschaftsforschung Gmbh. Hamerle, A., & Ronning, G. (1995). Analysis of discrete panel data. In G. Arminger, C. C. Clogg,

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& M. E. Sobel (Eds.), Handbook of statistical modeling for the behavioral sciences. (pp. 401-451). New York: Plenum. Heckman, J. J. (1981a). Statistical models for discrete panel data. In C. F. Manski & D. McFadden (Eds.), Structural analysis of discrete data with econometric applications (pp. 114-178). Heckman, J. J. (198 l b). The incidental parameters problem and the problem of initial conditions in estimating a discrete time-discrete stochastic process. In C. E Manski & D. McFadden (Eds.), Structural analysis of discrete data with econometric applications (pp. 179-195). Hsiao, C. (1986). Analysis of panel data. Cambridge, MA: Cambridge University Press. Keane, M. E, & Runkle, D. E. (1992). On the estimation of panel-data models with serial correlation when instruments are not strictly exogenous. Journal of Business & Economic Statistics, 10(1), 1-29. Kiefer, J., & Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Annals of Mathematical Statistics, 27, 887-906. KiJsters, U. (1987). Hierarchische Mittelwert- und Kovarianzstrukturmodelle mit nichtmetrischen endogenen Variablen. Heidelberg: Physica Verlag. McKelvey, R. D., & Zavoina, W. (1975). A statistical model for the analysis of ordinal level dependent variables. Journal of Mathematical Sociology, 4, 103-120. Muth6n, B. O. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115-132. Muth6n, B. O., & Arminger, G. (1994). Bayesian latent variable regression for binary and continuous response variables using the Gibbs sampler. Unpublished manuscript, UCLA, Graduate School of Education. Neyman, J., & Scott, E. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1-32. Rosett, R. N., & Nelson, F. D. (1975). Estimation of the two-limit probit regression model. Econometrica, 43, 141-146. Schepers, A., & Arminger, G. (1992). MECOSA: A program for the analysis of general mean- and covariance structures with non-metric variables, user guide. Fraunenfeld, Switzerland: SLI-AG. Schepers, A., Arminger, G., & Kiisters, U. (1991). The analysis of non-metric endogenous variables in latent variable models: The MECOSA approach. In E Gruber (Ed.), Econometric decision models: New methods of modeling and applications (pp. 459-472). Heidelberg: Springer-Verlag. Sobel, M., & Arminger, G. (1992). Modeling household fertility decisions: A nonlinear simultaneous probit model. Journal of the American Statistical Association, 87, 38-47. Stewart, M. B. (1983). On least squares estimation when the dependent variable is grouped. Review of Economic Studies, 50, 737-753. Tobin, J. (1958). Estimation of relationships for limited dependent variables. Econometrica, 26, 24-36. Wagner, G., Schupp, J., & Rendtel, U. (1991). The Socio-Economic Panel (SOEP)for GermanyMethods of production and management of longitudinal data (Discuss. Pap. No. 3 l a). Berlin: Deutsches Institut ftir Wirtschaftsforschung.

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Catastrophe Analysis of Discontinuous Development 9

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Hart L. J. van der Maas and Peter C. M. Mo/enaar University of Amsterdam The Netherlands

1. INTRODUCTION The theme of this book is categorical data analysis. Advanced statistical tools are being presented for the study of discrete things behaving discretely. Many relevant social science questions can be captured by these tools, but there is also tension between categorical and noncategorical data analysis which centers on ordinal, interval, and ratio scaled data. Zeeman (1993) presents an original view on this conflict (see Table 1). He distinguishes four types of applied mathematics according to whether things are discrete or continuous, and whether their behavior is discrete or continuous. Of special interest is Pandora's box. In contrast to the other three types of applied mathematics, discrete behavior of continuous things often gives rise to controversies. The mathematical modeling of music, the harmonics of vibrating strings, led to a major debate in the eighteenth century. The foundations of quantum theory are still controversial. A final example is catastrophe theory, which is concerned with the modeling of discontinuities. Researchers in developmental psychology are well aware of the controversial nature of discontinuities in, for example, cognitive development. Categorical Variables in Developmental Research." Methods of Analysis Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

77

78

H. L. J. van der Maas and P. C. M. Molenaar

TABLE 1 Four Types of Applied Mathematics Things Behavior

Discrete

Continuous

Discrete

Dice Symmetry DISCRETE BOX Finite probability Finite groups

Continuous

Planets Populations TIME BOX Ordinary differential equations

Music, harmony Light Discontinuities PANDORA'S BOX Fourier series Quantum theory Catastrophe theory Waves Elasticity CONTINUOUS BOX Partial differential equations

Note.

From Zeeman (1993).

Note that categorical behavior is not explicitly included in Zeeman's boxes. Categorical data, nominal measurements, should be labeled as discrete, whereas measurements on an ordinal, interval, or ratio level are covered by continuous behavior. Probably, the critical distinction here is in quantitative and qualitative differences in responding. If this distinction is crucial, then the definition of discreteness in catastrophe theory includes categorical data. We will not discuss further the classification of Zeeman. We hope that it illustrates the relationship between categorical data analysis and this chapter, the goal of which is to present our results, obtained by the application of catastrophe theory, to the study of conservation acquisition. Conservation acquisition is a kind of benchmark problem in the study of discontinuous cognitive developmerit. Discontinuous development is strongly associated with the stage theory of Piaget (1960, 1971; Piaget & Inhelder, 1969). The complexity of debate on this theory is enormous (the Pandora box), and in our presentation we will necessarily neglect a large body of literature on alternative models, other relevant data, and severe criticism on Piaget's work. It is common knowledge that Piaget's theory, especially concerning stages and discontinuities, has lost its dominant position in developmental psychology. Yet, we will defend that our results generally confirm Piaget's ideas on the transition of nonconservation to conservation. We start with a short introduction to catastrophe theory, issues in conservation research, and alternative models of conservation acquisition. We proceed by explaining our so-called "cusp" model of conservation acquisition and the possibilities to test this model. The last part of this chapter discusses the experi-

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ments that we conducted to collect evidence for the model. We first explain our test of conservation, a computer variant of the traditional test. Finally, data from a cross-sectional and from a longitudinal experiment are discussed in relation to three phenomena of catastrophe models: sudden jumps, anomalous variance, and hysteresis.

2. CATASTROPHETHEORY There are several good introductions to catastrophe theory. In order of increasing difficulty, we can recommend Zeeman (1976), Saunders (1980), Poston and Stewart (1978), Gilmore (1981), and Castrigiano and Hayes (1993). Catastrophe theory is concerned with the classification of equilibrium behavior of systems in the neighborhood of singularities of different degrees. Singularities are points where, besides the first derivative, higher order derivatives of the potential function are zero. The mathematical basis of catastrophe theory consists of a proof that the dynamics of systems in such singular points can be locally modeled by seven elementary catastrophes (for systems with up to four independent variables). The elementary behavior of systems in singular points depends only on the number of independent variables. In the case of two independent variables, systems characterized by singular behavior can be transformed (by a well-defined set of transformations) to the so-called cusp form. Our preliminary model of conservation acquisition is formulated as a cusp model. In contrast to the well-known quadratic minima, the equilibrium behavior of singular systems is characterized by strange behavior such as sudden jumps and splitting of equilibria. The quadratic minimum is assumed in the application of linear and nonlinear regression models. In each point of the regression line a normal distribution of observed scores is expected. In contrast, singular minima lead to bimodal or multimodal distributions. If the system moves between modes, this change is called a qualitative change (of a quantitative variable). The comparison with regression analysis is helpful. As in regression models, catastrophe theory distinguishes between dependent (behavioral) and independent variables (control variables). These variables are related by a deterministic formula which can be adjusted for statistical analysis. The cusp catastrophe is denoted by

V(X; a, b) = 1/4 X 4 - 1/2 a X

2 -

b X,

(1)

which has as equilibria (first derivative to zero): X 3 -

a X-

b = O.

(2)

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If we compare the latter function to regression models in the general form of

X=f(a,b),

(3)

we can see an important difference. The cusp function, Equation (2), is an implicit function, a cubic that cannot be formulated in the general regression form. This difference has far-reaching consequences. In the cusp, sudden jumps occur under small continuous variations of the independent variables a and b. In contrast, in the regression models, either linear or nonlinear small continuous variation of independent variables may lead to an acceleration in X but not to genuine discontinuities. Of course f, in the regression model, can be an explicit discontinuous threshold function, but then a purely descriptive position is taken. In catastrophe functions, the equilibrium surfaces are continuous (smooth), hence discontinuities are not built in. In catastrophe theory, qualitative changes in quantitative behavior are modeled in a way that is clearly distinguished from not only (nonlinear) regression models but also from Markov chain models (Brainerd, 1979) and special formulations of Rasch models (Wilson, 1989).

3. ISSUES IN CONSERVATION Conservation is the invariance of certain properties in spite of transformation of form. An example of a conservation of liquid task is shown in Figure 1. The liquid of one of two identical glasses (B) is poured into a third, nonidentical class (C). The child's task is to judge whether the quantity in C is now more than, equal to, or less than in A. A conserver understands that the quantities remain equal, a nonconserver systematically chooses the glass with the highest level as having more. A typical property of conservation tests is that the lowest test score is systematically below chance score.

FIGURE 1. A standard equality item of conservation of liquid. In the initial situation A and B have to be compared; in the final situation, after the transformation, A and C are to be compared (as indicated by the arrows below the glasses).

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Piaget and co-workers used conservation repeatedly as an example to explain the general equilibration theory. Piaget's idea of epigenetic development implies that children actively construct new knowledge. In conservation development, a number of events can be differentiated: (a) (b) (c) (d)

focusing on the height of the liquid columns only (nonconservation) focusing on the width of the liquid columns only the conflict between the height and the width cue the conflict between the height cue and the conserving transformation (e) constructing new operations, f and g (f) focusing on the conserving transformation (g) understanding of compensation (multiplication of height and width) In Piaget's model, the sequence of events is probably a, b, c, e, f, and g. In the Piagetian view, f and g are connected. Whether d should be included, or that d is connected to c, is unclear. Many authors give alternative sequences and deny the active construction of the new operations. Bruner (1967), for instance, argues that f and g already exist in the nonconservation period but are hampered by the perceptual misleading factor. Peill (1975) gives an excellent overview of the various proposals about the sequence of events a to g. Thousands of studies have been conducted to uncover the secrets of conservation development. Many models have been introduced which have been tested in training settings and in sequence and concurrence designs by test criteria varying from generalization to habituation. Mostly, consensus has not been reached. In our opinion, conservation (or nonconservation) cannot be reduced to a perceptual, linguistic, or social artifact of the conservation test procedure, although perceptual, linguistic, and social factors do play a role, especially in the transitional phase. In a later section, we will explain how we developed a computer test of conservation to overcome the major limitations of clinical conservation testing. An important approach to the study of conservation focuses on conservation strategies (also rules and operations). At present, the best methodology of assessing conservation and several related abilities has been introduced by Siegler (1981, 1986). He uses several item types to distinguish between four rules. These rules roughly coincide with Piaget's phases in conservation development. Rule 1: Users focus on the dominant dimension (height). Rule 2: Users follow rule 1 except when values of the dominant dimensions are equal, then they base their judgment on the subordinate dimension (width). Rule 3: Users do as rule 2 users except when both the values of the dominant and the subordinate dimensions differ, then they guess. Rule 4: Users multiply the dimensional values (in the case of the balance beam) and/or understand that the transformation does not affect the quantity of liquid. Multiplying dimensions (or compensation) is

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correct on conservation tasks too, but more difficult and rarely used (Siegler, 1981). Siegler's methodology has been criticized mainly for rule 3. Guessing (muddling through) is a rather general strategy of another order. Undoubtedly, it can occur, and should be controlled for. Several alternative strategies have been proposed, for instance, addition of dimensions (Anderson & Cuneo, 1978), maximization (Kerkman & Wright, 1988), and the buggy rule (van Maanen, Been, & Sijtsma, 1989). The validation of these alternatives will take a lot of test items but is not impossible. Another important concept in the study of conservation is that of the cognitive conflict. Piaget himself associated the concept of cognitive conflict to conservation. Most authors agree that conservation performance on Piagetian conservation tasks should be explained as (or can be described as) a conflict between perceptual and cognitive factors. Concepts such as salience, field dependency, and perceptual misleadingness have been used to define the perceptual factor in conservation (Bruner, 1967; Odom, 1972; Pascual-Leone, 1989). Siegler applies it in the dominant/subordinate distinction. We will use this factor as an independent variable in our model. The cognitive factor is less clearly understood. It can be defined as cognitive capacity, short-term memory, or as a more specific cognitive process such as learning about quantities. Also, the very general concept of maturation can be used. In cusp models, one has to choose two independent variables. In the case of conservation, this very important choice is difficult, which is one of the reasons to communicate our cusp model as preliminary. Flavell and Wohlwill (1969) took Piaget's view on the acquisition of conservation as a starting point. Piaget distinguished between four phases: nonconservation, pretransitional, transitional, and conservation. Flavell and Wohlwill (1969) formulated these Piagetian phases in the form of a simple equation: P( + ) = Pa * Pb l-k,

(4)

where P ( + ) is the probability that a child given person parameters Pa and k and item parameter Pb succeeds on a particular task. Pa is the probability that the relevant operation is functional in a given child. Pb is a difficulty coefficient between 0 and 1. The parameter k is the weight to be attached to the Pb factor in a given child. In the first phase, Pa and k are supposed to be 0, and consequently P ( + ) is 0. In the second phase, Pa changes from 0 to 1, whereas k remains 0. For a task of intermediate difficulty, (Pb = .5) P ( + ) = .25. According to Piaget, and Flavell and Wohlwill add this to their model, the child should manifest oscillations and intermediary forms of reasoning. Notice that this does not follow from the equation. The third phase is a period of stabilization and consolidation, Pa is 1 and k increases. In the fourth phase, Pa and k are 1 and full conservation has been reached.

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Of course, this model also has its disadvantages. It predicts a kind of growth curve that is typical for many models and it predicts little besides what it assumes. In addition, the empirical verification suffers from the lack of operationalization of the parameters. As Flavell and Wohlwill admit, task analysis will not easily lead to an estimation of Pb, Pa is a nonmeasurable competence factor, and k does not have a clear psychological meaning and therefore also cannot be assessed. Yet, the reader will notice that this 25-year-old model has a close relationship to modern latent trait models. The simplest one, the Rasch model, can be denoted as P ( + ) = 1/(1 +

e-(Pa-Pb)).

(5)

Pa is the latent trait and Pb is the item difficulty. Several modifications are known. Parameters for discrimination, guessing, and so on, can be added. Pa and Pb have in both models a similar meaning. In latent trait models, one cannot find a direct equivalent for k, although some latent trait models contain person-specific weights for item parameters. If we compare Equations (4) and (5), we can conclude that Equation (4) is a kind of early latent trait model. The theory of latent trait models has developed to a dominant approach in current psychometrics. Much is known of the statistical properties of these models. The most appealing advantage is that item parameters can be estimated independent of the sample of subjects and person parameters independent of the sample of items. Besides assumptions as local independence, some constraints on the item-specific curves are required. For instance, in the Rasch model, item curves have equal discriminatory power. More complex latent trait models do not demand this, but other constraints remain. The important constraint for the discussion here is that in all latent trait models, for fixed values of Pa and Pb, one and only one value of P ( + ) is predicted. The item characteristic curve suggested by catastrophe theory does not accommodate this last constraint. Hence, we can conclude that the behavior in the Flavell and Wohlwill model, similar to more advanced latent trait models (e.g., Saltus, proposed by Wilson, 1989), does not vary discontinuously. The behavior may show a continuous acceleration in the increase of P(+), but is not saltatory in a strict sense. A second important conclusion is that the model of Flavell and Wohlwill, contrary to their suggestion, does not predict the oscillations and intermediary forms of reasoning in the transitional phases. We will see that the cusp model explicitly predicts these phenomena. This latter point is also of importance with regard to Markov models of Brainerd (1979) and the transition model of Pascual-Leone (1970). In Molenaar (1986) it is explained why such models do not test discontinuities. Of course, many more models on conservation are proposed, often in a nonmathematical fashion; but as far as they concern discontinuity explicitly, it seems that they will be in

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a c c o r d a n c e with the c a t a s t r o p h e interpretation or they will not, that is, b e l o n g to the class o f m o d e l s d i s c u s s e d in this chapter.

4. THE CUSP MODEL In this section, w e explain the cusp m o d e l in general and its specification as a m o d e l o f c o n s e r v a t i o n . T h e type o f cusp m o d e l w e apply, typical by a rotation o f axes, is also f o u n d in Z e e m a n (1976). It is especially useful if the d y n a m i c s are u n d e r s t o o d in terms o f a conflict. In our application o f the cusp m o d e l , w e use p e r c e p t u a l m i s l e a d i n g n e s s and cognitive capacity as i n d e p e n dent variables and c o n s e r v a t i o n as a d e p e n d e n t variable. T h e m o d e l is r e p r e s e n t e d in F i g u r e 2.

FIGURE 2. The cusp model of conservation acquisition holds two independent and one behavioral variable: perceptual misleadingness, cognitive capacity, and conservation, respectively. The cusp surface is shown above the plane defined by the independent variables. The cusp surface has a folded form at the front that vanishes at the back of the surface. The folding implies that for certain values of the independent variables, three values of the behavioral variable, that is, three modes, are predicted. The mode in the middle is unstable and compelling and therefore called inaccessible. The remaining two modes lead to bimodality. The area in the control plane, defined by the independent variables, for which more behavioral modes exist is the bifurcation set. If the system is in its upper mode and moves from right to the left, a sudden jump will take place to the lower mode at the moment that the upper mode disappears. If the system then moves to the right, another jump to the upper mode will occur when leaving the bifurcation set at the right. This change in position of jumps is hysteresis. The placement of groups R, NC, TR, and C is explained in the text. Van der Maas and Molenaar (1992). Copyright 1992 by the American Psychological Association. Adapted by the permission of the publisher.

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A three-dimensional surface consisting of the solutions of Equation (2) represents the behavior for different values of the independent variables. The independent variables define the control plane beneath the surface. The bifurcation set consists of independent values for which three models of behavior exists. Two of the three models are stable, the one in the middle is unstable. Outside the bifurcation set, only one mode exists. Changes in independent variables lead to continuous changes in the behavior variable, except for places on the edges of the bifurcation set; there, where the upper or lower mode of behavior disappears, a sudden jump takes place to the other stable mode. In terms of conservation, we interpret this to mean that an increase in cognitive capacity, or a decrease in perceptual misleadingness, leads to an increase of conservation, in most cases continuously and sometimes saltatory. The reverse process is also possible, but is usually not the dominant direction in development. However, artificial manipulation of cognitive capacity (by shared attention tasks) or misleadingness (by stimulus manipulation) should lead to saltatory regressions. This cusp model suggests the discrimination of four groups of children. At the neutral point, the back of the surface, the probability of a correct response equals chance level. Both independent values have low values. At the left front, the scores are below chance level. Perceptual misleadingness is high, capacity is low. In the middle, both independent values are high. Two behavior modes are possible, above and below chance level. Finally, at the right front, the high scores occur. Here, perceptual misleadingness is low and capacity is high. Associated with these states are four groups which we call residual group (R), nonconservers (NC), transitional group (TR), and conservers (C), respectively. The residual group consists of children who guess or follow irrelevant strategies because of a lack of understanding of test instructions or a lack of interest. This group is normally not incorporated into models of conservation development. In this model, it is expected that this group is characterized by low values for both independent variables. The nonconserver and conserver children are included in each model of conservation development. Most models also include a transitional group, defined, however, according to all kinds of criteria. In the cusp model, the transition period is defined as the period when the subjects stay in the bifurcation set. In this period, the possibility of the sudden jump is present. The sudden jump itself is part of this transition phase. The transition includes more than just the sudden jump. There are at least seven other behavioral phenomena typical for the transition period (see section 4.2.).

