Advances in Psychology Volume 60 New Developments in Psychological Choice Modeling

Analysis of Covariance Structures

143

particular stimuli are compared. Krantz (1967) calls this condition “simple scalability”. However, numerous studies (Debreu, 1960; Krantz, 1967; Restle, 1961; Tversky & Russo, 1969; Rumelhart & Greeno, 1970; Tversky, 1972a, b; Sjoberg, 1977, 1980) reported violations of simple scalability in a variety of empirical situations. All stimuli are not equally comparable. The equal comparability holds only when stimuli to be compared are relatively homogeneous. When the stimuli are radically different on “irrelevant” dimensions (i.e., dimensions other than the one on which the comparison is supposedly made), they tend to be less comparable, and the choice probabilities tend to be less extreme (closer to 1D). If, on the other hand, the stimuli are similar, they are more comparable, and consequently more extreme choice probabilities tend to result (Krantz, 1967; Tversky & Russo, 1969; Rumelhart & Greeno, 1971). Thus differential degrees of similarity among stimuli give rise to context dependencies in the stimulus comparison process, called the similarity effect. This means that di, in Thurstone’s original model has its role to play. In particular, it has been shown (Halff, 1976) that d~ has distance properties, and di; satisfies the three metric axioms (minimality, symmetry and triangular inequality) required of the distance. The distance properties of dij make Thurstone’s general model considerably richer in its descriptive power than those models that assume simple scalability. Specifically, Thurstone’s general pair comparison model satisfies moderate stochastic transitivity (MST), but it can violate strong stochastic transitivity (SST), which is known to be equivalent to the simple scalability (Tversky & Russo, 1969). It is interesting to point out that di,, the distance between stimuli i and j , can be interpreted as a type of dissimilarity between the stimuli. Thus, dividing, mi - m, by di, in qi, in Thurstone’s general model is consistent with the empirical evidence (mentioned earlier) indicating that more dissimilar stimuli are less comparable. Sjoberg (1977) observed a high correlation between di; estimated from pair comparison judgments and a direct similarity rating between stimuli i and j separately obtained. The di, is thus not only theoretically expected to represent the stimulus dissimilarity, but there is also some empirical evidence to support the theory. The problem is how we may recover di; in Thurstone’s general model without overparametrizing it. Attempts to extend Thurstone’s pair

Takane

144

comparison model beyond Case 5 are almost as old as Thurstone’s original proposal of the model (Thurstone, 1927). For example, in Case 3 it is assumed that si, = 0 for all i and j, thereby reducing the number of parameters considerably. Case 4 was derived as a convenient numerical approximation to Case 3. However, in these cases differential comparability (d;,) between stimuli is exclusively atmbuted to individual uncertainties ($ and sf). Thus, they are rather restrictive as models of contextual effects in stimulus comparison processes. A couple of significant proposals were made in early 1980’s in the way of partially recovering di, in Thurstone’s model. Takane (1980) and Heiser and de Leeuw (1981) independently proposed the factorial model of pair comparisons (hereafter called the “HI., model), in which the covariance matrix between discriminal processes was assumed to have a lower rank approximation. That is,

s = ( S i j ) = XX‘

(5)

where X is an n by b (< n) matrix where n is the number of stimuli and b is the rank of mamx S. This amounts to assuming d$ = (Xi - Xj)’(Xi - Xi),

(6)

where xi and x, are i th and j th row vectors of X, since s; = x;‘x, and sf = x,’x,. That is, di, is assumed to be the Euclidean distance between stimuli i and j represented in a b dimensional Euclidean space. The X then represents the matrix of stimulus coordinates. An interesting development was due to Carroll (1980) and De Soete and Carroll (1983). The model is called the wandering vector model (WVM). In this model it is assumed that stimuli are represented as points in a b dimensional space where stimulus coordinates are given by X as in the THL model, that there is a random vector that varies over time, and that the projections of the stimuli onto this vector at a particular time determines the pair comparison judgment at the particular time. Under an appropriate distributional assumption on the vector we may derive the distribution of Yi - Y,, and the choice probability, pi,. Let u* denote the wandering vector, and let u* - N(v,I). Then Yi - Yj = (x;

- Xj)’U* - “(Xi - Xi)%, 4 1 ,

where dij is the same as in (6). It follows that

(7)

Analysis of Covariance Structures

145

pi, = Pr[(Xi - Xj)’U* > 01 ‘ij

$(z)dz = O(ri,),

= -00

where ri, = (xi - x,)’v/dij. It has been shown @e Soete, 1983) that the WVM is a special case of the THL model in which not only di, but also mi and m, are constrained in a special way; i.e.,

mi = xi’v and m, = x,’v.

(9)

Scale values of stimuli are represented in a particular direction in the space. Thus, although the THL model and the WVM were initially derived on the basis of entirely different rationales, they are quite similar to each other. Both the THL model and the WVM are designed to account for the differential comparability among the stimuli. However, these models strictly apply to either Case 1 or Case 2, where differences processes, Yi - Y,, and consequently observed choice probabilities, are assumed statistically independent across all pairs of stimuli. Both Takane (1980) and De Soete and Carroll (1983) developed parameter estimation procedures for their models. They both assume the statistical independence among the observations, while they use the data obtained by the multiplejudgment sampling. As has been discussed, the independence assumption is not tenable in the multiple judgment sampling. However, the assumption is made in virtually all previous estimation procedures for the Thurstonian pair comparisons models (e.g., Hohle, 1966; Bock & Jones, 1968; Arbuckle & Nugent, 1973; Takane, 1980; De Soete & Carroll, 1983; De Soete, Carroll, & DeSarbo, 1986). In order to account for the statistical dependencies among observations, pair comparison models had to await analysis of covariance structure formulations (Bloxom, 1972; Takane, 1985), to which we now turn. In closing of this section it might be pointed out that analogous developments (models of simple scalability to moderate utility models) can be traced in the Bradley-Terry-Luce (Bradley & Terry, 1952; Luce, 1959) type of constant utility model approach (Restle, 1961; Tversky, 1972a, b; Strauss, 1981). However, these developments are not readily amenable to the ACOVS formulations. See Indow (1975) and Luce

Takane

146

(1977) for insightful reviews of this line of development.

3. ACOVS Formulations In order to reformulate the THL model and the WVM in terms of analysis of covariance structures (ACOVS; Joreskog, 1970), let us first generalize the variance structure of these models. It was originally assumed that d2. ‘1 = (xi - x,)’(xi - x,) in these models. To this we may add 83 + gf + k$, where g? and gf are stimulus-specific uncertainties left unaccounted for by (xi - x,)’(xi - x,), and k$ represents uncertainty associated with a specific stimulus pair. These quantities represent amounts of specification error at two different levels. We now generalize this to covariance structures. Let t be a vector of ti, = Yi - Y, + ei, arranged in a specific order, where ei, is the error random variable associated with stimulus pair, ij. In a complete sampling design each subject makes judgments for all possible pairs of stimuli. In such a case t is a of dimensionality M = n(n - 1)/2, where n is the number of stimuli. Let A be an M by n design matrix for pair comparisons, whose rows are arranged in the same order as the elements of t. Each row of A corresponds with a specific comparison. If that comparison involves stimuli i and j and the direction of the comparison requires Yi - Yj (rather than Y, - Yi), the row has 1 in the ith column, -1 in the jth column and zeroes elsewhere. Let y be an n-component vector of Yi, and let e be an M-component vector of ei,. We assume

e

- N(O,K ~ )

(10)

where K2 is assumed to be diagonal with its diagonal elements denoted by k$. It may be further assumed k$ = k2 for all ij. Then t may be expressed, using mamx notation, as t = Ay+e.

(11)

The Ay takes differences between Yi and Y, in prescribed directions for all possible pairs of stimuli. We make a further structural assumption on y; namely, y = xu* + w*,

(12)

Analysis of Covariance Structures

147

where W*

=w

+m

with w

- N(0,G2),

(13)

and

u* = u + v

with

u - N(0,I).

(14)

Here m is the vector of mi (i = 1,..., n) and w is the random vector of stimulus specificities. The G2 is usually assumed to be diagonal with its ith diagonal element, g’, and indicates the degrees of stimulus specificities or uncertainties. The u* is the wandering vector introduced earlier. It follows that

t = A(XU* + w*) + e

-

N[A(Xv

+ m), A(XX’ + C2)A’ + K2].

(15)

When it is assumed that v = 0, then E (t) = Am, and since A m is the vector of mi - m,, this case corresponds with the THL model. If, on the other hand, it is assumed that m = 0 we obtain E(t) = AXV. This represents the mean structure, (xi- x,)’v, required of the WVM. The covariance structure, A(XX’ + G2)A’ + K2, remains the same for the both models. Note that diagonals of this covariance matrix are of the form, (xi - x;)’(xi - x ; ) + g’ + 8,” + k$, which is indeed the variance structure required of both the THL model and the WVM. Note also that offdiagonal elements of A(XX’ + G2)A’ + K2 are no longer zero, implying non-independence among the elements of t. It is interesting to note that the WVM is a random effect alternative to Bechtel, Tucker, and Chang’s (1971) scalar product model. In this model subjects are treated as fixed effects; i.e., for subject k, tk = AXvk and vk is explicitly estimated for each k. Analogous ACOVS formulations of classical Case 5 and Case 3 are also possible. Although these cases are not likely to provide satisfactory descriptions of pair comparison data, they may serve as good benchmark models. In Case 5 di, is assumed to be constant across all combinations of i and j . The simplest way this could occur is when s? and sj are constant, and si, is zero. In the ACOVS formulation this can be achieved by setting X = 0, and G2 = s21. Note that si, = 0 is not absolutely necessary to achieve di; = constant. It is sufficient to have sij = constant (Guttman,

148

Takane

1954). This case corresponds with X = Al, where 1, is an n-component vector of ones. However, this reduces to the previous case, since Al, = OM. Bock and Jones (1968), in their primitive attempt to incorporate systematic individual differences in Thurstone’s pair comparison model, present a model which is essentially equivalent to the ACOVS formulation of Case 5 in which K2 = 0 is also assumed. In Case 3 it is assumed that si, = 0 for all distinct pairs of i and j . This case can be obtained by X = 0 or X = Al,, and G2 being diagonal (not necessarily constant). Model (15) may be fitted to the data by the maximum likelihood or the generalized least squares method (Browne, 1974, 1984), when t is directly observed. In either case some existing programs, such as LISREL (Joreskog & Sorbom, 1981), EQS (Bentler, 1985) and COSAN (McDonald, 1980), may be used for actual computation. When only choices are observed, t has to be reduced to choice patterns. Correspondingly the distribution o f t must be converted into the probability dismbution of the choice pattern. Let h denote an observed pattern, and let f be the density function oft. Then Pr(h) = J f (t)dt

(16)

R

where R is the multidimensional rectangular region formed by the direct product of intervals Ri,, where Rij = (0, W) if stimulus i is chosen over stimulus j (rij > 0) and Ri, = (-, 0) is stimulus j is chosen over stimulus i (ti, c 0). Equation (16) is generally extremely difficult to evaluate due to nonzero covariances among the elements of t. However, the first and the second order marginal probabilities are relatively easily evaluated:

is the univariate marginal density of ri, - “(mi - m,), - x,) + g? + g j + k;)]. (The mi - mj must be replaced by (xi - x,)’v in the WVM.) Similarly,

where (xi

fii

- X,)’(Xi

Analysis of Covariance Structures

149

pi,,qr = Pr(i is chosen over j and q is chosen over r )

where fijqris the bivariate marginal density of ti, and tqr. Muthen (1984) developed LISCOMP, a computer program for the generalized least squares estimation of the ACOVS model for categorical data using the first and the second order marginal probabilities. It has been shown (Christoffersson, 1975; Muthen, 1975) that a loss of information incurred by ignoring higher order marginal probabilities in the estimation is relatively minor. Alternatively, LISREL may be used with tetrachoric correlations, but it only allows the simple least squares estimation. The ACOVS formulation of the WVM can be readily extended to the wandering ideal point (WIP) model recently proposed by De Soete et al. (1986). In the WIP model a subject is represented as a point which varies over time. The relative distances between stimulus points and the subject point at a particular time are supposed to determine preference relations observed at the particular time. The distribution of the subject point is assumed due to time-sampling of observations within a single subject. However, with the ACOVS formulation the model can be extended to the distribution of the ideal point over a population of subjects. Let u* be a random vector of coordinates of the subject point, and let U*

- N(v, D2),

where D2 is a diagonal matrix. (The D2 can be always made diagonal by rotating the space appropriately.) Let d(u*) be a vector of one half times squared Euclidean distances between stimulus points and the ideal point, i.e.,

where &u*) = (xi - u*)’(xi - u*). In the WIP model the distance is assumed inversely related to preference. Thus, we may set

150

y = -d(U*)

+w

in (1 l), where w is defined in (13). Then

t = A(-d(u*) + w) + e

+ +e

= A(Xu* - % x ( ~ ) w)

- N[A(Xv - % x ( ~ ) A(XD2X’ , + G2)A’ + K2],

(21)

where

x ( ~= ) diag(XX’)ln

=

Note that this model differs from the WVM in that it has the additional term in the mean structure XD2X‘ ( rather than XX’) in the covariance structure. Reparametrization by X* = XD will make the covariance structure identical in form to that of the THL model and the WVM. However, the mean will then be A(X*v* - Kdiag(X*’D-2X*)ln), so that we cannot get rid of D2 entirely. Vector has (xj’x, - xi‘xi) as its elements. Due to the nonlinear nature of this term, a special computer program is necessary to fit the ACOVS WIP model. An extension to choice data may be done in a manner similar to that in the WVM. 4. Possible Generalizations

A general method for analysis of covariance and mean structures (ACOVS with structured means) was given by Joreskog (1970). The method includes, among other things, conventional factor analysis, variancecomponent models, path analysis, linear structural equations, etc. Our approach is a special case of this general approach. Sorbom (1981) has shown how the ACOVS with structured means could be treated in a unified manner by analysis of moment structures (AMOMS) (see also Bentler, 1983). In our case the mean and covariance structures in (15) can be expressed as

M = A(X(VV’+ I)X’

+ mm’+ G ~ ) A +’ K~

(22)

Analysis of Covariance Structures

151

in terms of AMOMS, where it is further assumed that v = 0 or M = 0. Perhaps Bloxom (1972) was the first to note the importance of the ACOVS methodology in modeling pair comparison data. He developed his simplex model of pair comparisons (similar to Case 5 ) based on the ACOVS framework. Takane (1985), in an attempt to incorporate systematic individual differences into the THL model and the WVM, arrived at the ACOVS formulations of these models, which are similar in form to Bloxom’s simplex model. Working in the general ACOVS framework opens up an number of possibilities. First of all, a variety of interesting hypotheses (assumptions) can be tested explicitly. For example, G2 = a21 and/or K2 = b21 may be assumed and tested, or G2 = 0 and/or K2 = 0 may be assumed in (1 5 ) and their empirical validity tested. Bechtel et al.’s (1971) model corresponds with m = 0, G2 = 0 and K2 = 0. In the THL model we may relax XX‘+ G2 into a general positive definite matrix, S. We then have

E ( t ) = Am and V(t) = ASA’+ K2.