4.1. Routes Through Control Plane The classification of four groups suggests an expected sequence, but this does not directly follow from the cusp model. We need an additional assumption on

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the direction of change of independent variables as a function of age (or as a function of experimental manipulation). To get at a sequence of residual, nonconservation, transitional, and conservation, it should be assumed that at first, both factors are low, then, perceptual misleadingness increases, third, cognitive capacity increases too, and fourth, perceptual misleadingness decreases back to a low level. We discuss this assumption because from the literature we know that very young children indeed score at chance level, 4- to 6-year-olds score below chance level, and later on, they score above chance level (McShane & Morrison, 1983). The phases of the model of Flavell and Wohlwill (1969) coincide with the nonconservation, transitional, and conservation sequence. The phase in which very young children score at chance level is not included in their model nor in the model of Piaget. This is partly because of issues of measurement. Guessing is not helpful on the classical Piagetian test of conservation in which valid verbal arguments are required. Other test criteria (judgment-only criterion or measures of looking time and surprise) allow for false positive classifications, whereas Piaget's criterion suffers from false negatives. In the cusp model, we assume the use of the judgmentonly criterion of conservation. What should be clear now is that some assumption on the change of the independent variables is necessary to specify a sequence of behavior. Of course, this is also required in the model of Flavell and Wohlwill (1969).

4.2. Test of the Cusp Model: The Catastrophe Flags The transitional period, associated with the bifurcation set, is characterized by eight catastrophe flags derived from catastrophe theory by Gilmore (1981). Three of these flags can also occur outside the bifurcation set and may predict the transition. Together, they can be used to test the model. These catastrophe flags are sudden jump, bimodality, inaccessibility, hysteresis, divergence, anomalous variance, divergence of linear response, and critical slowing down. Some are well known in developmental research, others only intuitively or not at all. The sudden jump is the most obvious criterion. However, in the case of developmental research, quite problematic. Empirical verification of this criterion requires a dense time-series design. In spite of the many studies on conservation development, a statistically reliable demonstration of the sudden jump in conservation ability is lacking. In the following, we present data that demonstrate the sudden jump. Bimodality is also known in conservation research. In van der Maas and Molenaar (1992), we present a reanalysis of the data of Bentler (1970) which clearly demonstrates bimodality. Although this seems to be the most simple criterion, applicable to cross-sectional group data of the behavior variable only, some is-

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sues arise. In the prediction of bimodal score distribution, we combine two flags, bimodality and inaccessibility. Inaccessibility is implied by the unstable mode in between two stable modes. In a mixture distribution model (Everitt & Hand, 1981), this inaccessible mode misses. In these models, a mixture of two normal (or binomial, etc.) distributions is fitted on empirical distributions: F ( X ) = p N(Ixl, 0.1) + (1 - p) N ( I x 2 , 0"2) ,

(6)

where N is the normal distribution, p defines the proportions in the modes, and tx and 0. are the characteristics of the modes. This mixture model takes five parameters, whereas a mixture of binomials takes three parameters. The number of components can be varied and tested in a hierarchical procedure. An alternative formulation is found in the work of Cobb and Zacks (1985). They apply the cusp equation to define distributions of a stochastic cusp model: F ( X ) = ~. e -(z4-az2-bz)

Z-

sX-

l,

(7)

where a and b (independent values) define the form of the distribution, s and 1 linearly scale X, and h is the integration constant. The four parameters a, b, s, and 1 are estimated. It is possible to fit unimodal and bimodal distributions by constraints on the parameters a and b and compare the fits. The method of Cobb has some limitations (see Molenaar, this volume), and it is difficult to fit to data (e.g., in computing h). Yet, it takes into account the impact of the inaccessible mode and the computational problems are solvable. A comparison of possible forms of Equations (6) and (7) leads to the conclusion that their relationship is rather complex. In Equation (6), distributions can be fitted to data defined by IX~ = IX2 and 0.1 > > 0.2. Such distributions are not allowed in Equation (7), the modes must be separated by the inaccessible mode. Hysteresis is easily demonstrated in simpler physical catastrophic processes, but is probably very difficult in psychological experimentation. The degree of the hysteresis effect, the distance between the jumps up and down, depends on the disturbance or noise in the system (leading to the so-called Maxwell condition). Later on we discuss our first attempt to detect hysteresis in conservation. Divergence has a close relation to what is usually called a bifurcation. In terms of the chosen independent variables it means that if, in the case of a residual child, perceptual misleadingness as well as cognitive capacity are increased, the paths split between the upper and lower sheets, that is, between the high and low scores. Two children with almost similar start values (both independent variables very low) and following the same path through the control plane can show strongly diverging behavior. Again, this is not easily found in an experiment. The manipulation of only one independent variable is already difficult. Another choice of independent variables (e.g., a motivation factor or

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optimal conditions factor in another rotation of independent variables) may lead to an empirical test. Anomalous variance is a very important flag. Gilmore (1981) proves that the variance of the behavioral variable increases strongly in the neighborhood of the bifurcation set and that drops occur in the correlation structure of various behavioral measures in this period. In developmental literature, many kinds of anomalous behaviors, from oscillations to rare intermediate strategies, are known. In our view, this prediction, used here as a criterion, is what Flavell and Wohlwill call "manifestations of oscillations and intermediary forms of reasoning." Divergence of linear response and critical slowing down, the last flags, concern the reaction to perturbations. They imply the occurrence of large oscillations and a delayed recovery of equilibrium behavior, respectively. These flags are not studied in our experiments. Reaction time measures are possibly the solution for testing these two flags. The flags differ importantly in strength. Bimodality and the sudden jump, certainly when they are tested by regression analysis and mixture distributions, are not unique for the cusp model. It is very difficult to differentiate acceleration and cusp models by these flags. Inaccessibility changes this statement somewhat. It is unique, but as indicated before, usually incorporated in the bimodality test. Anomalous variance is unique, as other mathematical models do not derive this prediction from the model itself. In this respect, the many instances of oscillations and intermediary forms of reasoning count as evidence for our model. Hysteresis and divergence are certainly unique for catastrophe models. Divergence of linear response and critical slowing down seem to be unique, too. Such a rough description of the strength and importance of each flag is only valid in contrast with predictions of alternative models. In the case of conservation research, bimodality and the sudden jump are thought of as necessary but not sufficient criteria, the others as both necessary and sufficient. The general strategy should be to find as many flags as possible. The demonstration of the concurrence of all flags forms the most convincing argument for the discontinuity hypothesis. Yet, the flags constitute an indirect method for testing catastrophe models. However, the advantage is that, except for divergence and hysteresis, the flags only pertain to the behavioral variable. We suggested a perceptual and a cognitive factor in our cusp model of conservation acquisition, but the test of the majority of flags does not depend on this choice. These flags do answer the question whether a discontinuity, as restrictively defined by catastrophe theory, occurs or not. The definitions of hysteresis and divergence include variation along the independent variables. In the case of the cusp conservation model, we can use variation of perceptual cues of conservation items or divided attention tasks to

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force the occurrence of hysteresis and divergence. On the other hand, it is a pity that these appealing flags cannot be demonstrated without knowledge of the independent variables. On the other hand, by these flags, hypotheses about the independent variables can be tested.

4.3. Statistical Fit of Cusp Models A direct test of the cusp model of conservation acquisition requires a statistical fit of the cusp model to measurements of the dependent as well as the independent variables. The statistical fit of catastrophe models to empirical data is difficult and a developing area of research. In the comparison of Equations (2) and (3), the cusp models and (nonlinear) regression models, we did not discuss the statistical fit. It is common practice to fit regression models with one dependent and two independent variables to data and to test whether the fit is sufficient. Fitting data to the cusp model is much more problematic. There have been two attempts to fit Equation (2) in terms of Equation (3) (Guastello, 1988; Oliva, Desarbo, Day, & Jedidi, 1987). Alexander, Herbert, DeShon, and Hanges (1992) heavily criticize the method of Guastello. The main problem of Guastello's method puts forth the odd characteristic that the fit increases when the measurement error increases. Gemcat by Oliva et al. (1987) does not have this problem but needs an additional penalty function for the inaccessible mode. A different approach is taken by Cobb and Zacks (Cobb & Zacks, 1985). They developed stochastic catastrophe functions which can be fitted to data. In Equation (7), the basic function is shown. If one fits a cusp model, a and b are, for example, linear functions of observed independent variables. Examples of applications of Cobb' s method can be found in Ta' eed, Ta' eed, and Wright (1988) and in Stewart and Peregoy (1983). We are currently modifying and testing Cobb's method. Simulation studies should reveal the characteristics of parameter estimates and test statistics, and the requirements on the data. At present, we have not fit data of measurements of conservation, perceptual misleadingness, and cognitive capacity directly to the cusp model of conservation acquisition.

5. EMPIRICAL STUDIES 5.1. Introduction In the experiments, we have concentrated on the catastrophe flags. Evidence for bimodality, inaccessibility, and anomalous variance can be found in the literature (van der Maas & Molenaar, 1992). This evidence comes from

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predominantly cross-sectional experiments. Analyses of time series from longitudinal experiments have been rarely applied. The focus on cross-sectional research hampers the assessment of the other flags, most importantly, the sudden jump itself. We do not know of any study that demonstrates the sudden jump (or sharp continuous acceleration) convincingly. This is, of course, a difficult task, as dense time series are required. The presented evidence for anomalous variance is also achieved by crosssectional studies. In such studies, conservation scores on some test are collected for a sample of children in the appropriate age range. Children with consistent low and high scores are classified as nonconservers and conservers, respectively. The remaining children are classified as transitional. Then, some measure of anomalous variance is applied (rare strategies, inconsistent responses to a second set of items, inconsistencies in verbal/nonverbal behavior). By an analysis of variance (or better, by a nonparametric equivalent) it is decided whether the transitional group shows anomalous variance more than the other groups. In comparison with the strong demands, in the form of the catastrophe flags, that we put on the detection of a transition, this procedure of detecting transitional subjects is rather imprecise. However, it is a very common procedure in conservation research, and is also applied to decide whether training exclusively benefits transitional subjects. In light of the importance of anomalous variance for our verification, we want a more severe test. For these reasons, a longitudinal experiment is required. Special requirements are dense time series and precise operationalizations for the flags. This appears to be a difficult task when we rely on the clinical test procedure of conservation. Many retests in a short period creates a heavy load on our resources, as well as on the time of the children and the continuation of normal school activities. Moreover, this clinical test procedure has been heavily criticized. We chose to construct a new test of conservation. We describe this test and its statistical properties elsewhere (van der Maas, 1995), therefore only a short summary will be given here.

5.2. Instrument: A Computer Test of Conservation 5.2.1. Items In the clinical test procedure, the child is interviewed by an experimenter who shows the pouring of liquids to the child and asks a verbal explanation for the judgment. Only a few tasks are used and the verbal justification is crucial in scoring the response and detecting the strategy that is applied by the child. The rule assessment methodology of Siegler (1981), discussed previously, has a different approach. Siegler applies many more items that are designed to detect strategies on the basis of the judgments only. Although verbal justifications have an additional value, the level of conservation performance is determined by simple responses to the items only.

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The computer test is based on four out of six item types of Siegler's methodology. We call them the guess equality, guess inequality, standard equality, and standard inequality item types. The guess equality and guess inequality item types compare with Siegler's dominant and equal items. In the guess item types, the dimensions of the glasses are equal. In the equality variant amounts, heights and widths of the liquid are equal before and after the transformation. In the inequality variant, one of the glasses has less liquid, is equal in width, but differs in height. In these item types, the perceptual height cue points to the correct response and should therefore be correctly solved by all children. Consequently, these items can be used to detect children who apply guessing or irrelevant strategies. In view of the criticism of the judgment-only criterion concerning the possibility of guessing, items like this should be included. The standard item types, equality and inequality, compare with the conflict equal and subordinate items of Siegler. The standard equality item is shown in Figure 1. The dimensions of the glasses and liquids differ, whereas the amounts are equal. In the initial situation, the dimensions are clearly equal. Understanding the conserving transformation is sufficient to understand that the amounts are equal in the final situation, although multiplication of dimensions in the final situations suffice as well. The standard inequality item starts with an initial situation in which widths are equal but heights and amounts differ. In the final situation, the liquid of one of the glasses is poured into a glass of such width that the heights of the liquid columns are then exactly equal (see Fig. 3). 5.2.2. Strategies For all of these item types, a large variation in dimensions is allowed. For example, in the item in Figure 3, the differences in width can be made more or less salient. We expect that these variations do not alter the classification of children. Siegler makes the same assumption. A close examination of the results of Ferretti and Butterfield (1986) show that only very large differences in dimensional values have a positive effect on classification. Siegler uses six item types (each four items) to distinguish between four rules. We use instead the two standard item types (Figs. 1 and 3) and classify according to the following strategy classification schema (Table 2).

FIGURE 3. A standard inequality item of conservation of liquid.

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The conservation items have three answer alternatives: left more, equal, and right more. These three alternatives can be interpreted as the responses highest more, equal, and widest more in the case of equality items, and in the responses equal, smallest more, and widest more in the case of inequality items. The combination of responses on equality and inequality items is interpreted in terms of a strategy. Children who consistently chose highest more on equality items and equal on inequality items follow the nonconserver height rule (NC.h), or, in Siegler's terms, rule 1. Conservers (C) or rule 4 users chose the correct response on both equality and inequality items. Some children will fail the equality items but succeed on the inequality items (widest more). If they prefer the highest more response on the equality items, they are called NC.p, or rule 2 users in Siegler's terms. If they chose the widest more response, they probably focus on width instead of height. The dominant and subordinate dimensions are exchanged (i.e., from a to b in the list of events in conservation development). We call this the nonconserver width rule (NC.w). The nonconserver equal rule ( N C . = ) refers to the possibility that children prefer the equal response to all items, including the inequality items. We initially interpret this rule as an overgeneralization strategy. Children, discovering that the amounts are equal on the equality items, may generalize this to inequality items. If this interpretation is correct, N C . - m a y be a typical transitional strategy. The cell denoted by i l does not have a clear interpretation. Why should a child focus on width in the case of equality items, and on the height in the case of inequality items. This combination makes no sense to us. The same is true for the second row of Table 2, determined by the smallest more response on inequality items. This response should not occur at all, so this row is expected to be empty. The results of the experiments will show how much children apply the strategies related to the nine cells. If a response pattern has an equal distance to two cells, then it is classified as a tie. The number of ties should be small. Notice that ties can only occur when more than one equality and more than one inequality item is applied. The possibility of ties implies a possibility of a formal test of this strategy classification procedure.

TABLE2 Strategy Classification Schema Standard equality items Highest more

Equal

Widest more

Standard

Equal

NC.h/rule 1

NC. =

i1

Inequality

Smallest more

i2

i3

i4

Items

Widest more

NC.p/rule 2

C/rule 4

NC.w

4. Catastrophe Theory

93

The NC.=, NC.w, and the i l rules do not occur in Siegler's classification of rules, although they could be directly assessed in Siegler's methodology. Rule 3 of Siegler, muddling through or guessing on conflict items, does not occur in this classification schema. Proposed alternatives for rule 3, like addition or maximization, are also not included. Additional item types are required to detect these strategies. 5.2.3. Test Items are shown on a computer screen. Three marked keys on the keyboard are used for responding. They represent this side more (i.e., left), equal, that side more (right). First, the initial situation appears, two glasses filled with liquid are shown. The subject has to judge this initial situation first. Then, an arrow denotes the pouring of liquid into a third glass which appears on the screen at the same time as the arrow; the liquid disappears from the second glass and appears in the third. Small arrows below the glasses point to the glasses that should be compared (A and C). In the practice phase of the computer test, the pouring is, except by the arrow, also indicated by the sound of pouring liquid. This schematic presentation of conservation items is easily understood by the subjects. The application of the strategy classification schema is more reliable when more items are used. We chose to apply one guess equality, one guess inequality, three standard equality, and three standard inequality items in the longitudinal experiment, and two guess equality, two guess inequality, four standard equality, and four standard inequality items in the cross-sectional studies. Guessers are detected by responses to guess items and the initial situations of the standard items, which are all non-misleading situations. Two types of data are obtained, number correct and strategies. Apart from the preceding test items, four other items are applied. We call these items compensation construction items. On the computer screen, two glasses are shown, one filled and one empty. The child is asked to fill the empty glass until the amounts of the liquids are equal. The subject can both increase and decrease the amount until he or she is satisfied with the result. Notice that a conserving transformation does not take place, hence the correct amount can only be achieved by compensation. We will only refer briefly to this additional part of the computer test. It is, however, interesting from a methodological point of view as the scoring of these items is not discrete but continuous, that is, on a ratio scale.

5.3. Experiment 1: Reliability and Validity The first experiment is a cross-sectional study in which 94 subjects ranging in age from 6 to 11 years old participated. We administered both the clinical conservation test (Goldschmid & Bentler, 1968) and an extended computer test.

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H. L. J. van der Maas and P. C. M. Molenaar

The reliability of the conservation items of the computer test turned out to be .88. Four of 94 subjects did not pass the guess criteria; they failed the clinical conservation test as well. This clinical test consisted of a standard equality and a standard inequality item, scored according to three criteria: Piagetian, Goldschmid and Bentler, and judgment-only. The classifications obtained by these three criteria correlate above .96, hence to ease presentation, we only apply the Piagetian criterion. The correlation between the numbers correct of the clinical test and the computer test is .75. In terms of classifications, 79 of 94 subjects are classified concordantly as nonconservers and conservers. A more concise summary is given in Table 3. Here we can see that the classifications in NC.h and C do not differ importantly. For the NC.p, N C . = , and NC.b strategies, no definite statements can be made because of the small number of subjects. It seems that the ties appear among conservers and nonconservers on the clinical test. The percentages correct on both tests show that the tests do not differ in difficulty (62% vs. 59%, paired t-test: t = .98, df = 93, p = .33). We can state that the computer test seems to be reliable and valid (if the clinical test is taken as criterion).

5.4. Experiment 2: Longitudinal Investigation of the Sudden Jump and Anomalous Variance In the second study, 101 subjects from four classes of one school participated. At the beginning of the experiment, the ages varied between 6.2 and 10.6 years old. The four classes are parallel classes containing children of age groups 6, 7, and 8 years.

TABLE3 Classifications on the Computer Test Versus the Clinical Test Consisting of an Equality and an Inequality Item Clinical test scores

Strategy on c o m p u t e r test

=

:/:

NC.h

NC.p

NC.=

0

0

1

0

19

3

0

1

0

0

0

1

2

2

0

1

1

2

1

2

C

NC.b

Ties

1

0

4

3

30

2

0

0

1

4

3

0

0

0

7

40

1

7

0

53

Guess

Total

Note. Four score patterns of correct (1) and incorrect (0) can occur on the clinical test. These are compared with the strategies found on the computer test (strategies i l to i4 did not occur at all). The cells contain the number of subjects.