(23)

The goodness of fit comparison between this model and the original THL model tests the adequacy of the factorial decomposition of S into XX’+ G2. Two particularly interesting possibilities emerge, when stimulus information and/or subject information is available. Stimuli can be characterized by a set of externally supplied attribute values (Bock & Jones, 1968), by a set of features (Rumelhart & Greeno, 1971; Tversky & Sattath, 1979), or by a set of combinations of levels of manipulated factors (Sjoberg, 1975). Similarly, subjects performing the comparisons may be characterized by their background variables, such as sex, age, socioeconomic status, levels of education, etc. In the ACOVS framework these external variables can be incorporated in a relatively straightforward manner. Let B be an n by p (< n ) matrix of stimulus information. There are at least of couple of ways to incorporate this information. For example, we have

Takane

152

t = A(BS* + XU*

+ w) + e - N[A(Bm* + Xv), A(BD2B’ + XX‘ + G2)A’ + K2]

(24)

where s* - N(m*,D2). This model attempts to explain part of stimulus variability by B and the rest by X. This is analogous to Yanai’s (1970) approach to factor analysis with external criteria, in which whatever effects that can be explained by the external criteria are first partialed out, and factor analysis is applied to a residual covariance mamx. This is to see if there is anything interesting left unaccounted for by the external criteria. More simplified or complicated versions of this model may be obtained, as desired, by specializing s* in (24); e.g., s* = m*, s* = Pq* + r, etc. In either case it may be further assumed that v = 0 and/or X = 0. An alternative way to incorporate B is to constrain X by BQy where Q is analogous to regression coefficients. This amounts to assuming that all that has been explained by X can be explained by B. We then have

t = A(BQU* + w) + e

- N[ABQv,

A(BQQ’B’ + G2)A’ + K2].

(25)

A slight generalization of this model would replace Qu* by Qu* + s where s - N ( 0 , D2). We then have

v(t) = A(B(QQ’

+ D ~ ) B +’ G ~ ) A +‘ K

~ .

Subject information may also be incorporated in several ways. When the information is provided in nominal variables (e.g., male or female), one possibility is to partition the data into groups and to analyze them separately (Joreskog, 1971; Muthen & Christoffersson, 1981). This allows completely different covariance structures as well as mean structures across the groups. Of course, it is entirely permissible to constrain some elements in the covariance and mean structures to be equal across the groups. In fact, the gist of the general ACOVS method is that we may explicitly test the empirical validity of such constraints. Alternatively, subject information may be incorporated in a manner similar to regression analysis. Let zk be the q-component vector of the ktk subject’s background variables, and let mk and vk represent m and v in the THL model and the WVM, respectively, for subject k. We have

Analysis of Covariance Structures

153

two options. We may impose a regression structure on either mk or vk. In the first case, we have mk = Pzk and assume v k = 0, so that E (tk) = APzk and V(tk) = A(XX’ + G2)A’ + K2 or A(PD2P‘ + XX’ + G2)A’ + K2. (In either case XX’ + G2 may be replaced by a more general positive definite mamx, S.) In the second case we assume vk = P*zk while mk = 0, so that E(tk) = AXP*zk and v(tk) = A(XX’ + G ~ ) A + ’ K~ or A(x(P*D*~P*’+ I)X’ + c ~ ) A ’+ K ~ ) . (Again XX’ + G2 may be replaced by S.) Both stimulus and subject information can be simultaneously incorporated. Resulting models are combinations of those for the stimulus information and those for the subject information. All the generalizations discussed in this section carry over to the WIP model in a relatively straightforward manner. Assuming that we have both stimulus and subject information, X = BQ and vk = P*zk, we obtain, in the simplest case,

E(tk) = A(BQP*zk - !h diag(BQQ’B’)l,), with V(tk) = A(BQQ’B’

+ G2)A’ + K2.

5. Concluding Remarks In this paper we have shown that the ACOVS methodology is useful in probabilistic pair comparison modeling. No empirical examples are given, and the paper largely remained expository. An obvious follow-up is to exemplify the methodological ideas described in this paper through the analyses of actual data sets. Although some of the ACOVS models for pair comparisons presented in this paper can be fitted by existing programs (e.g., LISREL, LISCOMP), there are others that cannot. For example, no ready-made programs exist for parameter estimation for the ACOVS wandering ideal point models. The normality assumption on u and w, and consequently on t in (15), may not be adequate. In that case we may either transform the data or use a fitting criterion that does not assume normality. Asymptotically distribution free methods (Browne, 1984) may be useful in this context. It may appear that the proposed ACOVS models of pair comparisons have too many parameters to be estimated, particularly when the observed data are binary choices. This is indeed true for the general ACOVS

Takane

154

model. However, it is not true in our practical applications of the ACOVS model, since matrix A is always a fixed matrix in the pair comparison models. The number of parameters can be further reduced, if desired, by assuming that G2 and/or K2 are constant diagonal matrices. There are other possible generalizations that have not been explicitly discussed in this paper. For example, an extension to multiple choice situations seem to be rather straightforward. Also, treating subject’s background variables as random effects (rather than fixed effects) is already feasible in LISREL (Joreskog & Sorbom, 1981). This case corresponds with the error-in-variable regression analysis in the ACOVS framework. Our prospect of further developing the ACOVS methodology in connection with probabilistic choice models is thus bright, despite the fact that there are numerous tasks yet to be accomplished. Appendix How the ACOVS THL model and the WVM (15) may be fitted by LISREL is not so trivial. In this appendix we explain how this is done. We also explain how (24) and (25) can be fitted by LISREL. McArdle and McDonald (1984) provide a general framework for establishing the necessary correspondence. We appreciate Michael Browne’s help (personal communication) in clarifying the matter. The LISREL model consists of three submodels:

+ rt + 4 Measurement Model for y: i = Arfi + E Measurement Model for x: i = AXE + 8,

1. Structural Equation Model: 4 = B( 2.

3.

where the symbols with a tilde on top denote random vectors. Aside from its distributional assumption (i.e., multivariate normality) the model is completely specified by the following eight matrices: A,, Ax, B, r @ = E(&), Y = V ( r ) , 0, = V ( i ) and 0 6 = V ( s ) . (We stick with the notational convention used by Joreskog and Sorbom (1981) as much as possible.) Throughout this appendix it is assumed that Ax = I, @ = I, 0, = K2 (diagonal matrix) and 08 = 0 (zero matrix). The moment structure of i is then expressed as

Analysis of Covariance Structures

155

M =A~(I B)-’(TT’

+ Y ) ( I- B)-‘A,’ + K

~ .

(A-1)

The following results hold: Result I . The moment structure of (t in our notation) for the ACOVS THL model or the WVM is obtained by setting

Ay = [ A 01

k3 and

(Proof) A,(I - B)-’ = [A AX]. Thus, (A-1) becomes

= A(mm’

+ G2 + X(vv’ + 1)X’)A’ + K2,

which is identical to (22). Result 2 . The moment structure of to (24) is obtained by

(t in our notation) corresponding

A, = [ A AB* 01

and

Takane

156 r

1

where in order to avoid confusion our B is denoted by B*. (Note that in the above both A and B* are assumed known a priori, so that AB* can be evaluated a priori.) (Proof)Ay(I - B)-’ = [A AB* AX]. Thus, (A-1) becomes

M = [ A AB* AX]

= A(G2

[ 1 ii+ { !]] m+*’

[$i]+K2

+ B*(m*m*‘+ D2)B*’ + X(vv’ + 1)X’)A’ + K2,

which is identical to the moment structure required by (24). The above specification is apparently not unique. For example, setting

A, = [A 0 01

will give the same result. This latter specification may be more general than Result 2 in that it does not assume that both A and B* are known a priori. However, in Result 3 both A and B* have to be assumed known a priori. Result 3. The moment structure of (t in our notation) corresponding to (25) is given by setting A, = [ A AB* 01

B=

r= and

[!I

5 i]

Analysis of Covariance Structures

Y=

157

.I!

T!

0 0 1

!][

(Proof) Ay(I - B)-’ = [A AB* AB’Q]. Thus, (A-1) becomes M = [ A AB* AB*Q]

[ [“ “1 [c’ 0 0 0

0 0 VV‘

+

= A ( G ~ B*Q(VV’

+

+ I)Q’B*’)A’ + K

0 0 0 0 0 I

B!i’]+K2 Q’B*A‘

~ ,

which is identical to the moment structure required of (25). A slight generalization can be made by setting Y=

::]

0 D2 0 .

The moment structure then becomes

M = A(G2 + B*(Q(vv’ + 1)Q’ + D2)B*’)A’ + K2.

References Arbuckle, L., & Nugent, J. H. (1973). A general procedure for parameter estimation for the law of comparative judgment. British Journal of Mathematical and Statistical Psychology, 26, 240-260. Bechtel, G . G., Tucker, L. R., & Chang, W. (1971). A scalar products model for the multidimensional scaling of choice. Psychometrika, 36, 369-387. Bentler, P. M. (1983). Some contributions to efficient statistics in structural models: Specification and estimation of moment structures. Psychometrika, 48, 493-517. Bentler, P. M. (1985). Theory and implementation of EQS, a structural equations program. Los Angeles: BMDP Statistical Software. Bloxom, B. (1972). The simplex in pair comparisons. Psychometrika, 37, 119-136. Bock, R. D., & Bargman, R. E. (1966). Analysis of covariance structures.

158

Takane

Psychometrika, 31, 507-534. Bock, R. D., & Jones, L. V. (1968). The measurement and prediction of judgment and choice. San Francisco: Holden Day. Bradley, R. A. (1976). Science, statistics and paired comparisons. Biometrics, 32, 213-232. Bradley, R. A., & Terry, M. E. (1952). The rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika, 39, 324-345. Browne, M. W. (1974). Generalized least squares estimates in the analysis of covariance structures. South African Statistical Journal, 8, 1-24. Browne, M. W. (1984). Asymptotically distribution free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62-83, Carroll, J. D. (1980). Models and methods for multidimensional analysis of preference choice (or other dominance) data. In E. D. Lantermann & H. Feger (Eds.), Similarity and choice. Bern: Hans Huber. Christoffersson, A. (1975). Factor analysis of dichotomized variables. Psychometrika, 40, 5-32. Debreu, G. (1960). Review of R. D. Luce, Individual choice behavior: A theoretical analysis. American Economic Review, 50, 186-188. De Soete, G. (1983). On the relation between two generalized cases of Thurstone’s law of comparative judgment. Mathehatiyues et Sciences humaines, 21, 45-57. De Soete, G., & Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model. Psychometrika, 48, 553-566. De Soete, G., Carroll, J. D., & DeSarbo, W. S. (1986). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data. Journal of Mathematical Psychology, 30, 28-4 1. Guttman, L. (1946). An approach for quantifying paired comparisons rank order. Annals of Mathematical Statistics, 17, 144-163. Halff, H. M. (1976). Choice theories for differentially comparable alternatives. Journal of Mathematical Psychology, 14, 244-246. Heiser, W., & De Leeuw, J. (1981). Multidimensional mapping of preference data. Mathematiques et Sciences humaines, 19, 39-96. Hohle, R. H. (1966). An empirical evaluation and comparison of two

Analysis of Covariance Structures

159

models for discriminability. Journal of Mathemathical Psychology, 3, 174-183. Indow, T. (1975). On choice probability. Behaviormerrika, 2, 13-31. Joreskog, K. G. (1970). A general method for analysis of covariance structures. Biometrika, 57, 239-251. Joreskog, K. G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36, 409-426. Joreskog, K. G., & Sorbom, D. (1981). LJSREL VI user guide. Mooresville, IN: Scientific Software. Krantz, D. H. (1967). Small-step and large-step color differences for monochromatic stimuli of constant brightness. Journal of the Optical Society of America, 57, 1304-1316. Luce, R. D. (1959). Individual choice behavior: A rheoretical analysis. New York: Wiley. Luce, R. D. (1977). The choice axiom after twenty years. Journal of Mathematical Psychology, 15, 215-233. McArdle, J. J., & McDonald, R. P. (1984). Some algebraic properties of the Reticular Action Model for moment structures. British Journal of Mathematical and Statistical Psychology, 37, 234-251. McDonald, R. P. (1980). A simple comprehensive model for the analysis of covariance structures: Some remarks on applications. British Journal of Mathematical and Statistical Psychology, 33, 161-183. Muthen, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551-560. Muthen, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115-132. Muthen, B., & Christoffersson, A. (1981). Simultaneous factor analysis of dichotomous variables in several groups. Psychometrika, 46, 407419. Restle, F. (1961). Psychology of judgment and choice. New York: Wiley. Rumelhart, D. L., & Greeno, I. G. (1971). Similarity between stimuli: An experimental test of the Luce and Restle choice models. Journal of Mathematical Psychology, 8, 370-381. Sjoberg, L. (1975). Uncertainty of comparative judgments and multidimensional structure. Multivariate Behavioral Research, I I , 207-218.