4. Catastrophe Theory

g5

We placed a computer in each class and individually trained the children to use it. During 7 months, 11 sessions took place. Except for the first session, children made the test by themselves as part of their normal individual education. This method of closely following subjects is similar to what Siegler and Jenkins (1989) call the microgenetic method, except for the verbal statements on responses which they use as additional information. We can call our method a computerized microgenetic method. Its main advantage is that it takes less effort, because, ideally, the investigator has only to backup computer diskettes. 5.4.1. Sudden jump To find evidence for the sudden jump, we classified subjects in four groups, nonconservers, transitional, conservers, and residual on the basis of the strategies applied. The transitional subjects are subjects who applied both conserver and nonconserver strategies during the experiment. Twenty-four of the 101 subjects show a sharp increase in the use of conserver strategies. We corrected the time series for latency of transition points. The resuiting individual plots are shown in Figure 4. This figure demonstrates a very sharp increase in the conservation score. To judge how sharp, we applied a multiple regression analysis in which the conservation score serves as the dependent variable and the session (linear) and a binary template of a jump as independent variables. This latter jump indicator consists of zeros for sessions 1 to 10 and ones for sessions 11 to 19. Together, the independent variables explain 88% of the variance (F(2,183) = 666.6, p = .0001). The t-values associated with the beta coefficient of each independent variable is t = 1.565 (p - .12) and t = 20.25 (p = .0001), for the linear and jump indicators, respectively. Actually, the jump indicator explains more variance than a sixth-order polynomial of the session variable. Yet, this statistical result does not prove that this sharp increase is catastrophic. The data can also be explained by a continuous acceleration model, as the density of sessions over time may be insufficient. What this plot does prove is that this large increase in conservation level takes place within 3 weeks, between two sessions, for 24 of 31 potentially transitional subjects. For the remaining 70 subjects, classified as nonconservers, conservers, and residual subjects on the basis of strategy use, no significant increases in scores are found. The mean scores on sessions 1 and 11 for these subjects do not differ significantly, F(1,114)= .008, p = .92. The scores of these subjects stay at a constant level. The transitional group shows important individual differences. A few subjects show regressions to the nonconserver responses, some apply rare strategies during some sessions, but the majority show a sharp increase between sessions. One subject jumped within one test session, showing consistent nonconserver scores on all preceding sessions and conserver scores on all subsequent test sessions.

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4. Catastrophe Theory

97

5.4.2. Anomalous variance A special problem in the analysis of anomalous variance is that the conservation test consists of dichotomous items. The variance depends on the test score. We looked for three solutions for this problem using inconsistency, alternations, and transitional strategies. The inconsistency measure is achieved by an addition to the computer test, a repetition of four standard items at the end of the test. These additional items are not included in the test score. The inconsistency measure is the number of responses that differ from the responses on the first presentation. We do not present the results here. In summary, inconsistencies do occur in the responses of transitional subjects but occur more in the responses of residual subjects. The other two measures concern the application of strategies. The analysis of alternations did not indicate a transitional characteristic. The analysis of strategies, however, did. To explain this, we will now describe the results in terms of strategies. The other measures are discussed in van der Maas, Walma van der Molen, and Molenaar (1995). The responses to six standard items are analyzed by the strategy classification schema. The results are shown in Table 4. This table shows some important things. The number of ties is low, hence in the large majority of patterns, the classification schema applies well. The NC.h and the C strategy are dominant (80%). The other cells are almost empty or concern uncommon strategies. NC.p, rule 2 in Siegler's classification, N C . - , il, and NC.w make up 14% of the response patterns. Focusing on the subordinate dimension, NC.w is very rare. A statistical test should reveal whether the small number, 6 if guessers are removed,

TABLE 4 Responsesto Six Standard Items Analyzed by Strategy Classification Schema Equality items Ties 43 (8) Missing 265

Inequality items

Highest more Equal

NC.h

Smallest more

i2

Widest more

NC.p

417 (15)

Equal NC.=

Widest more

36 (9)

il

20 (5)

i4

2 (2)

4 (0)

i3

5 (3)

50 (11)

C

258 (4)

NC.w

11 (5)

Note. For 11 sessions, *101 subjects = 1111 response patterns can be classified. There were 265

missing patterns. Forty-three response patterns, ties, could not be classified because of equal distance to the ideal patterns associated with the nine cells. The 801 remaining patterns can be uniquely classified and are distributed as shown in the table. The number of guessers is displayed in parentheses. A response pattern is classified as a guess strategy if two guess items are incorrect or if one guess item and more than 25% of the responses to the non-misleading initial situations of the standard items are incorrect.

g8

H. L. J. van der Maas and P. C. M. Molenaar

can be ascribed to chance or cannot be neglected. Latent class analysis may be of help here. For the classification of time series, that is, the response patterns over 11 sessions, we make use of the translation of raw scores in strategies. The classification is rather simple. If NC.h occurs at least once, and C does not occur, the series is classified as NC. If C occurs at least once, and NC.h does not occur, the series is classified as C. If both C and NC.h occurs in the time series it is classified as TR. The remaining subjects and those who apply guess and irrelevant (i2, i3, and i4) strategies on the majority of sessions are classified as R. According to these criteria, there are 31 TR, 42 NC, 20 C, and 8 R subjects. Note that this classification of time series does not depend on the use of the uncommon strategies NC.p, N C . = , i l, and NC.w. Thus we can look at the frequency of use of these strategies in the four groups of subjects (see Table 5). NC.p, rule 2 in Siegler's classifications, does not seem to be a transitional characteristic; in most cases, it is used by the subjects of the nonconserver group. The strategies i l and NC.w are too rare to justify any conclusion. In 15 of 27 cases of application of N C . = , it is applied by the transitional subjects. The null hypothesis that N C . = is distributed evenly over the four groups is rejected (X2 = 11.3, d f = 3, p = .01). Furthermore, seven transitional subjects apply N C . = just before they start using the C strategy. This analysis demonstrates much more convincingly than cross-sectional studies that transitional subjects manifest what has been called intermediary forms of reasoning. Whether we can ascribe the occurrence of the NC.-- strategy to anomalous variance is not entirely clear.

TABLE 5 Application of Uncommon Strategies by Nonconservers, Conservers, Transitional, and Residual Subjects Group Strategy

NC

C

TR

R

NC.p NC.w NC.= il Other

21 (5.5%) 1 (0.3%) 5 (1.3%) 5 (1.3%) 349 (91.6%)

4 (2.5%) 0 (0%) 4 (2.5%) 0 (0%) 149 (95%)

7 (2.8%) 3 (1.2%) 15 (6.0%) 7 (2.8%) 217 (87%)

7 (11.8%) 2 (3.4%) 3 (5.1%) 3 (5.1%) 44 (74.6%)

Total

381 (100%)

157 (100%)

249 (100%)

59 (100%)

Note. Other strategies are C, NC.h, guess, ties, and irrelevant strategies (i2 to i4). Raw numbers and percentages of use in each group are displayed.

4. Catastrophe Theory

gg

Note that the search for anomalous strategies comes from the fact that traditional variance is not useful in a binomially distributed test score. In this respect, the compensation construction items are of interest. These measure conservation (in fact, compensation) ability on a continuous ratio scale (amount of liquid or height of liquid level). In this measure, a more direct test of anomalous variance may be possible.

5.5. Experiment 3: Hysteresis The last flag that we will discuss here is hysteresis. We build the cusp model of conservation acquisition on the assumption that perceptual and cognitive factors conflict with each other. Hysteresis should occur under systematic increasing and decreasing variation in these factors. Preece (1980) proposed a cusp model that is a close cousin of ours. Perceptual misleadingness plays a role in his model, too. We will not explain Preece' model here, but it differs from our preliminary model, for example, with respect to the behavior modes to which the system jumps. The difference can be described in terms of the conservation events (see section 3). Preece's model concerns event c, whereas our model concerns event d, resulting in a choice between the height more and weight more responses and between the height more and equal responses, respectively. So Preece expects a hysteresis effect between incorrect responses and we between correct and incorrect responses. The idea suggested by Preece is to vary the misleading cue in conservation tasks. In a conservation of weight test, a ball of clay is rolled into sausages of lengths 10, 20, 40, 80, 40, 20, and 10 cm, respectively (see Table 6). In this test, a child shows hysteresis if he or she changes the judgment twice; once during the increasing sequence and once during the decreasing sequence. If both jumps take place, hysteresis occurs either according to the delay (Hyst D) or to Maxwell convention (Hyst M). In the latter case, the jumps take place on the same position, for instance, between 20 and 40 cm. If only one jump occurs, we classify the pattern as "jump." This weight test was performed as a clinical observation test and was administered to 65 children. Forty-three subjects correctly solved all items. Two subjects judged consistently that the standard ball was heavier, and two judged the sausage as heavier. In Table 6 the response patterns of all subjects are shown, the consistent subjects are in the lower part. The codes - 1 , 0, and 1 mean ball more, equal weight, and sausage more, respectively. The first subject shows a good example of a hysteresis response pattern. When the ball is rolled into a sausage of a length of 10 cm, he judges the ball as heavier. Then, the sausage of 10 cm is rolled into a length of 20 cm, the child persists in his opinion. But when the sausage of 20 cm is rolled into a sausage of 40 cm, he changes his judgment, now the roll is heavier. He continues to think this when

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H. L. J. van der Maas and P. C. M. Molenaar

TABLE 6 Response Patterns of 65 Children in a Cross-Sectional Experiment on Hysteresis 10

20

40

80

40

20

10

nr

cm

cm

cm

cm

cm

cm

cm

Pattern

Alternation

39

-1 -1 -1 -1 0 0 0 -1 0 0 0 1 1 0 1 -1 1 1 1 0 0 0

-1

1

-1 -1

1 1

0 0 1 1 -1 0 0 1 -1 0 1 0 0 0

1 -1 0 0 0 1 0 0 1 -1 1 0 0 -1 -1 0 0 1 -1 1

hyst D hyst D' jump hyst M hyst D' hyst D hyst D' jump hyst D' hyst D' jump

inc/inc inc/inc inc/inc inc/inc c/inc c/inc c/inc c/inc c/inc c/inc c/inc

- 1 1 0

- 1 1 0

- 1 1 0

63 11 51 42 47 18

3 21 40 49 1 62 64 65 25 32 27

1 1

1

-1

-1

1 1 -1

1 -1

1

1

0 0 -1 -1 -1 -1 0 -1 -1 0 0 -1 1 -1 - 1 -1 0 1

-1 0 -1 -1 -1 -1 1 -1 -1 -1 0 1 1 0 1 -1 0 -1

-1 -1 -1 -1 -1 -1 1 1 1 0 0 1 0 -1 1 -1 0 0

1 1 0 -1 -1 -1 -1 -1 1 -1 1 0 0 0 -1 1 1 0 1 1

- 1 1 0

- 1 1 0

- 1 1 0

- 1 1 0

1

-1

0 0 0 -1

2 Subjects 2 Subjects 43 Subjects

Note. Explanation of abbreviations is given in the text.

the sausage reaches its maximum length of 80 cm. Then the sequence is reversed. The roll of 80 cm is folded to 40 cm, the child judges that the roll is still heavier. This continues until the roll is folded to 10 cm. On this last item the child changes his opinion, the ball is heavier again. The first subject is tested two times. The second time, hysteresis according to the delay convention again takes place. However, the jumps have exchanged relative position. The jump takes place earlier in the increasing rather than in the decreasing sequence of items. We denoted this strange phenomena as Hyst D'. This subject shows hysteresis between incorrect responses (inc/inc); subject 42 shows hysteresis between correct and incorrect responses (c/inc). This subject judges the ball and the sausage to be equal until the sausage is rolled into a

4. CatastropheTheory

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length of 80 cm. Then the ball is heavier. The child persists in this judgment when the sausage is folded to 40 cm. Finally, in the last two items, the child responses are "equal" again. Eleven instances of hysteresis and jumps occur among the tested children. The other patterns are not interpretable in these terms. The hysteresis results on the conservation of weight test are not conclusive. It would be naive to expect a convincing demonstration of hysteresis in the first attempt. The number of items in the sequence and the size of steps between items are difficult to choose. Perhaps, another kind of test is needed to vary misleadingness continuously. Furthermore, only the transitional subjects in the sample are expected to show these phenomena. In this regard, the results in Table 6 are promising. We also constructed a computer test of liquid for hysteresis. Only 2 instances of the jump pattern and four of the hysteresis pattern occurred in the responses of 80 subjects (from the longitudinal sample). Three of these subjects came from the transitional group. One of them responded as nonconserver until we applied the hysteresis test (between sessions 7 and 8) and responded as conserver in all subsequent sessions. These tests of hysteresis have some relationship to a typical aspect of Piaget's clinical test procedure of conservation. This aspect is called resistance to countersuggestion, which is not applied in the computer test of conservation. It means that, after the child has made his or her judgment and justification of this judgment, the experimenter suggests the opposite judgment and justification to the child. If the child does not resist this countersuggestion, he or she is classified as transitional. That countersuggestions work has been shown in many studies. Whether subjects who accept countersuggestion are transitional (or socially adaptive) is not always clear; as said before, cross-sectional studies have serious limitations in making this decisive. The method of countersuggestion can be interpreted in the cusp model as pushing the behavior in the new or old mode of behavior to see whether this mode still or already exists and has some stability. If countersuggestion is resisted, the second mode apparently does not exist. In the cusp model, the second mode only exists in the bifurcation set. Note that we defined the transition as being the period during which the system remains in this set. If the subject switches opinions there and back as a result of repeated countersuggestions, this behavior can be taken to be hysteresis. Whether the manipulation of countersuggestion yields an independent variable or not is less clear.

6. DISCUSSION In this chapter, we gave an overview of our work on the application of catastrophe theory to the problem of conservation development. Not all of the ideas

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H. L. J. van der Maas and P. C. M. Molenaar

or all of the empirical results are discussed, but the most important outcomes are presented. What can be learned from this investigation? One may think that catastrophe theory is a rather complex and extensive tool for something so simple as conservation learning. Some readers may hold the saltatory development of conservation as evident in itself (see Zahler and Sussmann, 1977). With respect to the first criticism, we state that there are theoretical and empirical reasons for using catastrophe theory. Theoretically, the reason for applying catastrophe theory is that the rather vague concept of epigenese in Piaget's theory can be understood as equivalent to self-organization in nonlinear system theory. Empirically, the lack of criteria for testing saltatory development, implies in the Piagetian theory, has been an important reason for the loss of interest in this important paradigm (van der Maas & Molenaar, 1992). The second criticism centers on the argument that our conservation items are dichotomous items, so behavior is necessarily discrete. The criticism is not in accordance with test theory. A set of dichotomous items yield, in principle, a continuous scale. On other tests in which a set of dichotomous items of the same difficulty are used, the sum score is unimodally distributed. In contrast, the sum scores on the conservation test are bimodally distributed. Yet, the use of dichotomous items complicates the interpretation of results. It raises a question about the relationship between catastrophe models and standard scale techniques, especially latent trait theory. This is a subject that needs considerable study. We suggested the use of compensation construction items to exclude radically these complications. This criticism of intrinsic discreteness is not commonly used by developmental psychologists. The traditional rejection of Piaget's theory assumes the opposite hypothesis of continuous development. With regard to this position the presented data are implicative. First, the sudden jump in the responses of a quarter of our longitudinal sample has not been established before. It is disputable whether this observation excludes alternative models that predict sharp continuous accelerations. We can argue that at least one subject showed the jump within a test session during a pause of approximately 30 s between two parts of the computer test. Hence, we demonstrated an immediate sudden jump for at least one subject. Second, evidence is presented for the occurrence of transitional strategies. Note that we, as this concerns a result in a longitudinal study, are much more certain that the subjects classified as transitional are indeed transitional. Transitional subjects appear to respond "equal" to both equality and inequality items. This may be interpreted to mean that they overgeneralize their discovery of the solution of standard equality items to the inequality items. Verbal justifications are required to decide whether this interpretation is correct. In our view the dem-

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103

onstration of a typical transitional strategy is quite important to Piaget's idea of knowledge construction, this result is confirmatory. It also makes an argument for using the rule assessment methodology in studying Piagetian concepts. We presented this result under the heading of anomalous variance. We relate the catastrophe flag anomalous variance to what Flavell and Wohlwill (1969) call oscillations and intermediary forms of reasoning. We suggest that the findings of Perry, Breckinridge Church, and Goldin Meadow (1988) concerning inconsistencies between nonverbal gestures and verbal justifications should be understood in terms of this flag, too. Third, the few instances of hysteresis response patterns suggest a choice for the cusp model of conservation. We admit that more experimentation is required here. The test should be improved and perhaps we need other designs and manipulations. We have already suggested using countersuggestion as a manipulation. Finally, evidence for other flags can be found in the literature, especially for bimodality and inaccessibility for which strong evidence already exists (Bentler, 1970; van der Maas, 1995). Divergence, divergence of linear response, and critical slowing down are not demonstrated. Although the presented results together make a rather strong argument for the hypothesis of discontinuous development, we will attempt to find evidence for these flags, too. The presented evidence for the catastrophe flags offers a strong argument for the discontinuity hypothesis concerning conservation development. That is, the data bear evidence for catastrophe models of conservation development. However, we cannot claim that this evidence directly applies to the specific cusp model that we presented. The choice of independent variables is not fully justified. The hysteresis experiment gives some evidence for perceptual misleadingness as independent variable, but this variable is, for instance, also used by Preece (1980). On the other hand, there are some theoretical problems for which the model of Preece (1980) can be rejected (see van der Maas & Molenaar, 1992). We did not put forward evidence for cognitive capacity as independent variable; in future studies, we will focus on this issue. However, it is in accordance with the views of many supporters and opponents of Piaget that conservation acquisition should be understood in terms of a conflict between perceptual and cognitive factors. We hope that we have made clear that a catastrophe approach to discontinuous behavior has fruitful implications. Catastrophe theory concerns qualitative (categorical) behavior of continuous variables. It suggests a complex relation between continuous and categorical variables that falls outside the scope of standard categorical models and data analysis methods. Yet, the catastrophe models are not unrelated to standard notions; but the question of how catastrophe models should be incorporated in standard techniques is far from answered.

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ACKNOWLEDGMENTS This research was supported by the Dutch organization for scientific research (NWO) and the Department of Psychology of the University of Amsterdam (UVA).

REFERENCES Alexander, R. A., Herbert, G. R., DeShon, R. E, & Hanges, E J. (1992). An examination of least squares regression modeling of catastrophe theory. Psychological Bulletin, 111(2), 366-374. Anderson, H. H., & Cuneo, D. O. (1978). The height + width rule in children's judgments of quantity. Journal of Experimental Psychology: General, 107(4), 335-378. Bentler, P. M. (1970). Evidence regarding stages in the development of conservation. Perceptual and Motor Skills, 31, 855-859. Brainerd, C. J. (1979). Markovian interpretations of conservation learning. Psychological Review, 86, 181-213. Bruner, J. S. (1967). On the conservation of liquids. In J. S. Bruner, R. R. Olver, & E M. Greenfield (Eds.), Studies in cognitive growth (pp. 183-207). New York: Wiley. Castrigiano, D. E L., & Hayes, S. A. (1993). Catastrophe theory. Reading, MA: Addison-Wesley. Cobb, L., & Zacks, S. (1985). Applications of catastrophe theory for statistical modeling in the biosciences. Journal of the American Statistical Association, 80(392), 793-802. Everitt, B. S., & Hand, D. J. (1981). Finite mixture distributions. London: Chapman & Hall. Ferretti, R. P., & Butterfield, E. C. (1986). Are children's rule assessment classifications invariant across instances of problem types? Child Development, 57, 1419-1428. Flavell, J. H., & Wohlwill, J. E (1969). Formal and functional aspects of cognitive development. In D. Elkind, & J. H. Flavell (Eds.), Studies in cognitive development: Essays in honor of Jean Piaget (pp. 67-120). New York: Oxford University Press. Gilmore, R. (1981). Catastrophe theory for scientists and engineers. New York: Wiley. Goldschmid, M. L., & Bentler, E M. (1968). Manual conservation assessment kit. San Diego, CA: Educational and Industrial Testing Service. Guastello, S. J. (1988). Catastrophe modeling of the accident process: Organizational subunit size.