160

Takane

Sjoberg, L. (1977). Choice frequency and similarity. Scandinavian Journal of Psychology, 18, 103-115. Sjoberg, L. (1980). Similarity and correlation. In E. D. Lantermann & H. Feger (Eds.), Similarity and choice. Bern: Hans Huber. Sorbom, D. (1981). Structural equation models with structured means. In K. G. Joreskog & H. Wold (Eds.), Systems under indirect observations: Causality, structure and prediction. Amsterdam: North-Holland. Strauss, D. (1981). Choice by features: An extension of Luce’s model to account for similarities. British Journal of Mathematical and Statistical Psychology, 34, 50-61. Takane, Y. (1980). Maximum likelihood estimation in the generalized case of Thurstone’s model of comparative judgment. Japanese Psychological Research, 22, 188-196. Takane, Y. (1985). Probabilistic multidimensional pair comparison models that take into account systematic individual differences. Transcript of the talk given at the 50th Anniversary Meeting of the Psychometric Society, Nashville. Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273-286. Thurstone, L. L. (1959). The measurement of values. Chicago, IL: University of Chicago Press. Tversky, A. (1972a). Choice by elimination. Journal of Mathematical Psychology, 9, 341-367. Tversky, A. (1972b). Elimination by aspects: A theory of choice. Psychological Review, 79, 28 1-299. Tversky, A., & Russo, J. E. (1969). Substitutability and similarity in binary choices. Journal of Mathematical Psychology, 6, 1-12. Tversky, A., & Sattath, S. (1979). Preference trees. Psychological Review, 86, 542-573. Yanai, H. (1970). Factor analysis with external criteria. Japanese Psychological Research, 12, 143-153.

New Developments in Psychological Choice Modeling G. De Soete, H. Feger and K. C. Klauer (eds.) 0 Elsevier Science Publisher B.V. (North-Holland),1989

161

TWO CLASSES OF STOCHASTIC TREE UNFOLDING MODELS J. Douglas Carroll AT&T Bell Laboratories, Murray Hill, NJ, U.S.A.

Wayne S. DeSarbo University of Michigan, U.S.A.

Geert De Soete University of Ghent, Belgium In this paper we propose two versions of stochastic choice models based on tree structure models - called “tree unfolding models”. These models can be viewed as discrete (tree structure) analogues of a recently proposed class of continuous (spatial) random utility models for paired comparisons choice data called the “wandering vector” and “wandering ideal point” models.

1. Introduction A class of continuous spatial models for paired comparisons choice data have been proposed by Carroll, De Soete, DeSarbo and others (Carroll, 1980; DeSarbo, De Soete, & Eliashberg, 1987; DeSarbo, De Soete, & Jedidi, 1987; DeSarbo, Oliver, & De Soete, 1986; De Soete & Carroll, 1983, 1986; De Soete, Carroll & DeSarbo, 1986; Schonemann & Wang, 1972; Wang, Schonemann, & Rusk, 1975; Zinnes & Griggs, 1974) These

Geert De Soete is supported as “Bevoegdverklaard Navorser” of the Belgian “Nationad Fonds voor Wetenschappelijk Onderzoek”. This paper is a substantially revised version of an article entilled “Stochastic tree unfolding (STUN) models’’ published in Communication & Cognition, 1987, 20, 63-76.

162

Carroll, DeSarbo, & De Soete

models are all variants of one form or other of what Carroll, De Soete and DeSarbo have called the “wandering vector model” and the “wandering ideal point model”. More generally this class of models can be referred to as multidimensional models for probabilistic choice, or, perhaps, more appropriately, stochastic multidimensional spatial choice models. They comprise an important subclass of a family of stochastic choice models called random utility models. References on this class of models can be found in Luce (1977) and McFadden (1976). In this paper we propose two versions of stochastic choice models based on tree structure models - called “tree unfolding models” (Furnas, 1980; De Soete, DeSarbo, Furnas & Carroll, 1984a, 1984b). For the history of the use of the term “unfolding” for this class of models see Carroll (1972, 1980), Furnas (1980). or Coombs (1964). Probably a better and more informative name for this class of models is “ideal point models”, since they generally assume preference is related to distance from an “ideal” stimulus point by a non-increasing monotonic function. From this point of view, the current class of models would be termed “tree ideal point” rather than “tree unfolding” models. However, in keeping with the historical use of the term “unfolding”, particularly in mathematical psychology and psychometrics, we shall continue to use the (perhaps) less informative but more “colorful” name “tree unfolding”. Thus we call these models, generically, Stochastic Tree UNfolding (or STUN) models. We shall deal in the current paper with two classes of tree structure models, differing in the way their respective “metric” is defined given the tree structure (ultrametric versus path length metric, sometimes called an “additive” metric) and with two different stochastic formulations (one entailing simple additive i.i.d. normal error, and the other entailing stochastic assumptions in which the structure of the tree becomes a central component). 2. The Two Structural Models

Given a fixed tree (i.e., a connected graph without cycles), there are (at least) two ways to define a metric on the objects represented as nodes of that tree. For now we restrict ourselves to the case where objects are placed only at terminal nodes. (See Carroll and Chang, 1973, for a

Stochastic Tree Unfulding Models

163

discussion of tree structure models in which objects are associated with, and distances defined between, all nodes, including internal as well as terminal nodes, of a tree.) The first of the two types of tree metrics we are considering in the present paper is the ultrametric, in which the distance between two (terminal) nodes is defined as what is often called the “height” (see Johnson, 1967) of their “least common ancestor” (1.c.a.) internal node. The 1.c.a. is the internal node at which the two first meet, or the “lowest” one which they share in common, in the hierarchy defined by the tree. It is important to note, here, that the ultramemc is dependent on the tree being a hierarchical tree; i.e., the ultramemc is based on a partial order being defined on the nodes (based on a subordinate-superordinate relationship between certain pairs of nodes). In particular, the distance between two (terminal) nodes is defined as the height of their 1.c.a. internal node. Generally speaking, these height values are assumed non-negative, and (more importantly) they are assumed to respect the same partial order as the (internal) nodes with which they are associated. That is, if A c B then h ( A ) 5 h ( B ) where A and B are two internal nodes, “c” can be interpreted as meaning that A is below B in the hierarchical order imposed on the tree (in set theoretical terms, “ A c B” has the usual meaning that the set [of terminal nodes contained in] A is a subset of [those contained in] B ) , and h ( . ) denotes the height values. These two conditions on the heights make the induced distance satisfy the ultrametric conditions. In particular, the ultramemc inequality can be stated as:

dik Imax (di,, d,k)

for all i, j , k ,

which can easily shown to be equivalent to saying that all triangles are

acute isosceles (isosceles, with the two longest sides equal). The ultrametric inequality (together with non-negativity) is a special case of (but much stronger than) the triangle inequality, which must be satisfied per definition by any memc. The ultramemc inequality (which, by itself, does not require non-negative distances) together with non-negativity comprise the ultrametric conditions. As pointed out by Johnson (1967) there is, in fact, a one-to-one relation (or isomorphism) between ultrametics and hierarchical trees, in the sense that, given an ultramemc, the hierarchical tree (and the non-negative height values of its internal nodes) are immediately defined, and vice versa (i.e., given a hierarchical

164

Carroll, DeSarbo. & De Soete

tree plus a set of height values satisfying non-negativity, a unique ultrametric is defined). In fact, we can generalize this by saying there is a one-to-one relation (isomorphism) between a set of dissimilarities (not necessarily distances, since they may not satisfy non-negativity) satisfying the ultrametric inequality (but not necessarily non-negativity) and hierarchical trees (with possibly non-negative height values). Finally, and most generally, an isomorphism exists between the set of all ordinally defined ultrametrics (i.e., rank orders, including ties, of dissimilarities such that any set of dissimilarities satisfying that rank order will satisfy the ultrametric inequality) and the set of all hierarchical trees (independent of the height values). Since any non-decreasing monotonic function of an ultrametric (preserving non-negativity) is also an ultrametric this is a very important property. Satisfaction of the ultramemc inequality, since it is based only on the ordinal properties of the distances or dissimilarities, is invariant under monotonic transformation of those distances (dissimilarities). A special case of such a monotonic transformation, of course, is addition of an additive constant, which can transform non-negative values into negative ones (or vice versa). However, given values satisfying the ultrametric inequality they can always be easily transformed into values satisfying non-negativity as well (and thus the full set of ultrametric conditions) by the simple device of adding a sufficiently large positive additive constant. The path length (or additive) metric is (superficially at least) quite different from the ultramemc. In this metric weights or lengths are associated with the links or the brunches of the tree (the edges in the graph connecting nodes of the tree to one another) and distance is defined as the length of the (unique) path joining the two nodes. Trees with this metric are sometimes called “free” or “unrooted” trees (see Cunningham, 1978) because they do not have a unique root (or “most superordinate” node) as do hierarchical (or “rooted”) ,trees. In the case of this path length metric it is quite important, however, that the branch lengths be nonnegative (otherwise the resulting “distances” may not satisfy the metric axioms - in particular the triangle inequality and/or non-negativity of the resulting “distances” may not hold). Unlike ultrametric distances, path length distances are not ordinally invariant - i.e., an increasing monotonic transformation (even if it retains non-negativity) of path length distances will not necessarily be path length distances, and, in fact, may not

Stochastic Tree Unfolding Models

165

be distances at all (Le., may not satisfy the metric axioms, and, in particular, the mangle inequality may be violated) (cf. De Soete, 1983). However, the path length property of these distances is invariant under addition of an additive constant, at least if the possibility of some negative lengths (and possibly negative distances) is allowed. In fact, the topology (i.e., the network structure) of the tree will not change as a result of addition of a (positive or negative) constant to these path length distances, but only some of the branch lengths (in particular, those of the branches linking terminal nodes to internal nodes of the tree, some of which may, in fact, become negative). In this sense, path length “distances” as well as ultrametric distances can be viewed as defined only on an interval scale. (The word “distances” is put in quotes here, since addition of a sufficiently large negative constant may lead to violations of the triangle inequality, or even of non-negativity, in which case these numbers will not be true distances at all, but only “dissimilarities”.) However, these two types of tree metrics are not as distinct as it may seem at first. The ultrametric can easily be seen to be a special case of the path length metric, obtained by defining the branch lengths in a particular way. In particular, assuming all lengths to be non-negative, the length of the branch connecting any two nodes can be defined to be the difference in height values between the superordinate (higher) and subordinate (lower) of the two nodes, where the “height” of a terminal node is defined to be zero. On the other hand, a set of path length or additive distances defined on a tree can be decomposed into the sum of an ultrametric distance defined on the same tree (defied by appropriately specifying heights for the internal nodes) and a second set of values which are of the form 6; = Ci + c, (i.e., are additively decomposable) for i # j. If the c’s in this second decomposition are nonnegative (and the decomposition can always be so defined that they are) these values (6;) can be viewed themselves as path length distances on a very special class of tree with only one internal node (to which all the terminal nodes attach) called (by graph theorists) a “star”, or (by numerical taxonomists) a “bush”. However, this decomposition is not unique. In fact, even the particular hierarchical tree with which the ultramemc component is associated is not unique, since a path length tree with n terminal nodes can be converted into an additive tree in n - 1 different ways, by “rooting” the (otherwise “unrooted”, or nonhiemchical) tree associated with the path length

Carroll, DeSarbo, & De Soete

166

-

metric at any one of n 1 different positions (at any one of n - 1 different internal nodes). Even if the particular hierarchy associated with the tree is specified, the heights, and thus the ultrametric component of the decomposition is not unique, but is defined only up to an additive constant. For a further discussion of the interrelations between these two types of tree metrics see Carroll (1976). Carroll and Pruzansky (1980), Furnas (1980) or Carroll, Clark, and DeSarbo (1984) for the three-way case. In the tree unfolding models for individual differences in preferences both the stimuli and the subjects are represented as terminal nodes in a tree. Like the spatial unfolding or “ideal point” models, a subject’s preferences are assumed to be inversely related to distances between the node representing that subject (the tree analogue of an “ideal point” for that subject) and the nodes representing the stimuli (the tree “stimulus points”). These distances are defined either as ultrametric or path length distances. In the papers by De Soete, DeSarbo, Furnas and Carroll (1984a, 1984b) a penalty function approach is described for fitting these models to a subjects by stimuli matrix of preference scores, sometimes referred to as a rectangular matrix of two-mode proximities between two sets of entities (in this case, subjects and stimuli). In the present paper we present stochastic versions of these models which are appropriate for paired comparisons preference data - the kind of data assumed in the “wandering vector” and “wandering ideal point” models. However, unlike many continuous stochastic preference models, where it is possible to fit the model to a single matrix of paired comparisons data (either replicated over subjects, or amalgamated over different individual subjects, but with those subjects treated as replications), at least one of the two classes of stochastic tree unfolding models (called SSTUN, for the “Simple” or “Special” STUN model) requires more than one subject (or paired comparisons matrix) to yield a non-trivial solution. (With only a single such matrix the tree obtained will always have the topology of a simple linear order corresponding to the best one dimensional solution for that preference matrix.) The second class of models (called GSTUN, for the “General” STUN model), however, can, in principle at least, recover the entire tree structure for stimuli, even from a single subject.

Stochastic Tree UMolding Models

167

3. The SSTUN Model In the simplest form of the Stochastic Tree Unfolding Model, called the SSTUN model, the paired comparisons are assumed to be generated from a process very closely related to Thurstone’s (1927) case V model, but with the “tree” preference scale values defined by the tree distances. That is, for individual i on trial t, in which stimulus j and k are to be compared, the probability p i , j k = P i (j> k) (the probability that i prefers stimulus j to k on trial r) is assumed to be generated from the following process:

with

d f k = djk

-+ &i,

(3b)

where ~f and I$ are independently normally distributed with mean zero and variance 02,and where ji, denotes the (ultrametric or additive) tree distance between the nodes representing subject i and stimulus j . This is exactly equivalent to the Thurstone case V model with subject i ’ s mean “discriminal process” equal to dj, and with all subjects having a common variance 2 of the discriminal process. Note, that we may drop in (2) the t superscript because p i , j k is independent of t, since the & J ’ s are assumed independent both of t and j . This property of independence of t is true of all the models to be assumed here, so that, henceforth, the t will generally be omitted. Note, also, that, without loss of generality, we may assume (3 equal to a constant, which can be taken, with appropriate scaling of the &,, as (J = I/*, in which case pj,jk

= @(ajk - j i j ) .

A preliminary version of a procedure which uses the De Soete, DeSarbo,

Furnas and Carroll penalty function procedure for fitting the structural tree unfolding model as a central component has been devised and successfully applied to some marketing data (cf. DeSarbo, De Soete, Carroll, & Ramaswamy, 1988).

168

Carroll, DeSarbo, & De Soete

This “simple” version of stochastic tree unfolding does not, however, utilize the tree structure in any inherent manner in generating the stochastic components. The distances, d i j , in this case (or other numbers assumed related in an inverse linear fashion to preference scale values) could have been generated by any process whatever - the structural model and the stochastic component are simply (as it were) “grafted” onto one another without any essential theoretical link interconnecting them. Furthermore, this leads to a Thurstone Case V model for each subject, which is known to entail strong stochastic transitivity for each subject; i.e., if p j , j k and pi,^ are both equal to or larger than 1/2, then P i , j l 2 max ( p i , j k , p , , ~ ) .There is considerable evidence in the literature, however, that in many realistic situations, strong stochastic transitivity does not obtain. At best, a weaker condition known as moderate stochastic transitivity [in which, under the same conditions, P i , j / 2 min ( p i , j k , pi,^)] can be expected to hold. The more general models to be discussed below, called General Stochastic Tree UNfoldinl (GSTUN) models satisfy moderate (but not strong) stochastic transitivity Furthermore, these models do indeed utilize the topology of the tree struc. ture in a central way. 4. The GSTUN Model

Given a fixed tree (hierarchical in the ultrametric case, or a rootless, nonhierarchical tree, or “free tree” in the path length case), we can define a matrix associated with that tree which we call the “Path Matrix”, and shall denote as P. We give this matrix that name because, in the case of a path length metric the matrix can be viewed as defining the unique path connecting every pair of terminal nodes i and j . In the case of an ultrametric it does not define such a path, but rather defines the “least common ancestor’’ node for every pair i and j , which can be viewed as defining the “path” connecting the two (if we think of the pair interconnecting directly via their 1.c.a. node). In the case of a path length metric the Path Matrix P is a matrix whose rows correspond to pairs of terminal nodes i and j , where (in the present case of the tree unfolding models) i corresponds to a subject and j to a stimulus. The columns of P correspond to branches in the tree. The general entry in P,which we shall designate as p ( i , h (for the entry in the

Stochastic Tree UMolding Models

169

row corresponding to node pair ( i , j ) and to branch 4) will be 1 if and only if branch 4 is included in the path connecting i to j , and 0 otherwise. This mamx is, thus, a binary “indicator” matrix indicating which branches are involved in the path interconnecting each pair of nodes i and I. Given a set of branch lengths h 1, h z ,..., h~ which we may represent as a Q-dimensional (column) vector h, the distances dij (i = 1,..., I, j = 1,..., J ) , which can be “packed” into another column vector of I x J components, d, can be defined via the mamx equation d = Ph

(4)

Now, let us suppose that h, rather than being a fixed vector of branch lengths, is a random variable. Furthermore, let us assume that, for individual i the dismbution of h is hi

- N ( P i , xi)

(5)

then, on a particular paired comparisons ma1 in which subject i is comparing stimulus j to stimulus k we have: pi,jk

PjU > k )

where Sj,jk = djk - di,. Under the assumptions we have made in this General Stochastic Tree Unfolding Model the dismbution of sj,,k is:

while

170

Carroll, DeSarbo. & De Soete

where p(i,) and p ( ~ r ( are ) the row vectors corresponding to the (i, j) and ( i , k) rows of the Path Matrix P, respectively. Since Zi is a covariance mamx, it is positive definite (or semidefinite) so that 6$j)(jk) is the squared generalized Euclidean distance between rows (i, j) and (i, k ) of P in the metric of Xi. Consequently, we obtain

where 0 denotes the standard normal distribution function. Since 6 is a (Euclidean) metric, model (10) is a moderate utility model (see Halff, 1976). It should be evident that exactly the same development will hold for the ultrametric case, except that the rows of the Path Matrix P correspond to Q internal nodes (rather than Q branches) with ~ ( i , )being ~ 1 if and only if node q is the 1.c.a. of i and j. It is clear, however, if one considers the structure of the path matrix, P, that the dismbution of the choice probabilities for subject i are dependent only on the distribution of those components of h that affect the distances from the node for subject i to the stimulus nodes. Those distributional parameters for subject i involving components of h not affecting these distances are indeterminate without further constraints.

5. Some Special Cases of GSTUN While the general model with pi and Zi completely unconstrained is of at least theoretical interest, this completely general model is much roo general for practical application to real data. As already noted, in some cases some of the parameters are intrinsically undefined. In all cases it has more parameters than observed data values, and therefore cannot be uniquely fitted. Thus, without further constraints, GSTUN should be taken as providing a broad theoretical framework within which a large number of interesting special cases can be viewed, rather than as a tractable statistical model in and of itself. Let us now, however, consider some cases which are of particular interest. Most importantly is the case in which pi = p and Zi = Z, for all i. Imposing these constraints is the simplest way of avoiding the intrinsic indeterminacies already alluded to above.

Stochastic Tree U@olding Models

171

These constraints on the p i ’ s and X i ’ s will, in fact, be assumed henceforth (unless otherwise indicated) in the present paper. Now, let us consider two further constraints of interest on Z (now assumed common over subjects). 1)

Z diagonal (i.e., the hq’s are independently normally distributed, but with different variances). In this case Z = diag (0;)so that 6 ( i , ) ( i k ) is the weighted Euclidean memc with weights 0;.Le.,

Q 66,) 2)

(ik)

=

0; b ( i j ) q

- P(ik)ql2.

q=l

Z = 021. This is case (1) above, with the variances of the hq’s all equal. We may, without loss of generality assume the common variance to be one, thus making Z = I, since we may absorb an arbitrary scale constant into the definition of p. In this case, of course, 6 ( i , ) ( i k ) is the ordinary Euclidean distance between p ( i , ) and p ( i k ) ; i.e.: 6tj)(ik)=

cQ

(P(ij)q

- P(ik)ql2,

q=l

It might be noted that the GSTUN model (and all its special cases) can be viewed as a special case of the wandering vector model (WVM) (Carroll, 1980; De Soete & Carroll, 1983) but with a diferenr Q dimensional stimulus space defined for each subject (corresponding to the submatrix of P associated with the distances from that subject’s node to the J stimulus nodes), with centroid vector p i , covariance matrix Z i for that subject (assuming here, for the moment, the most general form of GSTUN). This observation should be viewed, however, as being primarily of theoretical interest. In particular, it demonstrates (in a somewhat different way) that GSTUN “inherits” many important properties of the WVM. One notable feature of the GSTUN model in the ultrametric case is that (regardless of definition of Z) the variance term 6 $ , , ( i k , will be identically zero for any pair of pairs (ij) and (ik) sharing the same 1.c.a. nodes. This is because the rows p ( i j ) and p ( i k ) must be identical in such cases, and thus the distance between those rows (however, defined) will be zero. However, since we have already assumed that, for the i, j and k under consideration, subject i meets both stimuli j and k at the same internal

Carroll, DeSarbo, & De Soete

172

&,

node, it follows that, in this ultrametric case, = &. COIISeqUently dik - di, is zero, and pi,jk = @ - . Thus pi,jk is undefined, but could be

I:[

defined by fiat as equal to 1/2. It should be further noted, however, that since ultramemc distances can, as discussed earlier, be obtained as a special case of the path length metric, this makes it possible to fit the latter model but with constraints on the p’s that make the expected values of the distances ai, conform to the ultrametric conditions.

6. The Relation between the GSTUN and SSTUN Models

As the names imply the General STUN model, GSTUN, should include the Simple or Special STUN model, SSTUN, as a special case. It turns out this is true, under “general” conditions on the tree structure, for the GSTUN path length model (or the special case of that alluded to above in which the average distances are constrained to satisfy the ultrametric conditions). This relies on the fact that, in the case of this model the covariance matrix for individual i, Zi, can be so chosen as to make the covariances of the additive error components (added to the structural model) equal to an identity matrix (or a scalar times an identity matrix). (Note that we have now gone back to the most general case in which each subject is allowed a separate covariance matrix. We assume, however, that the centroid, p, of h is the same for all i.) Details follow. To demonstrate the fact that the SSTUN model is a special case of the GSTUN model (with path length metric and general covariance matrix Xi, different for each subject) let us first define Pi to be the submatrix of the path matrix P corresponding to those pairs of nodes including i (i.e., i paired with all J stimuli). Then, for given h, the J-dimensional (column) vector di defined as

.

di = Pih

(1 1)

contains the distances from i to the J stimuli. Under the assumptions made in the GSTUN model, the distribution of di will be di

where di

i) .

- N(&, PiZiPi’)

(12)

= Pip (the vector of expected values of tree distances for subject

Stochastic Tree UNolding Models

173

In order to make the GSTUN model equivalent to the SSTUN model (since we may choose P and p so that the expected values are already equal) it is necessary and sufficient to make the covariance matrices equal. The covariance matrix (of the distribution of “distances” over mals) in the SSTUN model is (without loss of generality, since we may choose 02 = 1)

ZfS’ = I

(13)

while as we have already seen, for the GSTUN model it is

z p = PiXiPi’

(14)

(where Zi is a Q x Q mamx of covariances of the branch lengths for subject i , while Cis) and XiG) are J x J matrices of covariances of the dij’s). Given a fixed tree (and memc) for subject i , and thus a fixed path matrix Pi, sufficient (but not, generally, necessary) conditions for Xic) = Xis) = I are that Pipi’ be nonsingular, so (Pipi’)-’ exists, (which is necessary) and that

zi = Pi’(PiPi’)-2Pi.