Psychological Bulletin, 103(2):246-255. Kerkman, D. D., & Wright, J. C. (1988). An exegenis of compensation development: Sequential decision theory and Information integration theory. Developmental Review, 8, 323-360. McShane, J., & Morrison, D. L. (1983). How young children pour equal quantities: A case of pseudoconservation. Journal of Experimental Child Psychology, 35, 21-29. Molenaar, E C. M. (1986). Issues with a rule-sampling theory of conservation learning from a structuralist point of view. Human Development, 29, 137-144. Odom, R. D. (1972). Effects on perceptual salience on the recall of relevance and incidental dimensional values: A developmental study. Journal of Experimental Psychology, 92, 185-291. Oliva, T. A., Desarbo, W. S., Day, D. L., & Jedidi, K. (1987). GEMCAT: A general multivariate methodology for estimating catastrophe models. Behavioral Science, 32, 121-137. Pascual-Leone, J. (1970). A mathematical model for the transition rule in Piaget's development stages. Acta Psychologica, 32, 301-345. Pascual-Leone, J. (1989). An organismic process model of Witkin' s field-dependency-independency. In T. Globerson, & T. Zelniker (Eds.), Cognitive style and cognitive development (Vol. 3, pp. 36-70). Norwood, Nd: Ablex.

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Peill, E. J. (1975). Invention and discovery of reality: The acquisition of conservation of amount. London: Wiley. Perry, M., Breckinridge Church, R., & Goldin Meadow, S. (1988). Transitional knowledge in the acquisition of concepts. Cognitive Development, 3, 359-400. Piaget, J. (1960). The general problems of the psychobiological development of the child. In J. M. Tanner, & B. Inhelder (Eds.), Discussions on child development (Vol. 4, pp. 3-27). London: Tavistock. Piaget, J. (1971). The theory of stages in cognitive development. In D. R. Green, M. E Ford, & G. B. Flamer (Eds.), Measurement and Piaget. (pp. 1-11). New York: McGraw-Hill. Piaget, J., & Inhelder, B. (1969). The psychology of the child. New York: Basic Books. Poston, T., & Stewart, I. (1978). Catastrophe theory and its applications. London: Pitman. Preece, E E W. (1980). A geometrical model of Piagetian conservation. Psychological Reports, 46, 143-148. Saunders, E T. (1980). An introduction to catastrophe theory. Cambridge, UK: Cambridge University Press. Siegler, R. S. (1981). Developmental sequences within and between concepts. Monographs of the society for Research in Child Development, 46(2), 84. Siegler, R. S. (1986). Unities across domains in childrens strategy choices. Minnesota Symposia on Child Psychology, 19, 1- 46. Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ: Erlbaum. Stewart, I. N., & Peregoy, P. L. (1983). Catastrophe theory modeling in psychology. Psychological Bulletin, 94(2), 336-362. Ta'eed, L. K., Ta'eed, O., & Wright, J. E. (1988). Determinants involved in the perception of the Necker Cube: An application of catastrophe theory. Behavioral Science, 33, 97-115. van Maanen, L., Been, E, & Sijtsma, K. (1989). The linear logistic test model and heterogeneity of cognitive strategies. In E. E. Roskam (Ed.). Mathematical psychology in progress (pp. 267-287). Berlin: Springer-Verlag. van der Maas, H. L. J., & Molenaar, E C. M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99(3), 395-417. van der Maas, H. L. J. (1995). Nonverbal assessment of conservation. Manuscript submitted for publication. van der Maas, H. L. J., Walma van der Molen, J., & Molenaar, E C. M. (1995). Discontinuity in conservation acquisition: A longitudinal experiment. Manuscript submitted for publication. Wilson, M. (1989). Saltus: A psychometric model of discontinuity in cognitive development. Psychological Bulletin, 105(2), 276-289. Zahler, R. S., & Sussmann, H. J. (1977). Claims and accomplishments of applied catastrophe theory. Nature (London), 269( 10):759-763. Zeeman, E. C. (1976). Catastrophe theory. Scientific American, 234(4), 65-83. Zeeman, E. C. (1993). Controversy in science, the ideas of Daniel Bernoulli and Ren~ Thom. Johann Bernoulli Lecture. Groningen, The Netherlands: University of Groningen.

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1. INTRODUCTION The current spectacular progress in the analysis of nonlinear systems has important implications for theories of development. Until recently, central theoretical constructs in developmental biology and psychology, such as epigenesis and emergence, seemed to defy any attempt at causal modeling. For instance, when addressing the problem of causal inference in experimental studies of development, Wohlwill (1973, p. 319) believed that it would be impossible to isolate sufficient causes of normal developmental processes which he described as "acting independently of particular specifiable external agents or conditions." As to this, the situation has changed considerably in that we now have available a range of nonlinear dynamical models that can explain the self-regulating mechanisms of normal development (such as "catch-up growth" after temporary inhi-

Categorical Variables in Developmental Research: Methods of Analysis Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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bition which served as the empirical example in Wohlwill's discussion of causal inference). Moreover, the mathematical theory of nonlinear dynamics has provided innovative, rigorous approaches to the empirical study of epigenetical processes (cf. Molenaar, 1986). This chapter is devoted to one of these approaches. The founders of modern nonlinear dynamics have always been quite aware of the relevance of their work to the analysis of developmental processes. Prigogine's work with Nicolis on applications of nonequilibrium thermodynamics is an excellent example (Nicolis & Prigogine, 1977), as is Prigogine's (1980) classic monograph on the physics of becoming. Haken (1983) applied his synergistic approach to self-organization in evolutionary processes. And last but not least, Thom's (1975) main treatise of catastrophe theory is entirely inspired by morphogenetics. A common theme in all of these applications is the causal modeling of self-organization in nonlinear developmental systems. Furthermore, this is accomplished by using the same basic paradigm: bifurcation analysis. In a bifurcation analysis, a nonlinear system is subjected to smooth variation of its parameters. One might conceive of this variation as the result of maturation, giving rise to slow continuous changes (so-called quasi-static changes) in the system parameters. Most of the time this only yields continuous variation of the behavior of the system, but for particular parameter values, the system may undergo a sudden change in which new types of behavior emerge and/or old behavior types disappear. Such a discontinuous change in a system's behavior marks the point when its dynamics become unstable, after which a spontaneous shift occurs to a new, stable dynamical regime. Hence, sudden transitions in the dynamics of systems undergoing smooth quasi-static parameter variation constitute a hallmark of self-organization. Notice that bifurcation analysis is a mathematical technique which is applied to a mathematical model of a given nonlinear system. For many developmental processes, however, no adequate mathematical model is available. For instance, it is unknown what would constitute a proper dynamical model of (Piaget's theory of) cognitive development. One could proceed by making an educated guess (called an "Ansatz") of what might be a plausible model and subject that model to a bifurcation analysis. This approach is commonly used in applied synergism (cf. Haken, 1983). Yet, in the context of cognitive development, even the formulation of a plausible Ansatz is a formidable endeavor. Far too little is known about the dynamics of cognitive systems, and their operation can only indirectly be observed. To study self-organization in such ill-defined systems, a mathematical technique is needed that does not require the availability of specific dynamic models. This is provided by catastrophe theory. Catastrophe theory deals with the critical points or equilibria of gradient systems, that is, systems whose dynamics are governed by the gradient of a socalled potential. A potential is a smooth scalar function of the state variables of a dynamical system. For the moment, a potential can be best conceived of as a

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convenient mathematical construct; later, we present possible interpretations in the context of developmental processes. Let x(t) denote the vector of state variables at time t (we will denote vectors and matrices by bold lower- and uppercase letters). The rate of change of the state variables in a gradient system, then, equals the gradient of a potential V[x(t)] with respect to these state variables: d l d t x ( t ) = grad~ V[x(t)], where grad~ = [8/8xl, ~/~X 2 ~/~Xn] and ~/~X i denotes the partial derivative with respect to the state coordinate x i. This implies that the equilibria of a gradient system are given by the zeroes of grad~ V[x]. Catastrophe theory, however, does not deal with an individual gradient system but yields a characterization of the equilibria of entire families of parameterized gradient systems: grad~ V[x; e] = 0, where c is a set of parameters. Hence catastrophe theory is the study of how the critical points of V[x; c] change as the parameters in e change. The main result of catastrophe theory can be summarized as follows: The equilibria of a subset of all potentials V[x; e] can be characterized in terms of a few canonical forms that depend only on the number of parameters in e and on the rank of the Hessian, that is, the matrix of second-order derivatives of V[x; c] with respect to the state variables. If the number of parameters in c does not exceed 7 and if the rank of the Hessian is not less than n - 2, where n is the dimension of the state vector x, then this result holds. The implications of this main result of catastrophe theory are profound: For gradient systems, one only needs to specify two numbers to give a characterization of the way in which their equilibria change as the parameters change. This is precisely the kind of result that is needed for the study of self-organization in ill-defined dynamical systems. In fact, Gilmore (1981) derived characteristic features of sudden changes in equilibria that do not even depend on these two numbers but instead require an absolute minimum of a priori information for their application. These so-called catastrophe flags are extensively discussed in van der Maas and Molenaar (1992) and in the chapter by van der Maas and Molenaar in this volume. A key question now presents itself: Are gradient systems general enough to capture the essential self-organizing characteristics of most developmental processes? Apart from the catastrophe flags which apply to any gradient (and conservative Hamiltonian) system, catastrophe theory proper applies to a subset of the gradient systems and the entire class of gradient systems itself only constitutes a subset of the set of all possible dynamical systems. Notice that a definite answer to this question can only be obtained by empirical means. For instance, whether or not a gradient system constitutes a satisfactory causal model of cognitive development (or, more realistically, of an aspect of cognitive development such as conservation acquisition) can only be determined in dedicated empirical research. Having said this, however, one can put forward some a priori considerations that are indicative of the viability of applications of catastrophe theory to biological and psychological developmental processes. As alluded to earlier, .

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these processes are in general ill-defined in that we are unable to give a complete specification of the underlying dynamics. If catastrophe theory is applied and the (unknown) true dynamics do not conform to a gradient system, then what is the error of approximation? More specifically, let the true dynamical equation be d/dt x(t) = F[x(t); e], where F now is an arbitrary vector-valued function. Then, according to Jackson (1989, p. 117): "If F[x; e] = 0 implies that grad x V[x; e] = 0, for some [scalar potential] V, then Thom's theorem [i.e., catastrophe theory] can be applied to V[x; el. It is not required that F = gradx V everywhere in the [state] space." Hence, if the equilibria of a nongradient system can be locally represented by the equilibria of a potential, then there is no error of approximation. The class of potential systems, that is, systems whose dynamics depend on the gradient of a potential, includes gradient systems, (conservative) Hamiltonian systems, and Hamiltonian systems with positive damping. Thus the class of potential systems is rather large, coveting many physical processes. Huseyin (1986) has extended the catastrophe theory program to this class of potential systems. Furthermore, Huseyin shows that the (so-called static) instabilities of autonomous nonpotential systems can be characterized in much the same way as those of potential systems (thus corroborating Jackson's observation). Taken together, these a priori considerations imply that the range of applicability of catastrophe theory may be rather large (although one would like to have a more detailed formal specification of the approximation error in the general case). This is a fortunate state of affairs because catastrophe theory may be the only principled method available to study self-organization in many ill-defined systems. In what follows, we will first present an outline of elementary catastrophe theory for metrical deterministic gradient systems. This presentation is based on Gilmore (1981) and follows a constructive approach that is distinct from the more abstract, geometrical approaches followed by most other authors (e.g., Thorn, 1975). Then, we move on to a consideration of catastrophe theory for stochastic metrical systems. After having presented a typology of stochastic systems to indicate the type of systems we are interested in, a concise overview is given of Cobb's innovative work in this area (e.g., Cobb & Zacks, 1985). It will be argued that Cobb's approach does not meet the basic tenets of catastrophe theory, and therefore a revision of his approach is formulated that agrees with the goals of elementary catastrophe theory. Even then, however, there remain some fundamental problems with extending the catastrophe theory program to metrical stochastic systems. We discuss these problems in a heuristic way and suggest a principled approach to deal with them. This part of the chapter draws on ongoing research and therefore the reader should not expect definite solutions. The same can be said about the final part of the chapter dealing with catastrophe theory for discrete stochastic systems. Here, we restrict the discussion

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to a characterization of these systems in a way that makes them compatible with the catastrophe theory program.

2. ELEMENTARYCATASTROPHETHEORY The basic tenet of catastrophe theory is to arrive at a classification of the possible equilibrium forms of gradient systems undergoing continuous quasi-static variation of their parameters. This implies a classification of the critical points of perturbed parameterized potentials V[x; c]. Critical points belong to the same class if there exists a diffeomorphic relationship between the corresponding potentials. Roughly speaking, a diffeomorphism changes x and/or e in a smooth manner (i.e., a local diffeomorphism at a point p is an invertible coordinate transformation for which derivatives of arbitrary order exist). Hence the program of elementary catastrophe theory is effected by smooth coordinate transformations. Actually the picture is more complicated, as will be indicated in a later section, but this suffices for our present purposes.

2.1. Coordinate Transformations We consider here transformations of a coordinate system [X1, X 2 . . . . . Xn] for processes in R'. Such coordinate transformations play an important role in effecting the program of elementary catastrophe theory and will also figure predominantly in the extension of this program to stochastic metrical systems. A transformation of the coordinate system [x~, x 2. . . . . x.] into [Yl, Y2. . . . . y.] then is defined by Yi = Yi(X1, X2 . . . . .

Xn) , i = 1 . . . . .

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(1)

at any point p in R" at which the Jacobian det 18Yi/Sxjl is nonzero. As Equation (1) is understood to be a local diffeomorphism, it is useful to express the new Yi coordinates in terms of a Taylor series expansion of the old x coordinate system. Suppose a given potential is represented in a particular coordinate system: V[x]. Then of course a coordinate transformation will change the functional form of this potential: U[y]. Yet the potential itself remains identical, hence U[y] = V[x]. In contrast, two different potentials will have different functional forms in the same coordinate system; for example, V[x] will not be equal to U[x]. However, in the words of Gilmore (1981, p. 18), these two potentials are "qualitatively similar if we can find a smooth change of coordinates so that the functional form of U, expressed in terms o f the new coordinates, is equal to the functional form of V in the original coordinate system." Hence if V[x] differs from U[x], but we can find a coordinate transformation so that U[y] = V[x],

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then these two potentials are qualitatively similar. In particular, the number and types of their critical points is the same. The way in which coordinate transformations are used in effecting the program of catastrophe theory can now be summarized as follows. A potential is expressed in a Taylor series expansion about its critical point. A coordinate transformation is sought that reduces this potential to a canonical form. All potentials that can be reduced to the same canonical form then constitute an equivalence class having the same type of critical point. This will be elaborated more fully in the next sections.

2.2. Morse Form To start, we consider a critical point x* of a potential V[x] at which the Hessian is nonsingular: grad x V[x] = 0 at x*;

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(2)

Only the quadratic and higher degree terms of the Taylor series expansion of V[x] about x* will be nonzero. It now can be proved that there always exists a coordinate transformation so that in a neighborhood of x*, V[x] can be expressed as U[y] -- ~i=

],n

)k/Yi2,

(3)

where h i a r e the eigenvalues of the Hessian matrix Vii. This simple quadratic form is the canonical Morse form for critical points of a potential at which the Hessian is nonsingular. The constructive proof given by Gilmore (1981, pp. 20-23) is elementary and consists of counting the number of disposable coefficients in the Taylor series expansion of a coordinate transformation that can be used to kill (i.e., transform to zero) coefficients in the Taylor series expansion of the transformed potential. From Equation (3) it follows that all critical points of potentials at which the Hessian is nonsingular belong to a single equivalence class. Hence, all of the equilibria of the arbitrary gradient systems for which the Hessian is nonsingular are qualitatively similar.

2.3. Them Forms If the Hessian at a critical point of a potential is singular (has zero eigenvalues), then it cannot be transformed to the Morse form, Equation (3). Suppose that the Hessian Vii of a potential V[x] has m zero eigenvalues (in technical jargon: has nullity m). Then, it follows from the so-called splitting lemma that this potential can be reduced to U[y] = Fdg[Y l, Y2 . . . . .

Ym] + ~i=m+ 1,n hiYi 2"

(4)

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Thus the potential can be reduced to a form that consists of two parts: a degenerate part Fdg associated with the m degenerate eigenvalues of the Hessian Vii and another part that is simply the Morse form associated with the nondegenerate eigenvalues of Vii. The y-coordinates associated with degenerate and nondegenerate directions in state space are split, hence the name of the lemma. Notice that in Equation (4) the part conforming to the Morse form already has a canonical form. Can the degenerate part Fdg also be transformed into a canonical form? To answer this question, we first note that, thus far, the parameters e have been left implicit because they play no role in the Morse form. That is, there is only one canonical Morse form, irrespective of the number of parameters in e. To arrive at canonical forms for Fdg, however, its dependence on c has to be made explicit: Fdg[Y 1. . . . . Ym; e*], where e* denotes the value of e at the critical point. We will see that canonical forms for Fdg depend on the number k of parameters in e as well as on m. Specifically, such canonical forms only exist if m is not larger than 2 and k is at most 7. To illustrate, let m = 1 and let there be k parameters. Then, the Taylor series expansion of Fdg about the degenerate critical point, as obtained from the splitting lemma, is given by

Fdg[Yl; C*] = ~itiYl i, i -- 3, 4 . . . . .

(5)

where it is understood that the critical point has been translated to y* = 0, and where the term for i = 0 is unimportant, the term for i = 1 is lacking because t~ = 0 at a critical point, and the term for i = 2 is lacking because of the degeneracy. It can be shown (cf. Gilmore, 1981, pp. 26-27) that coordinate transformations of both y~ and e can reduce Equation (5) to the following canonical form: CG[1; k] =

sz ~+2; S

=

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(6)

where, in general, CG[m; k] stands for the so-called catastrophe germ of a k-parameter potential whose Hessian at the critical point has m vanishing eigenvalues, and where s is a dummy variable (there are in fact two canonical forms, one is the negative of the other). The canonical form, Equation (6), is based on a Taylor series expansion in terms of the state variables of the potential. This implies that Equation (6) is valid in an open neighborhood of the critical point in state space. But it is only valid at the point e = e*, because no Taylor series expansion of the parameters has been considered. The extension of the validity of Equation (6) to a neighborhood of the critical point in parameter space amounts to considering the effect of perturbations of e* on the catastrophe germ. The constructive approach followed by Gilmore to establish this extension is the same as before, that is, determine which terms in the Taylor series expansion of the perturbed catastrophe germ can always be removed by a coordinate transformation. In this way, a canonical form of all possible perturbations of a catastrophe germ is obtained,

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which will be denoted by Pert[m; k]. Note that the canonical perturbations are different for different catastrophe germs and thus depend on m and k. The final result now reads, schematically, Cat[m; k] = CG[m; k] + Pert[m; k],

(7)

where Cat[m; k] stands for (elementary) catastrophe. The canonical forms of Equation (7) are called the Thom forms. In particular, if k = 2, then it can be shown that the canonical perturbation of Equation (6) is given by Pert[l; 2] = ClZ + c2z 2, yielding the cusp catastrophe which figures predominantly in the chapter by van der Maas and Molenaar: Cat[l; 2] = sz 4 + c2z 2 + ClZ; s = 1 o r - 1 .