(15)

The definition of Xi in eq. (15) is sufficient, but not necessary, as there will generally be other definitions of Xi, differing primarily in what generalized inverse of Pi’Pi is used to define X i . In this case, the mamx Pi’(PiPi’)-*Pi defining Zi is chosen as the Moore-Penrose generalized inverse. For the case of the GSTUN model with an ultramemc, the matrix Pipi’ will be singular, (p) and p> will be identical if ( i , j ) and ( i , j ’ ) share the same 1.c.a. node.) In fact, it can be proved that, under the conditions of this model (which we might call GSTUN(u) for GSTUN with an ultrametric) the mamx Pipi’ will always be singular, except for the relatively uninteresting case in which the tree has a structure equivalent to a (single) linear order for the stimuli, because of the fact that many rows in Pi will be replicated. This might lead one to assume that the ultrametric case of SSTUN cannot be generated as a special case of GSTUN(u). This, however, is not the case! In fact, GSTUN(u) with Z = a21 will be exactly equivalent (in terms of the predicted choice probabilities, in any case) to SSTUN, if we resolve the indeterminacy mentioned earlier “by fiat”, by defining pi,,k = 112 in the case in which

174

Carroll, DeSarbo, & De Soete

(subject) i meets (stimuli) j and k both at the same internal node. The choice probabilities predicted by GSTUN(u) in this case will then be identically the same as those predicted by SSTUN! Thus, in fact, GSTUN(u) (with a scalar covariance matrix) is indistinguishable from SSTUN. (Of course, GSTUN(u) is still more general, as it need not be restricted to a scalar covariance matrix, and, in this case, its properties are distinctly different from SSTUN - even if SSTUN is generalized to allow different variances or a general covariance matrix.) If we assume GSTUN(pl), i.e., GSTUN with path length metric, under very general conditions the matrix Pipi’ will be nonsingular. (In fact, it can be proved that it will always be nonsingular so long as no branches in the tree are omitted altogether; i.e., so long as there are no nodes that are identified and thus collapsed into a single node.) In fact, even weaker conditions are necessary - all that is needed to guarantee nonsingularity of Pipi’ is that no two stimuli are placed at the same terminal node. Even the “degenerate” star or bush tree mentioned earlier, with only a single internal node, leads to a nonsingular Pipi’ (in fact, in this case Pi is a square J x J matrix which itself is nonsingular). It thus would seem to follow that the GSTUN(p1) model is the only completely general model (in this sense), and thus, we feel, is to be favored over GSTUN(u) on theoretical grounds. As discussed earlier, it is possible to fit GSTUN(p1) with constraints making the expected distances ultrametric, so we do not feel that this theoretical argument favoring GSTUN(p1) necessarily rules out what is essentially an ultrametric model (in its “central tendency”). In fact the SSTUN model with ultrametric distances can be generated as a special case of GSTUN(p1) with ultrametric constraints on d E(d) (which implies certain constraints on p).

References Carroll, J. D. (1972). Individual differences and multidimensional scaling. In R. N. Shepard, A. K. Romney, & S . B. Nerlove (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences (Vol. 1). New York: Seminar Press. Carroll, J. D. (1976). Spatial, non-spatial and hybrid models for scaling. Psychometrika, 41 , 439-463. Carroll, J. D. (1980). Models and methods for multidimensional analysis

Stochastic Tree UNolding Models

175

of preferential choice (or other dominance) data. In E. D. Lantermann & H. Feger (Eds.), Similarity and choice. Bern: Huber. Carroll, J. D., & Chang, J. J. (1973). A method for fitting a class of hierarchical tree structure models to dissimilarities data, and its application to some body parts data of Miller’s. Proceedings of the 81st Annual Convention of the American Psychological Association, 8, 1097-1098. Carroll, J. D., Clark, L. A., & DeSarbo, W. S. (1984). The representation of three-way proximities data by single and multiple tree structure models. Journal of Classification, I , 25-74. Carroll, J. D., & Pruzansky, S. (1980). Discrete and hybrid scaling models. In E. D. Lantermann & H. Feger (Eds.), Similarity and choice. Bern: Hans Huber. Coombs, C. H. (1964). A theory of data. New York: Wiley. Cunningham, J. P. (1978). Free trees and bidirectional trees as representations of psychological distance. Journal of Mathematical Psychology, 17, 165-188. DeSarbo, W. S., De Soete, G., Carroll, J. D., & Ramaswamy, V. (1988). A new stochastic ultrametric tree unfolding methodology for assessing competitive market structure and deriving market segments. Applied Stochastic Models and Data Analysis, 4 , 185-204. DeSarbo, W. S., De Soete, G., & Eliashberg, J. (1987). A new stochastic multidimensional unfolding model for the investigation of paired comparison consumer preferencekhoice data. Journal of Economic Psychology, 8, 357-384. DeSarbo, W. S., De Soete, G., & Jedidi, K. (1987). Probabilistic multidimensional scaling models for analyzing consumer choice behavior. Communication & Cognition, 20, 93- 116. DeSarbo, W. S., Oliver, R. L., & De Soete, G. (1986). A probabilistic multidimensional scaling vector model. Applied Psychological Measurement, 10, 78-98. De Soete, G. (1983). Are nonmemc additive tree representations of numerical proximity data meaningful? Quality & Quantity, 17, 475478. De Soete, G., & Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model. Psychometrika, 48, 553-566. De Soete, G., & Carroll, J. D. (1986). Probabilistic multidimensional

176

Carroll, DeSarbo, & De Soete

choice models for representing paired comparisons data. In E. Diday, Y. Escouffier, L. Lebart, J. Pages, Y. Schektman, & R. Tomassone (Eds.), Data analysis and informatics IV (pp. 485-497). Amsterdam: North-Holland. De Soete, G., Carroll, J. D., & DeSarbo, W. S. (1986). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data. Journal of Mathematical Psychology, 30, 28-41. De Soete, G., DeSarbo, W. S., Furnas, G. W., & Carroll, J. D. (1984a). Tree representations of rectangular proximity mamces. In E. Degreef & J. Van Buggenhaut (Eds.), Trena3 in mathematical psychology. Amsterdam: North-Holland. De Soete, G., DeSarbo, W. S., Fumas, G. W., & Carroll, J. D. (1984b). The estimation of ultramemc and path length trees from rectangular proximity data. Psychometrika, 49, 289-3 10. Fumas, G. W. (1980). Objects and their features: The metric representation of two class data. Unpublished Doctoral Dissertation, Stanford University. Halff, H. M. (1976). Choice theories for differentially comparable alternatives. Journal of Mathematical Psychology, 14, 244-246. Johnson, S . C. (1967). Hierarchical clustering schemes. Psychometrika, 32, 241-254. Luce, R. D. (1977). Thurstone’s discriminal processes fifty years later. Psychometrika, 42, 461-489. McFadden, D. (1976). Quantal choice analysis: A survey. Annals of Economic and Social Measurement, 5, 363-390. Schonemann, P. H., & Wang, M.-M. (1972). An individual difference model for the multidimensional analysis of preference data. Psychometrika, 37, 275-309. Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273-286. Wang, M.-M., Schonemann, P. H., & Rusk, J. G. (1975). A conjugate gradient algorithm for the multidimensional analysis of preference data. Multivariate Behavioral Research, 10, 45-99. Zinnes, J. L., & Griggs, R. A. (1974). Probabilistic, multidimensional unfolding analysis. Psychometrika, 39, 327-350.

New Developments in Psychological Choice Modeling G. De Soete, H. Feger and K. C. Klauer (eds.) 0 Elsevier Science Publisher B.V. (North-Holland), 1989

177

PROBABILISTIC MULTIDIMENSIONAL ANALYSIS OF PREFERENCE RATIO JUDGMENTS Joseph L. Zinnes National Analysts, Philadelphia, PA, U.S.A. David B. MacKay Indiana University, U.S.A. A probabilistic multidimensional model is described for analyzing preference ratio judgments. This model combines the unfolding model of Coombs with the probabilistic model of Hefner, in which stimuli and individuals are represented by multivariate normal distributions. A simple procedure is described for approximating the maximum likelihood estimates of the location and variance parameters of the model. Two simulations show how well this procedure works, especially when there is considerable variability in the data. 1. Introduction

For some time now we have been confronted by a seemingly insolvable problem: how can be study, in a serious and convincing way, individual choice of interesting, multi-attribute stimuli when those stimuli are clearly identifiable. The problem with identifiable stimuli is that individual choices of those stimuli cannot be replicated a large number of times. It is easy enough, at least if one has sufficient reinforcers available, to ask subjects to indicate over and over again whether they can detect a signal buried in background noise, or whether they can identify which of two tones was presented, etc. But it is quite a different matter to ask subjects

This reseach was supported by National Science Grant SES-8120871. This paper is a revised version of an article publishcd in Cornrnitnicafion & Cognilion, 1987, 20, 17-43.

178

Zinnes & MacKay

over and over again which of two specific cars they prefer or which of two specific houses they would buy. Subjects can readily identify these stimuli and therefore can readily recall their previous responses. This is precisely the same problem that occurs in the testing field. Large numbers of replicated choices are important to the study of choice behavior because of the nature of the class of choice models that we believe are relevant. These models are inherently probabilistic and have large numbers of parameters to estimate. To estimate these parameters accurately and also to carry out sensitive statistical tests, namely those which discriminate between alternative choice models, generally require a considerable amount of replicated choices. At present we see only one way out of this predicament and that is to study individual choice behavior by collecting numerical judgments of preferences, rather than by obtaining choice data. Unlike choice data, these numerical judgments are obtained by having subjects indicate both which stimulus they prefer and by how much. The value of numerical judgment has been pointed out by numerous writers (Anderson, 1982; Eisler, 1982). They contain, under appropriate conditions, more information than simple choice responses, and therefore they make it possible, at least in principle, to obtain accurate parameter estimates using few if any replications of the individual judgments. This, at least, is our hope for the present. That hope does, of course, depend on a leap of faith: that numerical judgments and choice responses will be compatible, that the same underlying model will apply and therefore that the estimates of the parameters obtained by numerical judgments are precisely the same as those that would have been obtained had it been possible to replicate choice responses a large number of times. This is rather a large assumption. It is one we expect to investigate more fully in the future. Thus, in this paper, we pursue only the question of how to estimate, using numerical judgments, the parameters of a specific choice model, when it is reasonable to assume that that choice model is appropriate. The specific numerical judgment discussed in this paper is a ratio judgment, or what we call a preference ratio judgment. We assume that stimuli are presented pairwise to the subjects and that the subjects are asked to indicate how much they prefer one stimulus over another. The instructions that we have used in our own experimental work attempt to

Analysis of Preference Ratio Judgments

179

make it clear to the subject that what is wanted is a ratio judgment. The response of two, for example indicates that one stimulus is preferred twice as much as the other. To make sure that these instructions are understood, the subjects are given warm up trials involving very simple stimuli, such are lines of different lengths. The subjects then practice making ratio judgments concerning the relative lengths of pairs of lines. Although we confine ourselves in this paper to preference ratio judgments, it should not be concluded that this is the only type of judgment that could have been used to extract numerical information from subjects. In fact, the ubiquitous rating judgment has been used in the preference domain to do just this (Bechtel, 1976; Saaty, 1980; Scheffk, 1952; Sjoberg, 1967). Our use of the ratio judgment stems from our belief that it is the most appropriate judgment to use if our underlying preference model is indeed correct. In this model, the utility or desirability that a person has for a stimulus is represented by a Euclidean distance. Thus, choosing between two stimuli is conceptually equivalent to comparing two distances in a Euclidean space. Since distances in the model are determined only up to a multiplicative transformation, it would not make sense to compare the differences between two distances, as would be suggested by a rating judgment. This is the case, because differences are not invariant over multiplicative transformations. Their magnitude is thus totally arbitrary within the model. It would make sense, however, to determine the ratio between a pair of distances, because that value is invariant over a multiplicative transformation. Within the model, therefore, the preference ratio judgment is a meaningful judgment. There is another issue concerning our use of the ratio judgment. Even though we assume that subjects are carefully instructed to perform a ratio judgment, it does not follow that they will actually carry out those instructions. It has been shown (Birnbaum, 1982) that under some conditions subjects apparently respond to stimulus differences even when they are instructed to respond to their ratios. Birnbaum has, however, provided some evidence to indicate that when stimuli can be represented as distances, subjects appear to respond to the stimulus ratios when instructed to do so. It is, therefore, not unreasonable for us to assume that the preference ratio instruction does indeed generate a preference ratio response, that it does generate a response based on the ratio of two distances. However, the final determination of the precise conditions under which this

180

Zinnes & MacKay

assumption (and the others to be described in the following section) are valid, will have to wait for more detailed experimental tests. The preference model we use in this paper is a probabilistic, multidimensional version of Coombs’ unfolding model (Coombs, 1964). The probabilistic aspects are based on a model first put forth by Hefner (1958). The essential idea of Hefner’s model is to represent each stimulus in terms of a multivariate normal distribution and subjects’ decision processes in terms of random samples from these distributions. The Hefner model, or closely related models, have been used in connection with a number of different types of data. It has been used to explain same-different judgments (Zinnes 8z Kurtz, 1968; Zinnes 8z Wolf, 1977), choice response (Bijckenholt & Gaul, 1984; Croon, in press; De Soete, Carroll, & DeSarbo, 1986; Suppes & Zinnes, 1963; Zinnes & Griggs, 1974), similarity judgments (MacKay & Zinnes, 1981; Zinnes & MacKay, 1983) and recognition responses (Ashby 8z Townsend, 1986). The attraction of Hefner’s model is its conceptual simplicity. It is a very natural and powerful extension of the single dimensional choice models of Thurstone (1927). It is powerful, because the properties of the multivariate normal are well known and therefore one can answer in detail basic questions concerning the goodness-of-fit of the model and the invariance of its parameters over different experimental conditions. Our experimental work with preference ratio judgments has just begun (MacKay, Ellis, & Zinnes, 1986; MacKay & Zinnes, 1986). In these experiments, subjects made preference ratio judgments concerning residences that differed with respect to environment, location and economic characteristics. The Coombs-Hefner preference model discussed in this paper was applied to the data of these experiments and appeared to do well in explaining those data. In the following sections we focus on the problem of obtaining the maximum likelihood estimates of the parameters of the Coombs-Hefner preference model, when the data consist of preference ratio judgments. In Section 2, a simple approximation of the likelihood functions is developed. In Section 3, a simple expression for the initial estimates of the parameters is worked out. In the final two sections, two simulations are described, the purpose of which is to provide some idea of the accuracy and feasibility of the maximum likelihood estimates, especially when there is considerable variability in the data.

Analysis of Preference Ratio Judgments

181

2. The Preference Model

In the unfolding model of Coombs (1964), subjects and stimuli are both represented as points in an r-dimensional Euclidean space. The preference of the subjects are assumed to be determined by the distances between the subject points, called “ideal points”, and the stimulus points. The smaller the distance di, between ideal point i and stimulus point j , the more desirable is stimulus S, to subject Pi. To this deterministic model of Coombs we add the probabilistic assumptions of Hefner (1958). In particular, we let the r-dimensional random vectors Xi = (Xil,. . . ,Xi,), i = 1 ,..., m, be associated with the m ideal points and assume that they have an r-variate normal distribution with mean vector ui = (uilr . . . , ui,) and covariance mamx oT1,. Similarly, for the stimulus points, the r-dimensional random vectors X, = (X,l,. . . ,Xi,), j = m + l , . . . , m+n are associated with the n stimulus points and it is assumed that they also have an r-variate normal distribution with mean vector u, = @,I, . . . , u,,) and covariance matrix uf I,. The notation is intended to indicate that the variances of the components of each stimulus point do not differ from dimension to dimension, but that on any given dimension, the variances of the components of different stimuli may differ. Thus within a single dimension, these assumptions are precisely those of a Thurstone (1927) case 3 pair comparison model. The same is true for the variances of the ideal points. It would be desirable to formulate more general assumptions concerning the covariance matrices, but doing this might tend to increase the number of parameters that would have to be estimated. Under these assumptions, the interpoint distance di, is a random variable. On each mal, its value is determined by subject i sampling from the ith ideal and j t h stimulus dismbution and “calculating” the Euclidean distance between the two sample points. Thus, in terms of the rdimensional random vectors Xi and X,, the distance di, is given by d$ = (Xi - Xj)’(Xj - X j ) .

(1)

In contrast, the true distance Di, is not a random variable, but is defined in terms of the mean vectors ui and u, by

182

Zinnes

C ?

MacKay

It may also be noted that the true distance Di, does not correspond to the expected value of the distance di, and, in fact need not be monotonically related to it (Zinnes & MacKay, 1983). It will be useful to define the joint variance o$ by the equation

which can be conceptualized as the variance of the difference between the components x j k and x j k on each of the r dimensions. This term, which appears in many of the equations that follow, should not be confused with the variance of the distance di,. That variance, unfortunately, has a considerable more complex expression. To deal with preference ratio judgments, the experimental task of interest here, we use a direct adaptation of the decision rule of the Coombs unfolding model. It is assumed that subject i reports the preference ratio Ri,k when the ratio of the distances di, and dik equals Rijk, that is, when

Since the interpoint distances di,, i = 1, . . . , m, and j = m+l, . . . , m+n, are random variables, their values and that of the ratio Ri,k, can be expected to change with replications. The decision rule given in (4)only asserts that the subject accurately reports the ratio as it is perceived on each mal. It will be helpful to make one more assumption. This assumption concerns the independence of the distances di, and dik when the subject judges the stimulus pair S j and s k . We shall assume that the subject randomly selects two independent samples from his ideal point distribution, one of which is used to determine the distance dij and the other the distance dik. Under these conditions, the two distances dij and dik will be independent random variables. Whether this assumption is plausible or reasonable would depend on the specific details of the experimental procedure. If the two stimuli to be judged are presented sequentially or in widely separated spatial positions, the subject would have a tendency to evaluate each of the stimuli

Analysis of Preference Ratio Judgments

183

independently. This might also happen when the stimuli are complex, requiring the subject to spend a significant amount of time considering each stimulus separately. In any event, we assume in what follows that the two-sample, independence case applies and therefore that the ratio judgment Ri,k is based on the ratio of two independent random variables. Whether our results can be generalized to the one-sample, dependent case remains to be seen.

3. The Likelihood Function We consider first the problem of evaluating the probability density function of the ratio judgment Ri;k. This density function is needed because it forms the basis of the likelihood function that is to be maximized. Under the assumptions stated thus far, it follows that the "standardized" squared distance d $ / o $ has the noncentral chi-square distribution f 2 ( v , h i , ) , where the degrees of freedom equals the dimensionality of the space r and the noncentrality parameter h i ; equals

(Hefner, 1958; Zinnes & MacKay, 1983). Because of this and the independence assumption stated previously, we can immediately conclude that the ratio of the standardized squared distances

has the doubly noncentral F distribution F " ( v , , V k , h i , , X i k ) (Bulgren, 1971; Suppes & Zinnes, 1963; Zinnes & Griggs, 1974). The two noncentrality parameters of this distribution h i j and h i k are defined in (5), as they are for the noncentral chi-square distribution, while the degrees of freedom v, and v k are both equal to the dimensionality of the space r. These results indicate that there is a close relationship between the probability density function of the ratio judgment Ri;k and the probability density function of the doubly noncentral F distribution. Specifically, letting g (Ri;k) be the desired density function of R i , k , then

Zinnes & MacKay

184

where h”(.) is the density function of the doubly noncentral F distribution F ”(vj,Vk,hijhik). Equation (6) shows that if will be sufficient to focus our attention on developing a procedure for evaluating the function h”(.) of the doubly noncentral F distribution, in order to obtain a simple procedure for evaluating the density function g (&;k). We consider next, therefore, the F distribution. The exact expression of the density function of F ” distribution has been worked out (Bulgren, 1971; Kendall & Stuart, 1961, p. 252), but it is not expressible in closed form. It contains a doubly infinite series of terms, which for some values of the parameters - namely those in the tails of the distribution - converge extremely slowly. For practical applications, it is essential to find a simple, approximate expression of this density function. Two simple possible approximations immediately suggest themselves. One approach uses the central chi-square to approximate the noncentral chi-square distribution (Patnaik, 1949), this approach makes it possible to convert the F ” distribution to the central F distribution. The other approach uses a normal distribution to approximate the noncentral chisquare distribution. Although this latter approach has been used successfully to approximate the cumulative distribution function of the F ” distribution (Zinnes & Griggs, 1974), it did not seem to work as well to approximate the density function of this distribution. Consequently, our discussion here is confined to the former approach, based on using the central chi-square to approximate the noncentral chi-square distribution. From the Patnaik approximation, it follows that if sj has the noncentral chi-square distribution ~ ’ ~ ( v , , then h . ) s;/p; will have approximately the central chi-square distribution x il(v:) where the degrees of freedom v: I‘

equal

* v; =

(Vj

+ hj12

vj

+ 2h;

(7)

and the multiplicative factor p , is given by

v; + 2h; p i = v j + hi

-

Thus, to obtain the central F approximation of the F

”

distribution, we

Analysis of Preference Ratio Judgments

start with the distribution function of the F

185

distribution

Multiplying both sides of the inequality in (9) by p2v;/p1v;, use of (7), (8) and the definition

and making

reduces (9) to

Now we can make direct use of the Patnaik approximation. According to this approximation, the left-hand side of the inequality in (1 1) has approximately a central F distribution. Therefore, (1 1) can be written as follows

where H(*) is the distribution function of the central F distribution The degrees of freedom v; and vz, which in general will not be equal to each other and will have noninteger values, are given by (7). The final result is obtained by differentiating (12) with respect to f, which gives the approximation

F(v;,v;).

This equation expresses h", the density function of the F " distribution, in terms h, the density function of the central F distribution. It may be noted that the function h, which is the key element of (13), has a simple closed form expression and therefore (1 3 ) does indeed provide a straightforward procedure for evaluating h" for any of the values of its four arguments. To summarize, (13) which gives the approximation that is fundamental to evaluating the likelihood function to be maximized, replaces the density

Zinnes & MacKay

186

I L

n 1.50-

a

0.0

1.0

I .5

2.0

2.5

3.0

RANDOM V A R I A B L E

Figure 1. The central F approximation of the doubly noncentral F distribution. The degrees of freedom v1 and v2 are both equal to 2. The solid lines are the exact values, the dashed lines are the central F approximation. For curve A: hl = 1, h2 = 30, for curve B: hl = 1, hz = 4; for curve C: hl = 1, h2 = 1.

function of the F If dismbution, having equal degrees of freedom and integer values, with the density function of the central F distribution, having unequal degrees of freedom and noninteger values. Some idea of the accuracy of the approximation given in (13) is shown in Table 1 and Figure 1. Table 1 gives both the approximate and the exact values of the function hCf 1 vI,v2,hl,X2)where f = 1, v1 = v2 = 2, 4, 8 for a number of different values of and h2. The absolute and relative errors, given in columns 5 and 6 of the table, suggest that the approximation is quite good. The absolute errors do not exceed .02 and the relative errors do not exceed 6 percent. Furthermore, the larger relative errors seem to occur only for values in the tails of the distribution, where according to the last column of the table, convergence of the infinite series in the exact expression tends to be slowest. Figure 1 offers additional support for the approximation given in (13). Unlike Table 1, this figure attempts to show how the accuracy of the

Analysis of Preference Ratio Judgments

187

approximation affects the evaluations of the function g (Ri,k) for the entire range of values of the variable Ri,k = di,/dik. Three different distributions are plotted in this figure for the three different values of hi,: 1, 4 and 30. To highlight any major weakness of the approximation, the degrees of freedom v1 and v2 were both set equal to 2 for all three cases. Distributions with larger degrees of freedom tend to be more symmetric and therefore tend to be easier to approximate. Table 1. Exact and approximate values of the probability density function h(1 I v1 ,vz,hl,b)

11

h2

Exact

h (0 Approx.

.8 .I 1.O 1.o 1.o 1.o 1.o

1.o 1.o 2.0 3.0 4.0 8.0 12.0

.5184 .4920 S294 .5158 .4890 .3352 .1982

.5311 SO09 .5497 .5384 .5109 .3427 .I952

Degree of freedom

v, .8 .1 1.o 1.O 1 .O

1.o 1.0 1.o

.8 .1 1.O 1.0 1.0

1.o 1.O

1.o

Errof Percent

Kb

-.0128 -.0089 -.0203 -.0226 -.0218 -.@I74 -.0030

-2.46 -1.80 -3.94 -4.39 -4.47 -2.22 1.52

44 28 58 76 86 130 173

-.w4 -.0033 -.@I87 -.0122 -.0143 -.0108 -.0011 .006 1

-.57 -.44 -1.15 -1.65 -2.04 -2.14 -.34 6.23

52 33 70 86 97 149 192 288

-.0012

-.I1 -.09 -.27 -.48 -.71 -1.35 -.11 6.09

64 42 89 107 121 191 273 415

= v2 = 4

1.o 1.O 2.0 3.O 4.0 8.0 12.0 20.0

.7596 .7403 .7609 .7393 .7037 .5032 .3 138 .0978

.7640 .7436 .7697 .7515 .7180 S140 .3149 .0917

1.O 1.O

1.0980 1.0843 1.0937 1.0684 1.0283 .7246 .3815 .0973

1.0992 1.0852 1.0967 1.0736 1.0356 .7344 .3819 ,0913

2.0 3.O 4 .0 9.0 15.0 25.0

Absolute

-.OOO9 -.MI30 -.0051 -.0072 -.@I98

-.ooo4 ,0059

“The error equals the exact value minus the approximate value. bIndicates the number of terms summed to obtain the exact values of the density function given in the table.

188

Zinnes & MacKay

From the discrepancy between the exact and approximate values in this figure (the difference between the solid and dashed lines), it is evident that the approximation has its largest absolute error at the middle of the distribution, especially when the distribution is highly skewed. In general, however, the dashed lines (the approximate values) follow the solid lines (the exact values) quite closely, even in the tails of the distribution. For the level of accuracy typical of judgmental data, it would appear that (13) provides a reasonable level of accuracy. This is particularly encouraging, and to some extent surprising, since the approximation used in (13) is quite simple. 4. Starting Values

Even though a simple approximation was developed in the previous section for the density function of the observation Rijk, the likelihood function containing products of these density functions will still tend to be quite complicated. It is, therefore, not likely that a simple, closed-form solution exists that maximizes this function and consequently, it will be necessary to use iterative methods to obtain the maximum likelihood (ML) estimates of the unknown parameters. There are a number of standard iterative procedures that can be used (e.g., Chandler, 1969; IMSL, 1979). It does not seem to make a great deal of difference which one is selected, provided the iterative process starts with reasonably good parameter estimates. We consider next, therefore, procedures for obtaining good starting value (SV) estimates for both the coordinates and the uncertainty values of the stimulus and ideal points. Our concern here is to develop quick and simple procedures that can be expected to produce moderately accurate parameter values. SV estimates of the coordinates. For the purpose of obtaining these initial estimates, we assume that the joint uncertainty value = of + 0; is small relative to the distance DC. Well established metric, nonprobabilistic procedures can then be utilized. Although the accuracy of these SV estimates will depend very much on the validity of this assumption, it should be noted that these estimates are only the initial values of the iterative process. The final values, the ML estimates, do not require this assumption.