(8)

2.4. Discussion Our heuristic outline of elementary catastrophe theory focuses on the role of coordinate transformations in the derivation of canonical forms for the critical points of potentials. There are several reasons for choosing this particular point of view. The mathematical theory underlying Equation (7) is very abstract and therefore may not be readily accessible to developmental biologists and psychologists. The constructive transformational approach used.by Gilmore (1981) may fare better in creating a valid mental picture within the confines of a single chapter. Also, this constructive approach may be less known because most textbooks present a much more geometrically inspired perspective on catastrophe theory (e.g., Castrigiano & Hayes, 1993; Poston & Stewart, 1978; Thorn, 1975). Last but not least, as will be indicated in the next section, it turns out that the problems arising from the extension of catastrophe theory to stochastic metrical systems are closely linked to the way in which coordinate transformations for these systems are defined. As alluded to earlier, elementary catastrophe theory constitutes a powerful approach to the study of self-organization in ill-defined systems. Selforganization typically takes place in a "punctuated" way, in which the stability of the equilibrium of a system gradually decreases and becomes degenerate as a result of parameter variation over an extended period of time, after which a sudden shift to a new, qualitatively different equilibrium occurs. The ongoing behavior of a system is organized around its equilibrium states and therefore a qualitative shift in equilibrium will manifest itself by the emergence of new behavioral types. The results of catastrophe theory, based on mathematical deduction, show that under mild conditions (see the quotation of Jackson, 1989, in section 1) the occurrence of such stage transitions in epigenetical processes can be detected and modeled without a priori knowledge of the system dynamics. To apply elementary catastrophe theory, one only has to identify the degenerate

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coordinates of a system's equilibrium point and the parameters whose quasistatic variation induces the loss of stability. In the jargon of catastrophe theory, the degenerate coordinates are called the behavioral variables, and the parameters controlling system stability are called the control variables. Given a set of empirical measurements of candidate behavioral and control variables, and allowing for all possible coordinate transformations, the appropriate Thom forms then can be fitted to the data to determine whether a genuine stage transition is actually present and to which class it belongs. In closing this section, it must be noted that our outline of catastrophe theory, although providing a convenient stepping stone to the next section, is incomplete in almost all respects. It one thing to present a constructive approach, it is quite another to specify the necessary and sufficient conditions for such an approach to hold. Also, we do not give a complete list of the Thorn forms. All of this is contained in the excellent monograph by Castrigiano and Hayes (1993).

3. CATASTROPHETHEORY FOR METRICAL STOCHASTIC SYSTEMS Elementary catastrophe theory pertains to deterministic processes generated by gradient systems. In contrast, real developmental processes encountered in the biological and social sciences almost never can be forecasted with complete certainty and therefore do not constitute deterministic but stochastic processes. To extend catastrophe theory to the analysis of these stochastic processes, one first has to specify the relevant ways in which gradient systems can be generalized to include random influences. There are several distinct ways in which a gradient system can become stochastic (cf. Molenaar, 1990), but it will be sufficient for our purposes to restrict attention to two of them. First, consider the system: d/dt x(t) = grad x V[x] y(t) = x(t) + e(t),

(9)

where e(t) denotes random measurement noise. Here, the manifest process y(t) is stochastic, but the latent gradient system is still deterministic. The random measurement noise in (9) does not affect the system dynamics and thus prediction of x(t) by means of sequential nonlinear regression techniques will become perfect when t approaches infinity. In contrast, the role of random influences is entirely different in the following system: d/dt x ( t ) = grad x V[x] + w(t) y(t) = x(t),

(10)

where the random process w(t), called the innovations process, directly affects the system dynamics. This implies that x(t) can never be predicted with complete certainty, even if t approaches infinity.

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In what follows, we restrict attention to (10) where the system dynamics is intrinsically stochastic. This constitutes by far the most interesting (and, admittedly, also the most problematic) case. Although it is possible to allow for measurement noise e(t) in (10), we will avoid this complication because it is not essential to our purposes. Hence the second equation of (10) can be dropped because it involves a trivial identity transformation, after which x(t) will denote the manifest process. We are thus left with the first equation of (10), which constitutes an instance of a stochastic differential equation (SDE). Following the notational conventions appropriate for SDEs (to be explained in the next section), the starting point of our discussion is then given by the following metrical stochastic system: dx(t) = grad x V[x]dt + dw(t).

(11)

3.1. Stochastic Differential Equations The theory of SDEs can be formulated in a number of ways (e.g., employing martingale methods or semigroup methods; cf. Revuz & Yor, 1991). We will have to leave these elegant formal constructions for what they are, however, and concentrate on the mere presentation of a few basic notions that are of importance. The first of these is the Wiener process w(t). Consider the following process in discrete time: A (presumably drunken) man moves along a line, taking, at random, steps to the left or to the fight with equal probability. We want to know the probability that he reaches a given point on the line after a given elapsed time. This process is the well-known random walk. If now both the unit of time and the step size become infinitesimally small, then the one-dimensional Wiener process w(t) is obtained. Given that this Wiener process starts at the origin, w(0) = 0, the probability that it reaches point w at time t, p ( w ( t ) = w lw(0) = 0), can be shown to be Gaussian with mean zero and variance proportional to t. The sample paths of a Wiener process are continuous, that is, each realization is a continuous function of time. Yet, it can be shown (cf. Gardiner, 1990) that these sample paths are nondifferentiable. This implies that a realization of the Wiener process is exceedingly irregular, as is also reflected by the linear dependence of the (conditional) variance on elapsed time t. Of particular importance is the statistical independence of the increments of w(t). Let ~w(ti) = w(ti) - w ( t i - 1) denote these increments, where t i, i = 1, 2 . . . . . n is a partition of the time axis, then the joint probability of the 8 w ( t i ) is the product of normal distributions, each with mean zero and variance 8 t i - - t i - - t i _ 1. We now define the differential dw(t) by letting ~ti become infinitesimally small. But w(t) is nondifferentiable, hence dw(t) does not exist. On the other hand, the integral of dw(t) exists and is the Wiener process w(t). In particular:

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t

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(12)

o and this integral equation can be interpreted consistently. Thus dw(t) can be conceived of as a convenient abstraction to be used to arrive at a consistent definition of stochastic differential equations. We are now in a position to provide a definition of an n-dimensional SDE. Let w(t) denote an n-dimensional Wiener process with uncorrelated components: E[wi(t), wj.(t)] -= min(t, s)gij, where E denotes the expectation operator and ~ij is Kronecker's delta, that is, gij = 1 if i = j and gij = 0 otherwise. In addition, let a[x] denote an n-dimensional so-called drift vector and B[x] an (n • n)dimensional diffusion matrix. Both drift and diffusion are arbitrary functions of x(t). Then consider the SDE dx(t) = a[x]dt + B[x]dw(t),

(13)

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(14)

0

Note that if a[x] = grad x V[x] and B[x] = I,,, that is, the (n • n) identity matrix, then Equation (13) reduces to Equation (11). For our purposes, it suffices to consider two aspects of Equation (13). First, it should be noted that this SDE does not obey the usual transformation rules for differential equations. Instead, the famous Ito formula describes the way in which Equation (13) transforms under a change of variables. Specifically, let n - 1 for ease of presentation and consider an arbitrary smooth function of x(t): f[x]. Then, Ito's formula is given by

df[x] = { a[x]f'[x] + b[x]Zf"[x]/2 }dt + b[x]f'[x]dw(t),

(15)

where f'[x] and if[x] denote, respectively, the first- and second-order derivatives of fix] with respect to x(t). Second, it should be noted that, given initial conditions, Equation (13) specifies the evolution of the probability density function of x(t) as a function of time t. Because the drift and diffusion terms in Equation (13) do not explicitly depend on time, it follows that the probability density function of x(t) becomes stationary if t approaches infinity. That is, p(x, t) --) p(x)

as

t --~ oc

(16)

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Peter C. /t4. Molenaar and Pascal Hartelman

The stationary probability density p(x) is a function of the drift a[x] and the diffusion matrix B[x]. This functional dependence takes on a particularly simple form if x(t) obeys the so-called detailed balance condition. Heuristically speaking, a process satisfies detailed balance if in the stationary situation each possible transition (from x(t) = xl to x(t + d t ) = x 2) balances with the reverse transition (cf. Gardiner, 1990, pp. 148-170). If this is the case, that is, if a[x] and B[x] obey the formal criteria for detailed balance, then the stationary probability density is given by p(x) - exp(-~[x]),

(17)

where x

9 [x] = fdy.zta, B, y] zi[a, B, xl = ~ B~2[xl{2aktx] - ~ O/OxjB2j[x]}, k

i = 1, 2 . . . . .

n.

j

The dot in the second equation denotes the vector inner product (of the n-dimensional vectors dy and z). The important point conveyed by (17) is that under detailed balance, the stationary probability density of a homogeneous process x(t) is given by p(x) = exp(-~[x]), where ~[x] is a potential. Moreover, this potential ~[x] is a nonlinear function of the drift and diffusion terms in the SDE.

3.2. Cobb's Approach In a number of papers, Cobb has elaborated a stochastic catastrophe theory for metrical systems (e.g., Cobb, 1978, 1981; Cobb, Koppstein, & Chen, 1983). We will focus on the paper by Cobb and Zacks (1985) in which the link with the SDEs is explicitly addressed. What follows is a synopsis of the relevant parts of the latter paper. Let n = 1 and consider the following instance of Equation (13): dx(t) = grad x V[x]dt + b[x]dw(t),

(18)

which also can be regarded as a slight generalization of Equation (11). Now the first step in Cobb and Zack's approach consists of substituting an elementary catastrophe for the potential V[x]. In particular, V[x] is taken to be the cusp catastrophe given by Equation (8). In the second step, the stationary probability density of Equation (18) is determined. An application of (17) to (18) yields, after some rewriting, the stationary density in the form as given by Cobb and Zacks: x

p(x) = exp(2f {g[y]/bZ[y]}dy),

(19)

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where g[x] = V'[x] - b'[x]b[x], and the apostrophe denotes differentiation with respect to x. Cobb and Zacks then proceed by considering several functional forms for bZ[x]: a constant, a linear, or quadratic function of the state x, and so on. This yields distinct family types of stationary densities, one for each functional form of bZ[x] (and g[x]).

3.3. Issues with Cobb's Approach As alluded to earlier, the gist of catastrophe theory is to arrive at a minimal classification of the possible equilibrium forms of gradient systems undergoing quasi-static parameter variation. This is accomplished by transforming away all inessential details of particular instantiations of gradient systems and concentrating on the canonical forms thus obtained. In this way, an infinite variety of equilibrium forms is reduced to a minimal classificatory scheme that can be applied without specification of the underlying dynamical system equations. It seems, however, that Cobb's approach does not conform to this basic reductionistic tenet of catastrophe theory. The typology of stationary probability densities arrived at in his approach depends on b2[x] and, through g[x], on V[x]. This implies that each elementary catastrophe that is substituted for V[x] gives rise to a potentially infinite number of density types, one for each distinct form of b Z[x]. Hence, it would seem that Cobb' s approach creates a plethora of canonical forms, which reintroduces the need to consider the details of each system in order to determine its equilibrium type. Note, however, that this presumed state of affairs rests on the assumption that Equation (19) indeed represents the proper canonical form of stationary probability densities. That is, (19) should represent what is left after transforming away all inessential details of particular instantiations of stochastic gradient systems. But obviously this is not the case. It is always possible to find a change of variable y = f[x] such that application of Ito' s formula, Equation (15) to Equation (18), will yield a transformed SDE in which the diffusion term bZ[y] = 1. In fact, for each given instance of Equation (18), the diffusion term can be transformed into an infinite variety of forms by judicious application of Ito's formula. Consequently, the stationary density types in Cobb's approach are not canonical in the intended sense because they depend on the specific forms of diffusion terms, whereas these diffusion terms are not invariant under smooth coordinate transformations.

3.4. An Alternative Approach The classificatory scheme of Cobb depends on a bivariate feature vector: the potential V[x] and the diffusion term b2[x]. Given a fixed form of V[x], one can get different density types for different forms of b2[x]; and vice versa, given a

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fixed form of b2[x], different density types are obtained for different forms of V[x]. Pertaining to this, it was shown in the previous section that the diffusion term is not invariant under smooth coordinate transformations and therefore is not suitable to index canonical forms. Hence, one plausible alternative approach consists of trying to redefine the feature vector underlying the classificatory scheme so that it no longer depends on the diffusion term in the way envisaged by Cobb. This could be accomplished by simply removing the diffusion term from the feature vector which would result in a univariate feature vector composed only of the potential. Note, however, that this implies a redefinition of the featured potential itself. We address this important point first. The potential in Cobb's approach is defined as the entity whose gradient yields the drift term in Equation (18). This potential is a component of g[x], where g[x] itself is a component of the stationary density, Equation (19). One can only recover Cobb's potential from the stationary density (19) if the diffusion term b Z[x] is given, which is why Cobb's approach depends on a bivariate feature vector. Thus in order to arrive at a classificatory scheme of stationary densities that depends only on a potential, Cobb's definition of the featured potential will not do. Instead, we will have to turn to the definition of the stationary probability density given by Equation (17), taking n = 1: p(x) = e x p ( - ~ [ x ] ) . Note that the potential ~[x] in Equation (17) is entirely different from Cobb's potential V[x] underlying the drift term in Equation (18). In terms of Cobb's scheme based on (18), ~[x] is a nonlinear function of V[x] and the diffusion term b Z[x]. Moreover, ~[x] completely characterizes stationary densities and hence can be used as the univariate feature indexing canonical forms for these densities. In summary, it would seem that a minimal classificatory scheme for stationary probability densities can be based on the canonical forms of the potential ~[x] in Equation (17). It is then required that x(t) obey the detailed balance condition (which is automatically the case if n = 1). Taking qg[x] as the featured potential implies that, in contrast with Cobb's approach, no distinct reference is made to the drift term and/or the diffusion term in an SDE. This alleviates the problems in Cobb's approach, which, as alluded to earlier, arise from transformation of the diffusion term. We now can concentrate on the way q~[x] behaves under coordinate transformations in which the transformations concerned are given by the Ito formula, Equation (15). The main question then becomes: Does application of Ito's formula (15) to ~[x] in (17) (or ~[x] for n > 1) yield the same elementary catastrophes as the application of local diffeomorphisms, Equation (1), to the potential associated with a deterministic gradient system? In other words, is there a direct transfer of elementary catastrophe theory to metrical stochastic systems. The answer is, unfortunately, negative and leads to some rather fundamental problems that we have studied intensely for the past year.

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3.4.1. The m a i n p r o b l e m It can be shown that the collection of all possible potential forms in Equation (17) constitutes a single equivalence class (type). That is, each given form of ~[x] can be transformed into any other possible form by means of Ito's formula. In the words of Zeeman (1988), all potential forms in Equation (17) are "diffeomorphically measure equivalent." For instance, let ~[x] be the cusp catastrophe Cat[l;2] given by Equation (8), and let the values of the control variables c~ and c 2 be chosen so that (8) has three critical points (two of which are stable and one is unstable). Then, the associated stationary probability density p(x) = e x p ( - C a t [ 1 ;2]) has two modes corresponding to the two stable critical points of the cusp catastrophe. This bimodal density with cuspoid potential, however, can always be transformed by means of Ito's formula into a unimodal density for which the potential has Morse form (Hartelman, van der Maas, & Molenaar, 1995). Thus the distinction between Thom forms and Morse forms, a distinction that is central to the program of catastrophe theory, collapses at the level of modes of stationary densities of metrical stochastic systems. To illustrate this collapse at the level of modes of stationary densities, consider the simulated realization of dx(t) = (2x(t) - x(t)3)dt + dw(t), 0 < t < 100, shown in Figure 1. The drift term of x(t) is given by the gradient of the cusp catastrophe and the diffusion term is 1; in the notation of Equation (13), a[x] = 2x(t) - x ( t ) 3, and b[x] = 1. It then follows directly from Equation (17) that the stationary probability of x(t) is given by p(x) = c - l e x p ( x 4 / 4 - x2/2), where c-~ is a normalizing constant. Hence, the stationary density p(x) is bimo-

FIGURE 1. Simulated realization by means of numerical integration of a univariate SDE, dx(t) = a[x]dt + b[x]dw(t), 0 < t < 100, where the drift a[x] = 2x - x 3 and the diffusion term b[x] = 1.

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dal. A rough estimate of p(x) can be obtained by determining a histogram of the realized values shown in Figure 1. This histogram is presented in Figure 2 and clearly has the expected bimodal form. We next consider the following transformation of x(t): y(t) = f[x] = x(t) + x(t) 3. An application of Ito's formula (15) this time yields the drift and diffusion terms of the y(t) process: a[y] = 8x(t) + 5x(t) 3 - 3x(t) 5, b[y] = 1 + 3x(t) 2. Figure 3 shows a realization of this y(t)process, where again 0 < t < 100, and Figure 4 shows the histogram associated with this realization. It turns out that this histogram, and hence the stationary density of the transformed x(t) process, is now unimodal, illustrating the main problem discussed in this section. 3.4.2. Zeeman's solution The stationary density, Equation (17), constitutes a probability measure. Smooth coordinate transformations (diffeomorphisms) in the space of measures are defined by Ito's formula. In contrast, the potential associated with a deterministic gradient system constitutes a function. The space of continuous functions is dual to the space of measures, and smooth coordinate transformations in function space are defined by Equation (1). We have already seen that the crucial distinction in function space between Morse forms and Thom forms collapses in measure space because these forms are diffeomorphically measure equivalent. To avoid this collapse, Zeeman (1988) proposes to treat the measure defined by Equation (17) as a function. More specifically, he states (1988):

FIGURE 2.

Histogramof the realized x(t) values depicted in Figure 1.

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123

FIGURE 3. Simulated realization by means of numerical integration, 0 < t < 100, of the transformed x ( t ) process depicted in Figure 1. The transformation is y ( t ) = f [ x ] = x + x 3.

We regard the steady state u [our stationary density (17)] as a tool, a quantitative property of v [the deterministic gradient system acting as drift term in an SDE]. A n d then, having constructed the tool, we can use the tool in any way we please to study v . . . . Therefore we can apply the qualitative theory of functions to the tool to obtain a qualitative description of v. (p. 132)

200 180, 160. 140. 120, o

100, 80, 60,

4oi 2oi 0

-:~0

-15

-10

5

-5

10

15

y value

FIGURE 4. Histogram of the realized

y ( t ) values depicted in Figure 3.