05

Analysis of Preference Ratio Judgments

189

Metric analyses of the unfolding model start generally with a set of I scales. The I scale for single subject Pi consists of the set of distances between ideal point i and each one of the stimulus points. To obtain these I scales in the present case, some preliminary analysis of the ratio judgment Ri;k is necessary. Define Rbk = log Rijk

and

Then the problem of estimating the distances Di, 0’ = m + l , . . . , m+n) for the subject Pi consists of solving the linear systems of equations R t k = D t - D;k

0‘ = m + l , . . . , m+n)

(14)

for D t . The least squares solution of (14) is

-*

Dij = RC.

+ 0;.

where we have let

and have taken to be the least squares estimate of DC. To be precise, it should be made clear that the solution given in (15) is only an approximation, and that this is true even when the ratio judgment Ri,k is averaged over an infinite number of replications. This follows from the fact that the expected value of the log of the ratio judgment Ri;k does not equal the true ratio log(Di,lDik) and will only approach the true value in the limit when the ratios Di;loi; and Diklo& both increase without bounds. This latter point is discussed more fully in the next section. Several different metric approaches can be used to analyze these I scales (Bechtel, 1976; Carroll, 1972; Schonemann, 1970). We have found it very effective to use Schonemann’s procedure to solve simultaneously for the coordinates of the stimulus and ideal points, but then to discard the

Zinnes & MacKay

190

solution for the ideal points. With the coordinates of the stimuli now treated as known, it is possible to set up, for each subject, a linear regression problem to solve for the coordinates of the ideal points and, incidentally, for the subject-specific constant Di., which appears in (15). Both the unknown coordinates and the antilog of the unknown constant 0:. turn out to be equal to the regression weights of the linear regression equation (see, for example, (26) in Zinnes & MacKay, 1983). It should be mentioned that this procedure for solving for the stimulus coordinates has had to depend on group data. In essence we have had to assume that the stimulus coordinates do not differ from subject to subject. It would have been desirable to have been able to solve for the stimulus coordinates separately for each subject, but, under the assumptions of the present model, this does not seem to be possible. SV estimates of uncertainty values. To determine the SV estimates of the uncertainty values oi (i = 1, . . . , m) and 0,(j= m + l , . . . , m+n), it will also facilitate matters to proceed as we did in the previous section. We assume that the uncertainty values are small compared to the interpoint distances and we also make use of the log transformation. This transformation is especially useful here, because the expected value and variance of the log of a random variable having the central F distribution have, at least approximately, very simple expressions. In particular, if log f has the central F distribution F (v;,vg), then

1

E(1/2 logf) = 1 / 2 y -- 7

,:

Var(l/2 logf)

=

1/2

I:, :,I +

when v;, and v; are large (Kendall & Stuart, 1963, p. 379). These results are directly relevant here, because as we have seen in Section 3, the central F distribution is closely related to the distribution of the ratio judgment Rijk. We begin, therefore, with an attempt to use (18) and (19) to obtain an approximate expression for the mean and variance of RTjk. From the definition of R i j k and letting

Analysis of Preference Ratio Judgments

191

we can write

The notation hjk is appropriate here, because, as noted earlier, under the assumptions of the model, hjk has the doubly noncentral F distribution F ”(v1, V 2 , h i j l h i k ) . Equation (13) now provides the motivation to define

hjk =

aik

hjk aij

where, as in (lo),

because from (13) we know that hjk will have approximately the central F distribution F (v;,v;) with the degrees-of-freedom parameters determined, as in (7), by

Substituting (21) in (20) gives

which expresses the square of the ratio judgment R$k directly in terms of a variable having approximately a central F distribution. We are now in a position to apply the approximation of (18) and (19), along with (24) to obtain the mean and variance of log Ri,k. It is appropriate to apply (18) and (19) in the present case, because under our present assumption concerning the size of ai, relative to Di,, the noncentrality parameter hi, will be large. And, from (23), it is clear that then the degrees of freedom v i will also be large, and, in fact, in the limit

Zinnes & MacKay

192

*

vij

hij

= -. 2

This fullfils the conditions required by (18) and (19). Thus, applying the operator 1/2 log to both sides of (24), we obtain the expected value

which, from (18), (25) and the assumption that hi; is large, reduces approximately to

We can proceed similarly to obtain the variance of R;k, but this time making use of (19) instead of (18). Equation (24) then becomes Var(R ;k) = var( 1 / 2 log fijk),

Equation (27) is the basic result we need. It suggests that an estimate of the joint variance o$ can be obtained by solving a simple, linear system of equations, provided that we have estimates of the left hand side of (27), namely, the variance of R Q k and have estimates of the true distance Di;. The latter estimates present no problem, since these interpoint distances can be calculated directly from the SV estimates of the coordinates determined in the previous section. For an estimate of Var(R;k) we can make use of the fact that under our present assumptions, the approximation given in (26) is still valid and therefore in the limit,

Analysis of Preference Ratio Judgments

193

Consequently, we can use for an estimate of Var(R&)

where the summation is taken over the ni;k replications of Ri;k and the distances Di, and Dik are, as before, calculated from the SV estimates of the coordinates. Keeping these estimates in mind, we return to the system of equations given in (27). Since these equations are linear in o $ / D $ the least squares solution is 2 *l *l si;2 - D~;(s;;. - si.. ),

where we are using s$ as the estimate of o$ and

In order to arrive at estimates of the uncertainty values oi and a, for the subjects and the stimuli, the estimates of the joint variances s$ given in (29) can be carried out one step further. To do this, however, requires distinguishing between two cases. Case I. Assume that the subject and stimulus uncertainty values are unique to each subject. Then the best that can be done with the estimate of the joint variances s$ is to set the subject uncertainty estimate si equal to some small arbitrary value and to solve for the stimulus uncertainty s, using

There is, however, the possibility that one of the uncertainty estimates might turn out to be negative. This can be avoided by letting the uncertainty parameter for subject Pi be defined by

Zinnes & MacKay

194

si = 1/2 min si, . i

(33)

This solution has the convenient property of equating the uncertainty estimate for subject Pi with the smallest stimulus uncertainty estimate for this subject. The estimates of uncertainty parameters given in (32) and (33) have severe limitations. Because of their non-uniqueness properties, it is not meaningful to compare the subject and stimulus uncertainty values over different subjects. It would only be meaningful to compare the stimulus uncertainty values within a single subject. Case 2. Assume that the stimulus uncertainty values do not differ for different subjects. Possible subject differences would then be solely reflected by differences of the subject uncertainty values. Thus, for this case there are exactly m subject uncertainty values and n stimulus values to estimate. The relevant equations for doing this are the equations: s' = sf = sQ, i = 1 ,

. . . , m,

j = m+l,

. . . , m+n

(34)

which is just a simple linear system of equations in the unknowns sf and sf. Consequently, the least-squares solution is "2

where 1

z SQ

s3 = ( l / m )

i

This solution has the property of equating the average stimulus variance with the average subject variance, that is, letting

Analysis of Preference Ratio Judgments

195

While this solution is convenient, it also has the undesirable property of allowing some variance terms to take on negative values. To avoid this, the minimum subject variance could be equated with the minimum stimulus variance. This is analogous to the approach taken in Case 1 . It can be accomplished by adding and subtracting a constant to the variance estimates obtained from (35) and (36). Specifically, if we let

. . ,m = m + l , . . . , m+n

2

i = 1,.

(41)

2

j

(42)

mi = m i n s i , 1

m, = min s,, i

then if mi is less than m,, we can define the new estimate si2 and s;’ in terms of the previous estimates, obtained from (35) and (36), by the equations si2 = s’ s;‘

+ (1/2) I mi - m, I

= sf - ( 1 /2) I mi - m, I .

(43) (44)

If the converse should be the case, namely that m, is less than mi, the same result can be achieved by interchanging the plus and minus signs in (43) and (44). The fact that the estimates of the uncertainty values are nonunique, as they were in Case 1, means that here too there are limitations as to which uncertainty values can be meaningfully compared. In the present case, it would be meaningful to compare the stimulus uncertainty values among themselves and similarly to compare the subject uncertainty values among themselves. It would not, however, be meaningful to compare the stimulus uncertainty values with the subject uncertainty values. This is to say, that within the framework of the probabilistic unfolding model we have assumed, it is not possible to discriminated between the variability due to the subjects and that due to the ideal points. And this is even true in the present case, where we have assumed that the subjects do not differ with respect to the uncertainty values associated with the stimuli. 5. Simulation I

In the previous sections, we have been concerned with developing a simple procedure for obtaining ML estimates of the parameters of a probabilistic, multidimensional choice model, using as data pairs ratio

Zinnes & MacKay

196

judgments. This procedure has consisted primarily of obtaining a simple, although approximate, expression for the likelihood function that is to be maximized and of obtaining a simple, although approximate, expression for the estimates of the parameters that can be used as the initial values of an iterative process. To determine how effectively and accurately this procedure works, two simulation studies were performed.

Y

...

: T

A

1

I

1

B

I

Figure 2. The original and recovered configurations of Simulation 1, Series 4. Panel A shows the original configuration. The 12 stimulus points are labeled A through F and 1 through 6; the 12 ideal points are labeled 0 through 2. The 6 stimulus points on the inner hexagon had an uncertainty value of 1.2. The remainder 6 stimulus points and 12 ideal points had an uncertainty value of . l . Panel B shows the configuration recovered from a nonmetric analysis; panel C the SV configuration, and panel D the ML configuration.

I

Analysis of Preference Ratio Judgments

197

Simulation I contained 24 points, 12 of which were treated as stimuli and the remaining 12 as ideal points. The 12 stimulus points were located on the vertices of two hexagons, one of which was completely contained within the other. The 6 vertices of the inner hexagon lie on a unit circle, while those on the outer hexagon lie on a circle of radius 1.732. The ideal points were randomly located throughout both hexagons. This arrangement of both the stimuli and ideal points is shown in panel A of Figure 2. Values of the uncertainty parameter were also assigned to each of the 24 points. The 6 stimuli on the outer hexagon and the ideal points were given the uncertainty value of .l. The 6 stimuli on the inner hexagon were assigned a series of larger values. In Series 1, each one of the six points on the inner hexagon was assigned an uncertainty value of .3. In the remaining 3 series, 2-4, these points were assigned the values .6, .9, and 1.2, respectively. Since the points on the inner hexagon lie on a unit circle, it can be seen that in Series 3 and 4, these points actually had substantial amounts of uncertainty. These large uncertainty values were selected for the points on the inner hexagon, because such values tend to make it very difficult for the estimation procedure to recover the underlying stimulus configuration (Zinnes & MacKay, 1983). The simulated data in each of the four series consist of 12 sets of 66 pairwise ratio judgments of the 12 stimuli, one set for each of the 12 subjects. This corresponds to a complete set of data for each subject, when it is assumed that the subject do not replicate the judgments, that is, do not judge each set of stimuli more than once. Preference ratio judgments were constructed by randomly sampling from the bivariate normal distribution, this means invariances were determined in each of the four simulations as indicated previously. In some cases, however, it turned out to be highly desirable to make a slight modification of some of the preference ratios calculated by this process. There were a few instances in which the values of the interpoint distances obtained from these random samples prove to be extremely small or extremely large, thus resulting in either a very small or very large ratio. Since these extreme values had a strong biasing influence on the estimates of the uncertainty parameter, it was considered to be desirable to place upper and lower bounds on the numerical values of the preference ratios. Consequently, a lower bound of .1 and an upper bound of 10 was

198

Zinnes & MacKay

arbitrarily imposed on these ratios. Calculated ratio values below the lower bound of .1 were set equal to .1 and those above 10, were set equal to 10. The effect of using the upper and lower bounds is explored in the following section. The sets of data generated in each of the four simulations were analyzed using three different estimation procedures: the SV and h4L procedures discussed in the previous section, as well as by KYST, a typical nonmetric (NM) procedure (Kruskal, Young, & Seery, 1973). The KYST analysis was performed by converting the data to I scales and using the standard options of that program that are relevant for analyzing I scales. This includes the options: “Split by rows”, stress 2 (which gave better results that stress l), “lower corner matrix”, a value of STRMIN equal to .OO01, and a starting configuration determined by the TORSCA option, which gave lower stress values than those obtained using the true parameter estimates. In all cases, the KYST analysis terminated normally, before reaching 200 iterations. It will be recalled that there are two types of parameters to estimate: the coordinates and uncertainty values. Since the 12 stimuli and 12 ideal points are embedded in a 2-dimensional space, there are 48 + 3 or 51 parameters to estimate in all. However, the actual number of independent parameters is somewhat less than this, because of the uniqueness properties of these parameters. The preference ratio judgment is, in fact, invariant over translation, rotation and stretching of the coordinate axes. Because of this, the number of independent coordinates to estimate is actually equal to 47. Table 2 shows the degree to which the configuration of stimulus and ideal points was recovered by each of the estimation procedures. Two different measures of recovery, R and D 2 ,are shown in this table: R is the correlation between corresponding interpoint distances in the true and estimated configuration; D is the sum of squared differences between optimally aligned coordinates of the true and estimated configuration. The origins of the coordinate axes for both configurations were placed at the centroid. Both R and D 2 were calculated using the 12 stimulus points and the 12 ideal points. Table 2 makes it clear that the accuracy of the three estimation procedures, while quite good at low levels of uncertainty, deteriorates as the level of uncertainty increases. This is to be expected, because the higher