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An application of Zeeman's solution to avoid the collapse of distinct canonical forms seems to be straightforward. For a given stochastic process, one first determines the stationary probability density. Then the potential ~[x] associated with this stationary density can be conceived of as a function; that is, @[x] is treated in the same way as the potential V[x] associated with deterministic gradient systems and thus is amenable to smooth coordinate transformations defined by Equation (1). It then follows that the elementary catastrophes obtained for V[x] also yield the canonical forms for the critical points of ~[x] (and hence for the modes of the associated stationary density). 3.4.3. Discussion Zeeman's solution is presented in a fundamental paper (Zeeman, 1988) that introduces a new definition of the stability of dynamical systems. It is one of the rare technical publications dealing with catastrophe theory of stochastic systems, apart from Cobb's work, and presents a challenging and innovative point of view. Here, however, we have to restrict our discussion to the shift from measure space to function space proposed by Zeeman. It is clear that this shift prevents the collapse of Morse and Thom forms referred to in section 3.4.1. However, there are some issues that require further scrutiny. Perhaps the most important of these concerns the "timing" of the shift, that is, at which stage in the derivation of the stationary density in measure space does one shift to a function space perspective? To elaborate, suppose we are given a realization x(t), for t in the interval [0, T], of a homogeneous stochastic process. Then first the stationary density has to be estimated, for instance, by a cubic spline fit to the empirical histogram. Let the estimated potential obtained be denoted by ~[xl T]. Is this the potential that according to Zeeman's approach has to be treated as the function characterizing the observed process? Or should Ito's formula be applied to ~[xl T], yielding a transformed potential ~ [ f [ x ] IT], before the shift to a function space perspective is made? It would seem that Ito's transformation should not be made, because that would open up the door again for the problem discussed in section 3.4.1 (e.g., ~[x[ T] might be a Thom form which is Ito transformed to a Morse form ~ [ f [ x ] I T]). Accordingly, it appears to be the estimated potential ~[xl T] in the stationary density of the given realization x(t) to which Zeeman's solution applies. This assigns a special status to the actual measurements, to their operationalization, dimensionality, and scale. For instance, it may make a difference to Zeeman's solution whether the behavior of a system is measured in terms of amplitude or in terms of energy (which is proportional to the second power of amplitude). This special status of the given (chosen) measurements, however, will be problematic for the social sciences, where fundamental measures (dimensions) are rare and it is often a matter of taste as to which measurement scales are being used. It should be noted that Zeeman's aims in his 1988 paper are different from

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ours. Zeeman introduces SDEs with known properties (e.g., their variance e, where e > 0, is small) to define a so-called e-smoothing of the equilibria of deterministic gradient systems and thereby arrive at new definitions of equivalence and stability of gradient systems. In social scientific applications, however, such a priori information is almost always lacking. It then appears that Zeeman's solution presents some problems of its own that require further elaboration. Yet, a shift from the space of measures to function space as stipulated by Zeeman would seem unavoidable in order to prevent the collapse of canonical forms for the potentials in stationary densities. Moreover, the focus on stationary probability densities in stochastic catastrophe theory is plausible because these stationary densities constitute direct analogues of the equilibria of deterministic gradient systems that figure in elementary catastrophe theory. If, however, one were to change the focus on stationary densities, then one could look for an alternative characterization of stochastic processes for which the canonical forms in measure space keep the distinction between elementary catastrophes intact. In such an approach, it would no longer be necessary to shift from a measure space to a function space perspective. In closing this section, we present the outline of one such alternative approach. To reiterate, we are looking for an alternative measure that characterizes the equilibria of homogeneous stochastic systems while avoiding the problems with stationary probability density mentioned in section 3.4.1. Such a characterization should not cause a collapse into a single canonical form under Ito transformation, but instead keep intact in measure space the distinct canonical forms of elementary catastrophe theory. Recently, Hartelman has identified such an alternative characterization: the number of level-crossings nl(T) of a stochastic process, where l is the level and [0, 7] the interval of observation (cf. Florens-Zmirou, 1991, for a detailed discussion of levelcrossings in SDEs). It can be shown (Hartelman et al., 1995) that the number of modes of the measure nz(T) stay invariant under Ito transformation. This would seem sufficient to establish something approaching a one-to-one correspondence between canonical forms of nl(T) in measure space and elementary catastrophes in function space. Thus it appears to be possible to arrive at a sensible catastrophe theory of stochastic systems in measure space which does not require the shift to a function space perspective underlying Zeeman' s approach.

4. CATASTROPHETHEORY FOR DISCRETE STOCHASTICSYSTEMS Until now, we have been considering metrical deterministic and stochastic systems. That is, systems whose state (behavioral) variables are real-valued. Moreover, we have considered the evolution of these systems in continuous

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time under smooth quasistatic variation of the system parameters (control variables). In this setting of continuous variation of state variables, parameters, and time, it was shown that discrete changes (elementary catastrophes) emerge in the equilibrium forms of the systems concerned. These discrete changes in equilibrium type constitute a rather special instance of what is the main theme of this book, categorical variables. In this section, however, we consider categoricalvalued processes in another, more direct sense. In particular, we outline a possible extension of the program of catastrophe theory to stochastic systems with categorical state variables. The deliberations will remain at a rather general level, mainly concentrating on the possibility of deriving potentials for discrete stochastic systems as a prerequisite for the application of catastrophe theory. We start with potential theory for denumerable Markov chains in discrete time.

4.1. Potentials for Markov Chains Let S denote the set of categories making up the state space of a Markov chain. We denote representative elements of S by i, j, k, and so on. Let x,,, where n is integer-valued, represent the Markov chain. Then it holds that Pr[xn+ 1 = i[x, =j,x,_

1 = k .....

x o = l] = P r [ X , + l

= i[x~

=j],

(20)

where P r [ a [ b ] denotes the conditional probability of a given b. In addition, we assume that the Markov chain is homogeneous" Pr[xn+ 1 = i[x,

= j ] = P r [ x , + m + 1 = i[Xn+ m = j ] .

(21)

For the state space S, the complete set of conditional probabilities given by Equation (21) can be conveniently collected in a square, so-called transition matrix P. Then, given the starting probabilities P r [ x o = i], the measure on the state space S of a Markov chain is completely determined by the transition matrix P. Now the question we want to address is whether a potential can be defined for a Markov chain. Perhaps the most extensive treatment of this question can be found in a monograph by Kemeny, Snell, and Knapp (1976). They show that almost all results of potential theory generalize to Markov chains, although some require further assumptions that are analogous to the detailed balance condition for metrical stochastic systems alluded to earlier in section 3.1. More specifically, they present Markov chain analogues of potential theory concepts such as charge, potential operator, harmonic function, and so on. In particular, the Markov chain analogues of the potential are lira e~(I + P + . . .

+ P ' ) o r l i m (I + P + . . .

P')er,

(22)

where lim denotes the limit for n approaching infinity, e l is a finite normed row vector called left charge, and e~ a finite normed column vector called fight charge.

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In closing this section, note that Equation (22) gives two Markov chain analogues of the potential, one associated with a left charge cr and the other with a fight charge er Kemeny et al. (1976) refer to the analogue associated with the fight charge as the potential function, whereas the analogue associated with the left charge is denoted by the potential measure. To appreciate this difference, note that Kemeny et al. define Lebesgue integration on the denumerable state space S of a Markov chain by a vector inner product in which a nonnegative row vector constitutes the measure and a column vector constitutes the function (1976, p. 23). This would seem to imply that only the potential function associated with the fight charge can be conceived of as the Markov chain analogue of the potential in elementary catastrophe theory. But regarding the potential measure associated with the left charge, they state (Kemeny et al., 1976, p. 182): "Classically, potentials are left as point functions and are never transformed into set functions because such a transformation is frequently impossible. In Markov chain potential theory, however, every column vector can be transformed into a row vector by the duality mapping." Note that this resembles the shift from measure space to function space in Zeeman's approach to stochastic catastrophe theory for metrical stochastic systems (cf. section 3.4.2). At present, however, nothing more can be said about this interesting correspondence because a catastrophe theoretical analysis of the Markov chain potentials given by Equation (22) is still lacking.

4.2. Potentials for Markov Processes A continuous time homogeneous Markov process x(t) with denumerable state space S obeys conditions similar to Equations (20) and (21). For such processes, it is possible to derive a so-called master equation (e.g., Van Kampen, 1981). Denoting Pr[x(t) = i] by pi(t), the master equation is given by

dpi(t)/dt = Xj[wijpj(t) - wjpi(t)],

(23)

where wij is the transition probability per unit time from state j to state i. If for the state space S the complete set of transition probabilities is collected in a square matrix W, where the ith diagonal element of W is defined by wii = - E j wji for j unequal to i, then we obtain a representation that is similar to the one for Markov chains given in the preceding section. On the basis of this representation, Markov process analogues for potential theory concepts can again be derived. For this, we refer to the large literature on potential theory for Markov processes, including the momentous monograph by Doob (1984).

4.3. Discussion The message of the preceding two sections can be summarized in a simple statement: there exists a well-developed potential theory for denumerable Markov

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chains and processes. This would seem to provide us with a convenient stepping stone for the application of catastrophe theory to these types of stochastic systems. Unfortunately, such a catastrophe theoretical analysis has not yet been undertaken. At present, we do not know whether an application of catastrophe theory to the potential analogues of Markov chains and processes will prove to be fruitful. Perhaps alternative approaches may fare better, such as the approximation of denumerable Markov processes by stochastic differential equations to generalize the results presented in section 3. We have concentrated on the representation of homogeneous Markov chains and processes in terms of the transition matrices P and W, respectively. It should be noted that these representations are inherently linear and therefore would seem to be uninteresting for further catastrophe theory analysis in which the focus is on nonlinear potentials. Yet, the latter conjecture does not hold: for instance, Gilmore (1981) shows how the program of catastrophe theory is extended to matrices. More specifically, suppose that the transition rates wij in the master equation (23) are nonlinear functions in i, j and depend on the control variables c. Suppose also that the detailed balance condition holds, implying that the W matrix is symmetric. Then it follows that the stationary probability density of the Markov process is given by the eigenvector of W associated with the largest (zero) eigenvalue (cf. Van Kampen, 1981, p. 126). The components of this eigenvector are nonlinear functions in the states i, j, and will depend smoothly on the control variables in e (Gilmore, 1981). Hence, despite the linearity of the master equation in W, its stationary solution provides the proper setting for further catastrophe theoretical analysis.

5. GENERAL DISCUSSION AND CONCLUSION Catastrophe theory is an advanced mathematical subject, and so is the (potential) theory of stochastic processes. In this chapter, we have considered some forms of integration of these two theories. Of course, this is an enormous task and we could only present a few preliminary remarks about possible approaches. Yet, the task has to be addressed in order to arrive at principled applications of catastrophe theory to the stochastic processes typically observed in the biological and social sciences. As a first contribution to that, we presented two main results, one negative and one positive. The negative result pertains to Cobb's approach to catastrophe theory for metrical stochastic systems, which was shown to be flawed in one rather fundamental respect. The positive result concerns the presentation of principled ways to extend catastrophe theory to metrical stochastic systems, in particular Zeeman's approach. It should be remembered that in this chapter we introduced several assumptions of a restrictive nature. For instance, it was assumed that the processes un-

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der scrutiny are homogeneous and obey the detailed balance condition. Although these assumptions considerably eased the presentation, it remains to be established whether they are necessary and plausible. In fact, this also applies to the basic concept of a potential; we considered its possibly restrictive nature in section 1, but still have to address its interpretation in the context of developmental processes. We could take a formal route and refer to the stationary density of Equation (17): p ( x ) = exp(-~[x]). If ~[x] has a Morse form, then Equation (17) reduces to the (multivariate) Normal density. In the latter case, the potential is proportional to the quadratic form x ' ~ x , where 9 denotes the inverse of the covariance matrix of x, and hence this potential can be interpreted as an analogue of energy. More informally, we can use the interpretation given by Helbing (1994) of the potential in his model for the behavior of pairwise interacting individuals. According to Helbing, this potential can be understood as social field. Analogously, the potential in the stationary density (17) of developmental processes could be interpreted as developmental field. This would correspond nicely with the prominence of field concepts in mathematical biology and embryology (e.g., Meinhardt, 1982). In conclusion, the extension of the program of elementary catastrophe theory to stochastic systems leads to a number of deep and challenging questions which for the most part are still unexplored. The issues at stake not only pertain to the formal framework of catastrophe theory for stochastic systems but also relate to basic questions in developmental theory. For instance, the question of how to arrive at an integral conceptualization of intra- and interindividual differences in the timing and pattern (horizontal and vertical decalages) of discrete stage transitions marking the emergence of qualitatively new behavior. Or the specification of causal models of probabilistic epigenesis. We hope to have made clear that there exist principled approaches to resolve these questions. Thus we expect that further elaboration of the program of stochastic catastrophe theory will lead to fundamental progress in developmental theory.

REFERENCES Castrigiano, D. E L., & Hayes, S. A. (1993). Catastrophe theory. Reading, MA: Addison-Wesley. Cobb, L. (1978). Stochastic catastrophe models and multinomial distributions. Behavioral Science, 23, 360-374. Cobb, L. (1981). Parameter estimation for the cusp catastrophe model. Behavioral Science, 26, 75-78. Cobb, L., Koppstein, E, &Chen, N. H. (1983). Estimation and moment recursion relations for multimodal distributions of the exponential family. Journal of the American Statistical Association, 78, 124-130. Cobb, L., & Zacks, S. (1985). Applications of catastrophe theory for statistical modeling in the biosciences. Journal of the American Statistical Association, 80, 793-802.

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Doob, J. L. (1984). Classical potential theory and its probabilistic counterpart. New York: SpringerVerlag. Florens-Zmirou, D. (1991). Statistics on crossings of discretized diffusions and local time. Stochastic Processes and their Applications, 39, 139-151. Gardiner, C. W. (1990). Handbook of stochastic methods for physics, chemistry and the natural sciences (2nd ed.). Berlin: Springer-Verlag. Gilmore, R. (1981). Catastrophe theory for scientists and engineers. New York: Wiley. Haken, H. (1983). Advanced synergetics. Berlin: Springer-Verlag. Hartelman, E, van der Maas, H. L. J., & Molenaar, E C. M. (1995). Catastrophe analysis of stochastic metrical systems in measure space Amsterdam: University of Amsterdam. Helbing, D. (1994). A mathematical model for the behavior of individuals in a social field. Journal of Mathematical Sociology, 19, 189-219. Huseyin, K. (1986). Multiple parameter stability theory and its applications: Bifurcations, catastrophes, instabilities. Oxford: Clarendon Press. Jackson, E. A. (1989). Perspectives of nonlinear dynamics (Vol. 1). Cambridge, UK: Cambridge University Press. Kemeny, J. G., Snell, J. L., & Knapp, A. W. (1976). Denumerable Markov chains. New York: Springer-Verlag. Meinhardt, H. (1982). Models of biological pattern formation. London: Academic Press. Molenaar, E C. M. (1986). On the impossibility of acquiring more powerful structures: A neglected alternative. Human Development, 29, 245-251. Molenaar, E C. M. (1990). Neural network simulation of a discrete model of continuous effects of irrelevant stimuli. Acta Psychologica, 74, 237-258. Nicolis, G., & Prigogine, I. (1977). Self-organization in nonequilibrium systems. New York: Wiley. Poston, T., & Stewart, I. N. (1978). Catastrophe theory and its applications. London: Pitman. Prigogine, I. (1980). From being to becoming: Time and complexity in the physical sciences. San Francisco: Freeman. Revuz, D., & Yor, M. (1991). Continuous martingales and Brownian motion. Berlin: Springer-Verlag. Thom, R. (1975). Structural stability and morphogenesis. Reading, MA: Benjamin. van der Maas, H. L. J., & Molenaar, E C. M. (1992). Stagewise cognitive development: An application of catastrophe theory. Psychological Review, 99, 395-417. Van Kampen, N. G. (1981). Stochastic processes in physics and chemistry. Amsterdam: NorthHolland. Wohlwill, J. E (1973). The study of behavioral development. New York: Academic Press. Zeeman, E. C. (1988). Stability of dynamical systems. Nonlinearity, 1, 115-155.

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1. INTRODUCTION Sometimes developmental models involve qualitatively different subgroups of individuals, such as learning disabled versus normal learners, children who have been abused versus children who have not, or substance use abstainers versus experimenters versus regular users. Other developmental models hypothesize that individuals go through a sequence of discrete stages, such as the stages in Piaget's theories of development (Piaget, 1952) or Kohlberg's (1969) stages of moral development. Both types of models are characterized by the involvement

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Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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of discrete latent variables. The first models can be tested by means of a measurement model for discrete latent variables, namely, latent class analysis (Clogg & Goodman, 1984; Goodman, 1974; Lazarsfeld & Henry, 1968). Stage sequential models can be tested by means of latent transition analysis (LTA) (Collins & Wugalter, 1992; see also Hagenaars & Luijkx, 1987; van de Pol, Langeheine, & de Jong, 1989), which is an extension of latent class models to longitudinal data. Typically, in latent class and latent transition models, data are analyzed in the form of frequencies or proportions associated with "response patterns." Conceptually, a response pattern is a set of possible subject responses to the manifest variables. For example, suppose the variables are two yes/no items. Then, No,No is one response pattern, No,Yes is another response pattern, and so on. Technically, a response pattern proportion is simply a cell proportion, in which the cell is part of the cross-classification of all the manifest variables involved in the problem of interest. Latent class models usually involve data collected at one time only. For example, let Y = {i, j, k} represent a response pattern made up of response i to Item 1, response j to Item 2, and response k to Item 3. Then, C

P(Y) = ~_~ ~lcPilcPjlcPklc, c=l

where ~/r is the proportion of individuals in latent class c, and Dilc is the probability of response i to Item 1, conditional on membership in latent class c. These parameters are conceptually similar to factor loadings in that they reflect the strength of the relationship between the manifest variables and the latent variable. They are different from factor loadings in that they are estimated probabilities, not regression coefficients. In latent class models, a strong relationship between the latent and manifest variables is reflected in a strong p parameter, that is, a parameter close to zero or one. We refer to the p parameters as measurement parameters. Latent transition models can involve some variables that are measured only once (because they are unchanging, such as an experimental condition) and other variables that are measured longitudinally (because they are expected to change overtime, such as stage membership). Let Y = {m, i,j, k, i',j', k', i",j", k"} represent a response pattern that is made up of a single response to a manifest indicator (m) of an exogenous variable and responses to three items at times t (i, j, and k), t + 1 (i', j', and k'), and t + 2 (i", j", and k"). Then, the estimated proportion of a particular response pattern P(Y) is expressed as follows for a first-order model): C

S

S

S

c=l p=l

q=l

r=l

v(r) = s s s s

~ c P m l c g p I cD i l p, cPj l p, cPk l p, c'r q l p, cP i ' I q, cPj' I q, cDk' q,c~r r l q, cP i" l r,cOj" l r,cPk" I r,c,

6. Latent Class and Latent Transition Parameters

135

where "Vc is the proportion in latent class c. In this model, individuals do not change latent class membership over time. The factor 8p Ic is the proportion in latent status p at time t conditional on membership in latent class c. This array of parameters represents the distribution across latent statuses at the first occasion of membership. Latent status membership can and does change over time. The factor Tqlp, c is an element of the latent transition probability matrix. These parameters represent how the sample changes latent status membership over time. The factor Pilp, c is the probability of response i to item 1 at time t, conditional on membership in latent status p at time t and on membership in latent class c. These measurement parameters are interpreted in the same way as their counterparts in latent class models. The purpose of this chapter is to address two practical issues related to parameter estimation in latent class and latent transition models. These practical issues arise frequently in developmental research. The first issue has to do with parameter estimation when the number of subjects is small in relation to the number of response patterns. As latent class and latent transition models become used more extensively, it is inevitable that researchers will attempt to test increasingly complex models. In our work, for example, we test models of early substance use onset in samples of adolescents. These models often involve four or more manifest variables used as indicators, measured at two or more time points. The cross-classification of all of these manifest variables measured repeatedly can become quite large. For a problem involving four dichotomous variables measured at two times, there are 28 - 256 cells in the cross-classification. If there are more than two occasions of measurement, if the items have more than two response options, and/or if there are additional items, the table becomes much larger. The size of the cross-classification is not a problem in and of itself, but it often indirectly causes a problem when the sample size is small in relation to the number of cells, that is, under conditions of "sparseness." There is a general, informal consensus that a minimum sample size in relation to the number of cells (N/k) is needed to perform latent class analyses successfully. However, to our knowledge, there are no firm guidelines, either about what the minimum N or N/k should be or about what are the likely consequences if one performs analyses on data that are too sparse. The source of the concern about sparseness is partly the well-established literature on the difficulties of conducting hypothesis testing under sparseness conditions. This literature has shown in numerous settings (see Read & Cressie, 1988) that when the data matrix is sparse, the distributions of commonly used goodness-of-fit indices diverge considerably from that of the chi-squared, making accurate hypothesis testing impossible. Collins, Fidler, Wugalter, and Long (1993; see also Holt & Macready, 1989) showed that under conditions of sparseness in latent class models, the expectation of the likelihood ratio statistic can

136

L. M. Collins, P. L. Fidler, and S. E. Wugalter

be considerably less than the expectation of the chi-square distribution, and that although the expectation of the Pearson statistic is much closer to that of the chi-square, its variance is unacceptably large. Collins et al. (1993) recommended using Monte Carlo simulations to estimate the expectation of the test statistic under the null hypothesis. The demonstrated problems with hypothesis testing under sparseness conditions have left researchers uneasy about parameter estimation under these conditions. It is not known what the effects of sparseness are, if any, on parameter estimation. One possible consequence of sparseness is bias, that is, the expectation of the parameter estimates could be greater than or less than the parameter being estimated. Another consequence could be an unacceptably large mean squared error (MSE), indicating a lack of precision in parameter estimation. Also, it is not known what the limits of sparseness are, that is, how small can the number of subjects be in relation to the number of cells if parameters are to be estimated with acceptable accuracy? Another important question related to sparseness is whether, given a fixed N, adding a manifest indicator is a benefit, because it increases the amount of information, or a detriment, because it increases sparseness. The second practical issue addressed by this chapter is the estimation of standard errors for parameters. Estimation for latent class and latent transition models is usually performed using the EM algorithm (Dempster, Laird, & Rubin, 1977). Less often, estimation is performed using Fisher's method of scoring (Dayton & Macready, 1976). The EM algorithm has the advantage of being a very robust estimation procedure. Its disadvantages are that it is slow compared with Fisher's method of scoring, and that it does not yield standard errors of the parameters it estimates. Unlike the Fisher scoring method, the EM algorithm does not require computation of the information matrix, hence standard errors are not a by-product of the procedure. It has been suggested (e.g., Rao, 1965, p. 305) that standard errors can be obtained by inverting the information matrix computed at the final iteration of the EM procedure. This sounds reasonable, but just how well this work has never been investigated. It also remains to be seen how well standard errors can be estimated in very sparse data matrices.

2. METHODS To investigate the issues presented in the first section, we performed a simulation study. The purpose of the simulation was to generate data based on latent class models with known parameters while systematically varying sparseness and other factors. We estimated the parameters in the data we generated and examined whether the factors we varied affected parameter estimation.

6. Latent Class and Latent Transition Parameters

137

The following independent variables were selected as likely to affect the accuracy of estimation: 1. Number of items. We used four-, six-, and eight-item models. Because dichotomous items were used, this resulted in data sets with 16, 64, and 256 response patterns, respectively. 2. Size of p. We reasoned that the size of p might be an important factor because all else being equal, latent class models with extreme values (close to zero or one) for the p parameters generate sparser data matrices than latent class models with less extreme p parameters. This factor had two levels: p = .65 and 9=.9. 3. Sparseness. This was operationalized as N/k, that is, the number of subjects divided by the number of cells (in this case, response patterns). Four levels of N/k were crossed with the other factors in the study: 1, 2, 4, and 16, with N/k of 1 being a level that many researchers would feel is too low and N/k of 16 being a level that most researchers would agree is "safe." Thus the design as discussed here involves the following fully crossed factors: four levels of N/k, three levels of number of items, and two levels of size of p, or 24 cells. To investigate extreme sparseness, four more cells were run with N/k = .5. One cell had six items, p = .65; one had six items, p = .9; one had eight items, p = .65; and one had eight items, p = .9. We did not cross N/k = .5 fully with number of items, because the four item conditions would have involved only eight subjects. Table 1 shows the design, including the number of artificial "subjects" in each data set, computed by multiplying N/k times the number of response patterns. As Table 1 shows, the design was set up so that there would be multiple cells with the same N in order to make it possible to investigate the effects of adding items while holding N constant. TABLE 1 Design of Simulation and Numbers of Subjects Used to Generate Data N/k .5

1

2

4

16

N of items

p = .65

4 6 8

* 32 128

16 64 256

32 128 512

64 256 1024

256 1024 4096

p = .90

4 6 8

* 32 128

16 64 256

32 128 512

64 256 1024

256 1024 4096

*Cell not included in design.

138

L. M. Collins, P. L. Fidler, and S. E. Wugalter

One thousand random data sets were generated for each cell. Parameters of the known true model were estimated in each data set using the EM algorithm (Dempster et al., 1977). All of the data sets were generated using models with two latent classes, so one ~ parameter was estimated in each data set. In the four-item data sets, all eight p parameters were freely estimated. To keep the total number of parameters estimated constant across data sets, constraints were imposed on the p parameters in the six- and eight-item conditions. In the sixitem models, the p parameters associated with the fifth and sixth items were constrained to be equal to those for the first and second items, respectively. In the eight-item models, the p parameters associated with the fifth through eighth items were constrained to be equal to those associated with the first through fourth items, respectively. Thus each data set involved estimation of one -,/parameter and eight p parameters.

1.1 Data Generation There is a finite set of possible response patterns that can be generated for a given set of items, and each latent class model produces a corresponding vector of predicted response pattern probabilities. We used a total of six vectors associated with the three numbers of items and the two measurement strengths. A single vector was used to generate data for each of the four levels of N/k. The data were generated using a uniform random number generator written in FORTRAN. Data were generated for a single subject by randomly selecting a number from zero to one from the uniform distribution. This number was then compared with the cumulative response pattern probability vector which determined a particular subject's response pattern. A mastery model was used to generate the data in which one of the latent classes in each model was a "master" latent class, where the probability of passing the item was large for each item, and the other was a "nonmaster" latent class, where the probability of passing was small for each item. Of course, in individual data sets the parameter estimates did not always come out this way. In a few instances, the two latent classes did not seem distinct at all, so that it was difficult or impossible to denote one class as a mastery class and one as a nonmastery class. An example is a solution where the probability of passing each item is estimated as .9 for one latent class and .6 for the other. Under these circumstances, it cannot be determined whether a p parameter is conditional on membership in the mastery latent class or on membership in the nonmastery latent class. Because the true values for the p parameters that are conditional on the mastery latent class are different from those conditional on the nonmastery latent class, when this occurs it is impossible to compare the parameter estimates to their true values, and bias and MSE cannot be assessed. We examined each replication to identify instances of this indeterminacy. Table 2 shows the number of these solutions in each cell. Most of the indeter-

139

6. Latent Class and Latent Transition Parameters

TABLE2

Number of Indeterminate Solutions Out of 1000 Replications Per Cell N/k

p = .65

p = .90

.5

1

2

4

16

N of items 4 6 8

* 34 1

87 19 0

59 7 0

35 1 0

0 0 0

4 6 8

* 0 0

0 0 0

0 0 0

0 0 0

0 0 0

*Cell not included in design. minate solutions occurred when p = .65 and there were four items, with the largest number (87) occurring when N/k = 1, which was the sparsest matrix included for four items. There were no such solutions when p = .9, and none when N/k = 16. There was only one instance of indeterminacy when there were eight items, occurring when N/k = .5 and p = .65. Data sets yielding indeterminate solutions were r e m o v e d from the simulation and new random data sets were generated until a total of one thousand suitable solutions was obtained for each cell. The remaining results are based on this amended data set.

1.2. Parameter Estimates We evaluate the parameter estimates in terms of bias and MSE. Table 3 shows bias and M S E for the ~/parameter, and Table 4 shows these quantities for one of the p parameters. The pattern of results was virtually identical for all eight p parameters, so only one set of results is shown here. In the population from which these data were sampled, the latent class parameter equals .50 and the P parameter equals either .65 or .90. There is some bias present in the sample-based estimates of both types of parameters, particularly when p = .65. The overall amount of bias is quite small for the latent class parameter; in no cell is it larger than .026. There is considerably more bias in the estimation of the P parameters, where bias is more than .03 in six cells. For both parameters, bias virtually disappears when p = .90, even for conditions where N/k is as low as .5. Bias generally decreases as N/k increases and as the number of items increases, although there is some slight nonmonotonicity in the bias associated with the latent class parameter. Because certain groups of these parameters sum to one and the parameters are bounded by zero and one, posi tive bias in the estimation of one parameter in a set implies negative bias somewhere else in the set, and vice versa. For example, the 9's associated with a

Cr~

c~

t~

0

c~

~

0

0

~

II

c~

CD CD ,-.I

::j

ml a,1

CD

w-.i,,

0.,,,.k

C~

Dhl t,,.i.

I'rl C~

C~ ,m.

.,m.

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r~ 0

c~

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rrl

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C~L

s~

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141

6. Latent Class and Latent Transition Parameters

particular item and a particular latent class must sum to one across response categories. This means that although the bias in Table 4 is positive overall, if we e x a m i n e d the c o m p l e m e n t of this parameter, we would find that the bias is negative overall. Although there can be a substantial amount of bias when N/k is small, holding N/k constant and increasing the n u m b e r of items generally tends to decrease the bias. (Note that to maintain a constant N/k, the overall N must be increased when more cells are added.) This pattern is not completely consistent for the latent class parameter. Essentially the same pattern holds for the MSE. The M S E is much smaller overall for both the latent class parameter and the p parameter when p = .90 than it is when p = .65. The M S E generally decreases as N/k increases, and as the n u m b e r of items increases, although there are some inconsistencies evident in the M S E s for the latent class parameter. Even with N/k = .5, the M S E is small when eight items are used. G i v e n a particular N, is there an increase in bias or M S E when items are added to a latent class model? Tables 5 and 6 show that as the n u m b e r of items

TABLE 5

Bias in Estimation of the ~ Parameter as a Function of N N of items 4

N of subjects 32 64 p = .65

256

.026 (.041) .008 (.051) .003 (.058)

1024

32 64 p = .90

256

.000 (.OO9) .001 (.005) -.000 (.001)

1024

6

.016 (.047) .024 (.054) -.001 (.029) .002 (.OO7)

-.002 (.015) -.000

(.003)

-.000

(.OO8) .003 (.004) -.001 (.OOl) -.000

(.000)

Note. Mean squared errors are given in parentheses.

8

-.001 (.o01) -.ooo

(.00o)

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L. M. Collins, P. L. Fidler, and S. E. Wugalter

TABLE 6

Bias in Estimation of p Parameter As a Function of N N of items 4 N of subjects 32 64

p = .65

256

.053 (.050) .043 (.039) .O38 (.020)

.040 (.052) .031 (.035) .010 (.009) .002 (.OO2)

-.002 (.008) .001

.003 (.006) .002

1024

32 64 p = .90

256

6

(.003)

(.003)

-.001 (.001)

-.000 (.OO1) -.001 (.000)

1024

8

.007 (.005) .002 (.001)

-.003 (.OO1) .000 (.000)

Note. Mean squared errors are given in parentheses.

increases for a fixed N, bias and MSE sometimes do increase slightly, at least when the overall N is 64 or less. Under conditions when N is larger than this, generally bias and MSE remain about the same or even decrease slightly as the number of items is increased.

1.3. Estimation of Standard Errors Each cell of the simulation contains a sampling distribution of each parameter estimated. This sampling distribution is made up of the 1000 replications in each cell. The standard deviation of a parameter estimate across the 1000 replications is our definition of the true standard error of the parameter. To estimate the bias and MSE associated with estimating the standard errors, we subtracted this standard deviation from the estimate of the standard error obtained by inverting the information matrix at each replication. Table 7 contains bias and MSE for the estimation of the standard error of the latent class parameter, and Table 8 contains this information for the estimation of the standard error of the p parameter.

6. Latent Class and Latent Transition Parameters

143

TABLE 7 Bias in Estimation of the Standard Error of the ~/Parameter N/k

N of items 4

1

2

4

16

*

.284 (.714) -.035 (.022) -.023 (. 001 )

.167 (.131) -.066 (.011) -.005 (. 000)

.146 (.150) -.037 (.004) -.004 (. 000 )

-.023 (.015) -.004 (.000) .000 (. 000 )

.222 (.979) .000 (.000) -.001 (.000)

.011 (.004) -.001 (.000) .000 (.000)

-.003 (.000) .000 (.000) .000 (.000)

.001 (.000) -.000 (.000) .000 (.000)

.018 (.039) -.049 (. 004 )

p = .65

p = .90

.5

4

*

6

-.002 (.000) -.000 (.000)

8

Note. Mean squared errors are given in parentheses. *Cell not included in design.

TABLE 8

Bias in Estimation of the Standard Error of the p Parameter N/k

N of items 4

1

2

4

16

*

.056 (.157) -.016 (.026) -.009 (.000)

.057 (.089) -.023 (.009) -.003 (.000)

.057 (.104) -.014 (.001) -.001 (.000)

-.003 (.013) -.001 (.000) .001 (.000)

.086 (.366) -.002 (.000) -.000 (.000)

-.008 (.007) -.002 (.000) .000 (.000)

-.005 (.000) .000 (.000) .000 (.000)

.000 (.000) .000 (.000) .000 (.000)

-.016 (.037) -.020 (.001)

p = .65

p = .90

.5

4

*

6

-.013 (.OO2) -.002 (.000)

8

Note. Mean squared errors are given in parentheses. *Cell not included in design.

144

L. M. Collins, P. L. Fidler, and S. E. Wugalter

Tables 7 and 8 show that although generally the standard errors are estimated well, there is considerable bias in the estimation of the standard error for the latent class parameter when p -- .65, there are four items, and N/k is low. Under these conditions, inverting the information matrix tends to produce a standard error that is larger than the standard deviation of the empirical sampling distribution, particularly for the three lowest values of N/k. The MSE is also large in these conditions, especially when N/k = 1. For both the latent class and p parameters bias tends to be positive in the four-item conditions and negative in the six- and eight-item conditions. Inverting the information matrix produces highly accurate estimates of the standard error in almost all of the p = .9 conditions. The one exception is the four-item, N/k = 1 condition where there is considerable positive bias and the MSE is unacceptably large. The overall pattern of the results is the same for the p parameters, except that there is less bias in the estimate of the standard error and the MSEs are smaller in most cases.

3. DISCUSSION The results of this study are very encouraging for the estimation of latent class models. They suggest that highly accurate estimates of latent class parameters can be obtained even in very sparse data matrices, particularly when manifes titems are closely related to the latent variable. Even when N/k was .5 or 1, bias was negligible for estimating the latent class parameter and only slight for estimating the p parameters. Both bias and MSE were smallest when p = .9, indicating that in circumstances in which the items are strongly related to the latent classes, a smaller N is needed for good estimation compared with circumstances in which the items are less strongly related to the latent classes. Usually, a researcher has only so much data to work with, that is, a fixed N, and is debating how many indicators to include. With measurement models for continuous data, there is usually no debate; more indicators are better. But with latent class and latent transition models, the researcher is faced with a nagging question: Will adding an indicator be a benefit by increasing measurement precision, or will it be a detriment by increasing sparseness? The results of our study suggest that when the p parameters are close to zero or one, adding additional items has little effect on estimation. When the p parameters are weaker, adding items generally decreases the MSE for overall Ns of 256 or greater, and may increase it slightly for smaller Ns. The effect of the addition of items on bias is less consistent, but generally small. It is important to note that in the design of this simulation, we used constraints so that when additional items were added the total number of parameters estimated remained the same. This amounts to treating some items as replications of each other; conceptually, it is similar to constraining factor loadings to be equal in a confirmatory factor analysis. With-

6. Latent Class and Latent Transition Parameters

145

out such constraints, as items are added, more parameters are estimated. If the addition of items is accompanied by estimation of additional parameters, this may change the conclusions discussed in this paragraph. This study also indicates that standard errors of the parameters in latent class models can be estimated well in most circumstances by inverting the information matrix after parameter estimation has been completed by means of the EM algorithm. However, there can be a substantial positive bias in the estimate of the standard error, particularly when p is weak and N/k is small. This approach to estimation of standard errors probably should not be attempted when N/k is one or less or there are four or fewer manifest indicators. Serendipitously, this study also revealed a little about indeterminate results, that is, results for which the latent classes are not clearly distinguished. We should note that such results are not indeterminate if they reflect the true model that generated the data. However, in our study, the data generation models involved clearly distinguished latent classes. We found that, again, strong measurement parameters were very important; none of the indeterminate cases occurred when p = .9. It was also evident that more items and a greater N/k helped to prevent indeterminate solutions. It is interesting that strong measurement showed itself to be so unambiguously beneficial in this study. On the other hand, p parameters close to zero or one are analogous to large factor loadings in factor analysis, clearly indicating a close relationship between manifest indicators and a latent variable. On the other hand, all else being equal, p parameters close to zero and one are also indicative of more sparseness. Given a particular N, the least sparse data would come from subject responses spread evenly across all possible response patterns. When the p parameters are such that some responses have very high probabilities and others have very low probabilities, subject responses will tend to be clumped together in the high-probability response patterns, whereas the low-probability response patterns will be empty or nearly empty. For this reason, it might have been expected that strong measurement would tend to result in more bias or larger MSEs. This simulation shows that the sparseness caused by extreme measurement parameters is unimportant. Like all simulations, this one can only provide information about conditions that were included. We did not include any conditions where there were more than two latent classes. As mentioned previously, we controlled the number of parameters estimated rather than let the number increase as more items were added. However, in many studies, researchers will wish to estimate p parameters freely for any items they add to a model, which will result in an increase in the total number of parameters to be estimated. It would be worthwhile to investigate the effects of this increased load on estimation. Finally, we did not mix the strengths of the measurement parameters. Each data set contained measurement parameters of one strength only. Of course, in empirical research set-

146

L. M. Collins, P. L. Fidler, and S. E. Wugalter

tings different variables would be associated with different measurement strengths, so the effect of this should be investigated also.

ACKNOWLEDGMENTS This research was supported by National Institute on Drug Abuse Grant DA04111.

REFERENCES Clogg, C. C., & Goodman, L. A. (1984). Latent structure analysis of a set of multidimensional contingency tables. Journal of the American Statistical Association, 79, 762-771. Collins, L. M., Fidler, P. L., Wugalter, S. E., & Long, J. D. (1993). Goodness-of-fit testing for latent class models. Multivariate Behavioral Research, 28, 375-389. Collins, L. M., & Wugalter, S. E. (1992). Latent class models for stage-sequential dynamic latent variables. Multivariate Behavioral Research, 27, 131-157. Dayton, C. M., & Macready, G. B. (1976). A probablistic model for validation of behavioral hierarchies. Psychometrika, 41, 189-204. Dempster, A. P., Laird, N. L., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1-38. Goodman, L. A. (1974). Exploratory latent structure analysis using both indentifiable and unidentifiable models. Biometrika, 61, 215-231. Hagenaars, J. A., & Luijkx, R. (1987). LCAG: Latent class analysis models and other loglinear models with latent variables: Manual LCAG (Working Paper Series 17). Tilburg, Netherlands: Tilburg University, Department of Sociology. Holt, J. A., & Macready, G. B. (1989). A simulation study of the difference chi-square statistic for comparing latent class models under violations of regularity conditions. Applied Psychological Measurement, 13, 221-231. Kohlberg, L. (1969). Stage and sequence: the cognitive-developmental approach to socialization. In D. A. Goslin (Ed.). Handbook of socialization theory and research. Chicago: Rand McNally. Lazarsfeld, P. F., & Henry, N. W. (1968). Latent structure analysis. Boston: Houghton Mifflin. Piaget, J. (1952). The origins of intelligence in children. New York: International Universities Press. Rao, C. R. (1965). Linear statistical inference and its applications. New York: Wiley. Read, T. R. C., & Cressie, N. A. C. (1988). Goodness-of-fit statistics for discrete multivariate data. New York: Springer-Verlag. van de Pol, F., Langeheine, R., & de Jong, W. (1989). PANMARK User Manual: PANel analysis using MARKov chains. Voorburg: Netherlands Central Bureau of Statistics.

Contingency Tables and Between-Subject Variability Thomas D. Wickens

University of California, Los Angeles Los Angeles, California

1. INTRODUCTION An important problem in the analysis of contingency tables is how to treat data when several observations come from each subject. How should these dependent responses be combined into a single analysis? The problem is, of course, an old one; I found reference to the effects of heterogeneity of observations going back at least to Yule's textbook in 1911. The dangers of pooling dependent observations in the Pearson statistic are well known, although many researchers are not really sure what to do about it. For example, Lewis and Burke pointed the problem out in 1949 in their discussion of the use and misuse of chi-square tests, and yet Delucchi (1983), in his reexamination of the outcome of their recommendations, noted that errors are still too common. The introduction of new statistical techniques for frequency data has both helped and hurt. On the one hand, we now have both the procedures to represent and analyze the between-subject component of variability in these designs and the computer power needed to implement these procedures. On the other hand, the variety of new techniques, and the incomplete understanding many users have of them, has partially obscured the problem. For example, I have

Categorical Variables in Developmental Research: Methods of Analysis Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

147

148

Thomas D. Wickens

spoken to several researchers who believe that using "log-linear models" somehow eliminates the problems of combining over subjects. However, this is not really the case. The issue was brought to my attention some time ago when a colleague asked me to look at a paper he was reviewing and to comment on the statistics. The data in question contained results from a group of subjects with several observations per subject, and thus involved a mixture of between-subject and withinsubject variability. They had been analyzed with standard log-linear techniques. The analysis bothered me, and eventually I identified the problem as a failure to treat the between-subject variability correctly. I also realized that the issue was more general than this particular instance, and that many treatments of categorical designs for researchers, my own included (Wickens, 1989), did not discuss how to analyze such data. In particular, they gave no practical recommendations for the everyday researcher. Thinking about the problem, and about what to recommend, led me to explore several approaches to this type of analysis (reported in Wickens, 1993). Here I describe, and to some degree expand upon, that work. My thoughts are influenced both by the statistical problems and by the need to make recommendations that are likely to be used.

2. ASSOCIATION VARIABILITY To begin, remember that there are two common situations in which we are called on to test for association between two categorizations. In one, each subject produces a response that is classified in two dimensions, producing one entry in a contingency table. In the other situation, there are two or more types of subjects, each generating a response that is classified along a single dimension. Again, a two-way table results, although the underlying sampling models are different. Essentially, the difference between the two sampling situations is the same as that between the correlation model and the regression model. I will concentrate on the first situation here, but will mention the second situation again later. ~ Consider the following example, loosely adapted from the design that called the problem to my attention. Suppose that you are interested in determining whether there is a link between the behavior of a caregiver and that of an infant. You want to know whether a certain caregiver behavior is followed by a particular target behavior of the infant. You observe the interaction of a caregiver-infant dyad for some fixed duration. You divide this period into a se-

IThere is a third possibility in which the design constrains both marginal distributions--Fisher's (1973) lady testing tea--but is less often used and I will not discuss it here.

149

7. Between-Subject Variability

ries of short intervals and select certain pairs of intervals, say one pair every 5 min, for examination. In the first of any pair of two consecutive intervals, you record whether the caregiver made the antecedent behavior and in the second whether the infant made the target behavior. Aggregating over the pairs, you obtain a 2 • 2 table that describes the contingency of the two behaviors: Infant Yes No

I

Caregiver NoYeS 4213 2817

I

Each count in this table records the outcome from one pair of intervals. You can discover whether the behaviors are contingent by looking for an association between the row categorization and the column categorization. For a single dyad, these data are easy to analyze. You just test for an association in the 2 • 2 table. There are many ways to conduct this test, all of which have similar properties. I do not consider their differences here; an ordinary Pearson statistic suffices. Of course, you would not normally run the study with only one dyad. You would obtain data from several caregiver-infant pairs and want to generalize over these pairs. Table 1 shows some data, simulated from the model that I describe in the next section. These data form a three-way contingency table, the dimensions of which are the antecedent behavior, the target behavior, and the subject (classification factors A, B, and S, respectively). The goal of an analysis is either to test the hypothesis that there is no association between the behavior classifications or to estimate the magnitude of this association. Two sources of variability influence these observations, one occurring within a dyad, the other between dyads. The within-dyad source is embodied in the

TABLE 1 Simulated Data for Eight Dyads Responses

Log odds ratio

Pair

YY

YN

NY

NN

Yk

sk2

1 2 3 4 5 6 7 8

42 44 42 29 42 24 38 45

17 30 9 29 15 24 24 16

13 12 34 23 24 27 15 19

28 14 15 19 19 25 23 20

1.634 0.526 0.698 -0.187 0.780 -0.076 0.868 1.064

0.190 0.204 0.222 0.162 0.180 0.157 0.174 0.183

The estimated log odds ratios Yk and their sampling variance s2 are obtained from Equations (1) and (2).

Note.

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scatter of the responses within each 2 x 2 table; the between-dyad source is embodied in the differences among these tables. Some between-dyad scatter is expected because of within-dyad response variability, but there is more variability among the observations in Table 1 than can be attributed to such within-dyad influences. Considered in terms of these sources of variability, there are three ways that one might test a hypothesis of independence or unrelatedness. The most general approach is to find a method that explicitly uses estimates of both the betweensubject and the within-subject components of the variability. Presumably, the tests in this class should be the most accurate. However, such tests are relatively complicated and not well incorporated into the conventional statistical packages. Their complexity creates a problem in their application. Many researchers and some journal editors pay little attention to new statistical methods unless a very strong case can be made for them, preferring to fall back on more familiar procedures. Therefore it is also important to look at tests based on simpler variability structures. The more familiar types of tests are based on estimates of variability of only one type, either between-subject or within-subject. If there is much betweensubject variability, then the class of tests that estimate and use it should be the next best alternative, although they might be inferior to the tests that use both sources. Tests that use only the within-subject variation are likely to be the least accurate. Although these notions make sense, I was unclear of the size and nature of the biases and of the relative advantages of each class. The simulations I describe in this chapter address these similarities and differences. My goal is to make some practical recommendations.

3. THE SIMULATION PROCEDURE To look at the characteristics of the different testing procedures, I simulated experiments with various amounts of between-subject and within-subject variability, applied a variety of test statistics to them, and looked at the biases and power of these tests. The first step in this program was to create a model for the data that included both components of variation. The model I used has between-subject and within-subject portions. The within-subject portion is conventional. The kth subject (or dyad) gives a 2 • 2 table of frequencies Xok. In a design in which each subject gives a doubly classified response, these frequencies are a multinomial sample of in k observations from a four-level probability distribution with probabilities "rr0k. The multinomial distribution is, of course, a standard model for categorical data analysis. The between-subject portion of the model provides the probabilities ~ijk and the number of observations m k. The probabilities derive from a set of parame-

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151

ters that are tied to the processes being modeled. A 2 • 2 table of probabilities "rrij can be generated by the two marginal distributions and the association. Specifically, the marginal distributions are expressed by a pair of marginal logits ~l - log "rrll -I- 'rrl2

and

~2 = log Trll nt- 'rr21,

"1721 -Jr- "ri'22

'7r12 + ,rt'22

and the association by log odds ratio: xI - log 31"11'13"22. 'IT 12"rr21

In the example, the parameters ~l and ~2 m e a s u r e how likely the caregiver and infant are to emit the antecedent and target behaviors, respectively. They are essentially nuisance parameters in that example, although in other research their relationship is the principal focus of the work. As in many studies of association, one wants to make an inference about xl, either estimating it or testing the null hypothesis that it equals zero. The between-subject portion of the model specifies the distribution of these quantities over the population of subjects. I represented them as sampled from normal distributions, with unknown means and variances: ~l "~ ~'t/'(~L~,,

0"~1)'

~2 "-" ~ I/'(~2, 0"~2), and

11 "~ ~l]~lxn, (r2).

Properly speaking, the triplet (~l, ~2, ~) should have a trivariate distribution to allow intercorrelations, but I will not follow up on this multivariate aspect. For one thing, the values of the correlations depend closely on the particular study being modeled and I wanted to be more general; for another, I suspect that their effects are secondary to those that 1 discuss here. In a given experiment, n subjects are observed, each corresponding to an independently sampled triplet of distribution parameters. Over the experiment, there are n of these triplets (~lk, ~2k, ilk), which I have shown by adding a subscript for the subject and placing a tilde above the symbol, implying that, from the point of view of the study as a whole, they are realizations of random variables and not true parameters. For any given experiment, the mean fi of the log odds ratio is not, except accidentally, equal to the mean population association; typically,

n

The three subject parameters ~lk, ~2k, and fik determine a set of probabilities ~rij for the four events. The other portion of the model concerns the item sample sizes m k. In most of my simulations m~, was fixed, as if it were set by the experimental procedure.

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In the example, this situation would occur if the dyads were sampled at regular intervals for a fixed period of time. A few simulations let m k vary. Suppose that instead of taking observations at fixed times, every instance of a particular type of caregiver behavior was classified, along with the subsequent behavior of the infant. The number of observations would then depend on the caregiver's behavior rate. I modeled this situation by assuming that instances of the behavior were generated by a Poisson process (like the multinomial, a standard sampling model) and that the rate parameter of this process varied over subjects according to a gamma distribution. This mixture model implies that m k has a negative binomial distribution (e.g., Johnson, Kotz, & Kemp, 1992). Both the mean and the variance of this distribution must be chosen. This model was the basis of the simulations. Each condition was based on values of the mean and variance for the distributions of ~k, ~2k, and xlk, and on the value of n and the distribution of m k. For each of the n subjects in the "experiment," I sampled an independent random triplet (~k, ~2k, ilk) according to the relevant normal distributions and converted it to cell probabilities ~ijk. If m k was random, I generated its value from a negative binomial distribution, and for either the fixed or random case, I generated m~ multinomially distributed responses. I then analyzed these data using a variety of test statistics, some based only on the within-subject variability, some only on a between-subject estimate of variability, and some that separated the two sources. To get stable estimates of such characteristics as the probability of rejecting a null hypothesis, I replicated the "experiment" between 2,000 and 10,000 times, depending on the particular part of the study, and calculated the observed proportion of null hypotheses rejected at the 5% level. Over the course of the study, I varied several characteristics of the experiments. Two of these characteristics concerned properties of the experiment: the fixed sample size n and the (possibly random) observation sample size m~. Two other characteristics concerned the ancillary characteristics of the subject population: the degree to which the marginal distribution of responses was skewed toward one alternative and the amount of between-subject variability. Finally, I varied the size of the association effect, so that I could investigate both Type I error rates and power. I also looked at some studies with more than two response categories. I will not describe all of the simulations and their details here, nor will I mention every statistic that I tried. Some of these are reported in Wickens (1993), others I have not reported formally but they are essentially similar to the ones I describe here.

4. TESTS BASED ON MULTINOMIAL VARIABILITY The most familiar class of test statistics contains tests that are based on the within-subject multinomial variability alone. The simplest procedure is to col-

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153

lapse the data over the subjects (obtaining the marginal distribution of Table 1) and test for association in the resulting two-way table. This approach is unsatisfactory for several reasons. First, it violates the rules about the independence of observations in "chi-square tests." Second, collapsing the three-way table into its margin can artificially alter the association if, as is likely, the relative frequencies of the two behaviors are related to the subject classification--this is the effect that in its more pernicious manifestation gives Simpson's paradox. This test also shows all of the same biases as the other tests in this class, so I will not linger on it, but will turn instead to more plausible candidates. The second test procedure is a likelihood-ratio test based on a pair of hierarchical log-linear models fitted to the three-way A by B by S table. One model expresses the conditional independence of the responses given the subject; in the bracket notation that denotes the fitted marginals it is [AS][BS], and in loglinear form it is log [,LijCIk-- ~k + ~kA + ~kB + ~kS + ~k~kS + ~kjBS. This model allows for relationships between the subjects and the behavior rates, but excludes any association between the two behaviors. The second model is of no interaction in the association. It adds to the conditional-independence model a parameter that expresses a constant level of association; in bracket notation it is [AS][BS][AB], and in log-linear form it is log [.Lij N] --- ~k -~- h A + ~B + ~k~k"_~_ ~kAB ..[_ ~kiAS _~_ ~kjBS. These models are hierarchical, and the second perforce fits the data better. The need for an association term is tested with a likelihood-ratio test: ,, N I

AG~

~-

2 ce~llsXij k log ~[.Lijk ,,CI

----

G[AS][BSI -- G[ABI[AS][BS]' 2 2

where the ~ are fitted frequencies. This statistic is referred to a chi-square distribution with 1 df Because this test explicitly includes the subjects as a factor in the analysis, it appears to solve the problems associated with their pooling. The likelihood-ratio test is appropriate only when the more general model adequately fits the data, so some authorities recommend testing the no-interaction model [AB][AS][BS] first and only proceeding to the difference test when it cannot be rejected. Retention of the no-interaction model implies that there are few differences in association among the subjects. The recommendation to use this test as a gatekeeper is not very useful, however. For one thing, the goodness-offit test is neither easy to run nor particularly powerful when the per-subject frequencies mk are small. More seriously, it leaves the researcher stranded when homogeneity of association is rejected. In practice, I think the recommendation is ignored more often than followed. The third test in this class approaches the problem through an estimate of

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the log odds ratio. Any of several estimates can be used here. I describe one based on Woolf's average of the log odds ratios (Woolf, 1995) in some detail because I will be using an extension of it here; another one, which I do not describe, uses the Mantel-Haenszel estimate (Mantel, 1963). For a single subject, the log odds ratio TIk is estimated by the ratio

( ')(,) (xl2~+~1)(l) x21k+~ X l l k -Jr- "~

Yk = log

X 2 2 k "Jr" "~

.

(1)

The 1 added to each frequency reduces bias and prevents unseemly behavior when an observed frequency is zero (Gart & Zweifel, 1967). With multinomial sampling, the sampling variance of Yk approximately equals the sum of the reciprocals of the cell frequencies, again with the 1 adjustment: 2

1

n t-

S k -= Xll k +

1

1

+

+ ~

1

(2)

1

1

1

1

--

X 1 2 k -q- _

X21 k "-1- _

X 2 2 k nt- _

2

2

2

2

Table 1 gives these estimates for the individual subjects. Aggregation over subjects uses a weighted average with weights equal to the reciprocal of the multinomial sampling variability: Yw

"~wky k s k

with

wk

1 sk2"

(3)

The sampling variance of an average that uses these weights is the reciprocal of the sum of the weights: 1

var(~w) = Ew k. The hypothesis that the population value corresponding to Yw equals zero is tested by the ratio of Yw to its standard error, or equivalently by referring the ratio X2 =

~2 varfPw)

(4)

to a chi-square distribution with 1 df Test statistics in this class are unsatisfactory in the presence of betweensubject variability. Figure 1 shows the proportion of null hypotheses rejected at the nominal 5% level as a function of the association standard deviation for experiments with n = 10 subjects, m = 100 observations per subject, and marginal distributions that were nearly symmetrical (tx~ = 0.3 and o'~ = 0.2, giving ap-

7. Between-Subject Variability

155

FIGURE 1. Type I error rates at the nominal 5% level for tests based on multinomial variability. Subject sample size n = 10, fixed response sample size mk = 100, and near-symmetrical marginal distributions. Each point is based on 10,000 simulated experiments.

proximately a 5 7 : 4 3 distribution in the marginal categories). If the tests were unbiased, then these lines would lie along the 5% line at the bottom of the figure. Instead, the proportion of rejected hypotheses increases with the intersubject variability. The bias is substantial when the association varies widely among subjects. One might now ask how the biases in Figure 1 depend on the sample size. In particular, conventional lore about sample sizes would suggest that large samples of subjects or items would reduce or eliminate the bias. Unfortunately, this supposition is wrong, as Figure 2 shows for the statistic AG28 (the other statistics are similar). The bias of the test increases (not decreases) with the number of observations per subject and is essentially unaffected by the number of subjects. The problems with these tests are too fundamental to be fixed by larger samples.

FIGURE 2. Type I error rates at the nominal 5% level for the statistic AG2B as a function of subject and item sample sizes. The association standard deviation is 0"0 = 1.0 and each point is based on 10,000 simulations.

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Thomas D. Wickens

The difficulty with this class of tests, and the source of the biases in Figures 1 and 2, is that the wrong null hypothesis is being tested. There are three null hypotheses that could be tested 9 The strongest of these asserts that there is no relationship at all between the behavior categories. In terms of the variability model, this hypothesis implies that every glk is zero 9 It is a compound hypoth2 are esis, as it implies that both the mean association txn and the variability orn zero: Hol 9 p~ = 0

and

2=0. ~rn

This hypothesis is equivalent to a hypothesis of the conditional independence of the two responses given by the subjects and is tested by the fit of the log-linear model [AS][BS]. A much less restrictive null hypothesis is that there is no association between categories in the sample: Ho2: ql = 0 .

This hypothesis focuses attention on the particular sample. It does not involve the population parameters. The third null hypothesis refers only to the mean association: Ho3: I~,q = 0.

It does not preclude intersubject variability and is also less restrictive than Hol. 2 = 0, these three When there is no intersubject variability, that is, when ~rn null hypotheses are equivalent. When crn2 > 0, they differ. The statistics that incorporate only the multinomial variability test hypothesis H02. Although these statistics are satisfactory ways to test this hypothesis, when the subjects vary, as they usually do, a researcher probably wants to test the population hypothesis H03 instead of the sample hypothesis H02. The researcher's intent is to generalize to a population parameter, not to restrict conclusions to the particular sample. A test of H03 must accommodate the subject variation.

5. TESTS BASED ON BETWEEN-SUBJECTVARIABILITY There is a handy rule for treating between-subject effects that I have often found useful. If you wish to analyze anything that varies across subjects, you first find a way to measure it for each individual, ideally to reduce each subject's data to a single number. These numbers can then be investigated using a standard statistical procedure that accommodates their variability. 2 In this spirit, I consider from this class statistics that test the distribution of the log odds ratios for a difference in mean from zero. All these statistics are based on the observed variability of the Yk.

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157

The prototype for a test based on an empirical estimate of variability is the t test. To apply it here, the association between categories for each subject is estimated by Yk. The observed mean Yu and variance s u2 are calculated, and the hypothesis that the population mean corresponding to these observations is equal to any particular value is tested with a single-sample t test; for Ho3 tu =

~

YO

(5)

.

The ordinary t test might prove unsatisfactory for a variety of reasons. One potential problem is heterogeneity of variability. The sampling variability of the log odds ratio Yk depends both on the underlying distribution -?rijk (and through it on the subject-le