Analysis of Preference Ratio Judgments

199

Table 2. Hexagon example: recovery of distances and coordinates

0

.3 .6 .9 1.2

sv .960 .94 1 365 .785

Correlation (R)' ML

Nonmetric

.998 .994 .986 .968

.953 .916 .659 ,579

Squared differences (D 2 ) b .3 .6 .9 1.2

352 1.353 3.630 6.875

.067 .153 .354 .926

1.201 2.04 1 1.329 9.312

R is the correlation between corresponding interpoint distances of the true and estimated configurations. It includes points of both stimuli and individuals. D 2 is the sum of squared differences between optimally aligned coordinates of the true and estimated coordinates. It includes the points of both stimuli and individuals. levels of uncertainty would tend to produce ratio judgments having a higher degree of variability. It is therefore reasonable to expect the standard errors of the coordinate values to be a function of the uncertainty values. It is also evident from this table that the rate of deterioration differs for the three estimation methods. The accuracy of the ML estimates seem to decrease only slightly with increases in the level of uncertainty. The deterioration of the SV estimates is somewhat greater and that of the NM (the nonmetric estimation procedure) greater still. These conclusions are consistent with both the R and D 2 statistics. In general, it appears that the MI., estimates of the coordinates are actually quite good, even when the levels of uncertainty are substantial. These conclusions are also evident from the plots shown in Figure 2. This figure shows the configurations recovered from the three different estimated procedures for Series 4, the one containing the highest level of uncertainty. The plots in this figure show the locations of both the 12

Zinnes & MacKay

200

stimulus and the 12 ideal points. For comparison purposes, Figure 2 also shows, in panel A, the location of the stimulus and ideal points in the true configuration. Except for stimulus point 2, located on the inner hexagon, it can be seen that the ML configuration, is as expected from Table 2, exceedingly accurate, even at this high level of uncertainty. The SV and the NM configurations are, as expected from Table 2, considerably less accurate at this high level of uncertainty. This is particularly true of the NM configuration, where the inner hexagon is quite highly distorted. The positions of stimulus points 1 and 2 are in fact reversed from their true position, while stimulus points 1 and 5 actually coincide in the NM configuration. The ideal points, in the three recovered configurations shown in Figure 2, have properties that are very similar to those of the stimulus points. The accuracy of the ideal points in the ML, configuration is substantially better than that in the SV and NM configurations. In fact, the locations of the ideal points in the NM configuration are especially poor. The ideal points T, X, Y and Z, while located within the outer hexagon in the true configuration, have actually been placed well outside this configuration in the NM configuration. In addition, the ideal points U, Q, S, and 0, while having very distinct positions in the true configuration, actually coincide in the NM configuration. Table 3. Hexagon examplc: rccovery of unccrtainly valucs

.3 .6 .9

1.2

.1

.I ,I .1

.368 .44 1

.so0 SO7

,095

.I13 ,152 .293

.303 ,606

384 1.123

.OX8 .089 .089

.088

Note: 0 1 and 0 2 arc thc uncertainty valucs of thc coordinatcs

of thc inncr and outer hexagons, rcspcctivcly.

Table 3 gives some indication of how well the uncertainty values are recovered by two of the estimation procedures, the SV and the ML

Analysis of Preference Ratio Judgments

201

procedures. Estimates using nonmetric methods are not shown in this table because those methods are purely deterministic and therefore do not provide for an estimation of variances. The uncertainty values shown in this table are consistent with the coordinates that were obtained in the process of calculating the sum of squared differences D 2 . In other words, as the same multiplicative factor was applied to the joint variances were applied to the coordinates in the process of optimally aligning the estimated configuration with the true configuration. This is appropriate, are only determined up to a multiplicative because the joint variances transformation. To arrive at the uncertainty values of the stimuli and the ideal points, the standardization discussed in the previous section was used. The minimum uncertainty values of the stimulus was set equal to the minimum uncertainty value of the ideal points. In the present case, this effectively means setting the uncertainty value of the ideal points equal to the uncertainty value of the stimulus points on the outer hexagon. Table 3 shows that the accuracy of the uncertainty estimates depends on the magnitude of the true value. The large uncertainty values are not estimated as well as the smaller values, although the ML estimates of the larger uncertainty values are actually quite good. In contrast, the SV estimates do deteriorate substantially at the higher levels of uncertainty. This is to be expected, since they were derived by assuming that the interpoint distances are large relative to the sizes of the joint variances.

05

05

6 . Simulation I1 In the previous simulation, lower and upper bounds were placed on the ratio judgments, to avoid the effects of extreme values biasing the parameter estimates. To determine whether, in general, such limits should be used, an additional simulation study was performed, one in which the number of replications of the preference ratio judgments was varied. Four levels of replications were used: stimuli on the inner hexagon were set equal to .3. As in the previous simulation, the stimuli on the outer hexagon were assigned an uncertainty value of .l, as were the ideal points also. The configuration of stimuli and ideal points used in this simulation was identical to the one used in the previous simulation.

Zinnes & MacKay

202

The data from the simulation were analyzed using two different approaches. In the No Limits approach, the ratio judgments obtained from random samples of coordinates were not modified, even when under some conditions they produced extreme values. In the Fixed Limits approach, lower and upper bounds of .1 and .10 were imposed on the ratio judgments. Table 4. Hexagon example: SV estimates of the Uncertainty parameter for different number of replications

Replications

1 2 4 8

No limits"

.252 .238 .300 .293

.187 .145

.116 .112

Fixed limits'

.368 .338 .321 .334

.095

.WO .070

.068

Note: The correct values of 01 and 0 2 are .3 and . I , respectively. No upper limit was placed on the ratio judgments to be

analyzed. Lower and upper limits of the ratio judgments to be analyzed were set at .1 and 10, respectively. The results for both approaches are shown in Table 4, for both the SV and ML estimation procedures. The values in this table show quite unmistakably that under the No Limits approach, the estimates of the uncertainty values become increasingly more accurate as the number of replications increase. This is not the case, however, for the Fixed Limits approach. Although the estimate of 01 improved slightly with increases in replication, the estimate of 0 2 becomes appreciably worse. Furthermore, although the No Limits approach is slightly better than the No Limits approach at low levels of replication, it is clearly the case that the No Limits approach is superior at the higher levels of replication. Thus, a concern for possible biasing effects of extreme values of the preference ratio appears to be justified only when there are few if any replications of the preference ratio judgments.

Analysis of Preference Ratio Judgments

203

These results seem very encouraging. Even though the data of these simulation studies are highly artificial, they seem to show that the maximum likelihood estimation procedure does amazingly well in recovering the location and the uncertainty values of the stimuli and the ideal points, even in the presence of a high degree of uncertainty. Of course, these results depend upon the validity of the underlying probabilistic model with which we are working. If the model were inappropriate in some situation, one could not expect the maximum likelihood estimation procedure, or any estimation procedure, to accurately estimate the unknown parameter values. It should also be recalled that the SV procedure was only intended to provide the starting values for the ML iterations. The fact that the SV estimates are as good as they are, given their relative simplicity, is also encouraging. Thus under normal conditions, when the data do not contain substantial amounts of variability, the ML iterations should converge fairly rapidly.

References Anderson, N. H. (1982). Methods of information integration theory. New York: Academic Press. Ashby, F. G., & Tonwsend, J. T. (1986). Varieties of perceptual independence. Psychological Review, 93, 154-179. Bechtel, G. G. (1976). Multidimensional preference scaling. The Hague: Mouton. Birnbaum, M. H. (1982). Controversies in psychological measurement. In E. Wegener (Ed.), Social attitudes and psychological measurement. Hillsdale, NJ: Erlbaum Biickenholt, I., & Gaul, W. (1984). A multidimensional analysis of consumer preference judgments related to print ads. Methodological advances in marketing research in theory and practise. EMACESOMAR Symposium, Copenhagen. Bulgren, W. G. (1971). On representations of the doubly non-central F distribution. Journal of the American Statistical Association, 66, 184186. Carroll, J. D. (1972). Individual differences and multidimensional scaling. In R. N. Shepard, A. K. Romney, & S . B. Nerlove (Eds.),

204

Zinnes & MacKay

Multidimensional scaling: Theory and applications in the behavioral sciences (Vol. 1). New York: Seminar Press. Chandler, J. P. (1969). STEPIT - Finds local minima of a smooth function of several parameters. Behavioral Science, 24, 81-82. Coombs, C. H. (1964). A theory of data. New York: Wiley. Croon, M. (in press). A comparison of statistical unfolding models. Psychometrika. De Soete, G., Carroll, J. D., & DeSarbo, W. S. (1986). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data. Journal of Mathematical Psychology, 30, 28-41. Eisler, H. (1982). On the nature of subjective scales. Scandinavian Journal of Psychology, 23, 161- 171. Hefner, R. A. (1958). Extensions of the law of comparative judgment to discriminable and multidimensional stimuli. Doctoral dissertation, University of Michigan. IMSL (1979). IMSL library reference manual. New York: International Mathematical and Statistical Libraries, Inc. Kendall, M. G., & Stuart, A. (1961). The advanced theory of Statistics (Vol. 2). New York: Hafner. Kendall, M. G., & Stuart, A. (1963). The advanced theory of statistics (Vol. 1, 2nd 4.).New York: Hafner. Kruskal, J. B., Young, F. W., & Seery, J. B. (1973). How to use KYST, a very flexible program to do multidimensional scaling and unfolding. Bell Telephone Laboratories, Murray Hill, NJ. MacKay, D. B., & Zinnes, J. L. (1981). Probabilistic scaling of spatial judgments. Geographical Analysis, 13, 21-37. MacKay, D, B., & Zinnes, J. L. (1986). Probabilistic multidimensional scaling of spatial preferences. In R. Golledge & H. Timmerman (Eds.), Behavioral modeling: Approaches in geography and planning. New York: Croom-Helm. MacKay, D. B., Ellis, M., & Zinnes, J. L. (1986). Graphic and verbal presentation of stimuli: A probabilistic MDS analysis. Advances in Consumer Research, 13, 529-533. MacKay, D. B., & Zinnes, J. L. (in press). Probabilistic multidimensional scaling of residential preferences: An experimental evaluation. Cahiers de Gebgraphie de Besuncon.

Analysis of Preference Ratio Judgments

205

Patnaik, P. B. (1984). The non-central chi-square and F distributions and their approximations. Biometrika, 36, 202-232. Saaty, T. L. (1980). The analytic hierarchy process. New York: McGraw-Hall. Scheffd, H. (1959). The analysis of variance. New York: Wiley. Schdnemann, P. H. (1970). On metric multidimensional unfolding. Psychometrika, 35, 349-366. Sjoberg, L. (1967). Successive intervals scaling of paired comparisons. Psychometrika, 32, 297-308. Suppes, P., & Zinnes, J. L. (1963). Basic measurement theory. In R. D. Luce, R. R. Bush, & E . Galanter (Eds.), Handbook of mathematical psychology (Vol. 1). New York: Wiley. Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273-286. Zinnes, J. L., & Kurtz, R. (1968). Matching, discrimination, and payoffs. Journal of Mathematical Psychology, 5, 392-421. Zinnes, J. L., & Griggs, R. A. (1974). Probabilistic multidimensional scaling unfolding analysis. Psychometrika, 39, 327-350. Zinnes, J. L., & Wolff, R. P. (1977). Single and multidimensional samedifferent judgments. Journal of Mathematical Psychology, 16, 30-50. Zinnes, J. L., & MacKay, D. B. (1983). Probabilistic multidimensional scaling: Complete and incomplete data. Psychometrika, 48, 27-48.

This Page Intentionally Left Blank

New Developments in Psychological Choice Modeling G . De Soete, H. Feger and K. C. Klauer (eds.) 0 Elsevier Science Publisher B.V. (North-Holland), 1989

207

TESTING PROBABILISTIC CHOICE MODELS Patrick M. Bossuyt Erasmus University, Rotterdam, The Netherlands Edward E. Roskam University of Nijmegen, The Nctherlands A framework for the concepts “model”, “theory” and “data” within the context of probabilistic choicc models is presented. What is commonly called a test of a model comes down to an asscssment of the goodness-of-lit relation between a model of a theory and a model of data. A distinction is made between theories that lead to ordinal restrictions on choice probabilities (Type I), and theories that specify a functional relation between utilities and choice probabilities (Type 11). The study of probabilistic choice behavior could benefit from an assessment of the goodness-of-fit of Type I theories with statistical methods usually reserved for Type I1 theories.

1. Introduction

Probabilistic choice theory, and with it, the use of probabilistic choice models, originated from the psychophysical laboratories. The primary aim was the measurement of subjective sensations in a way that could be theoretically defended. Gradually, the ideas became intertwined with notions from classic algebraic utility theory and axiomatic measurement theory. During the past decades we have seen a further development of probabilistic choice theory in psychology, largely separated from a

This research was made possible by grant 00-40-30 from the Dutch Foundation for the Advancement of Pure Science Z.W.O. This paper is a revised version of an article published in Cornrnunicalion & Cognition, 1987,20,5-16.

208

Bossuyt & Roskam

growing field of behavioral decision studies. On the other hand, some of the basic ideas have been taken up by economists, which led to the development of a research area known as (probabilistic) discrete-choice modeling. Despite this growing tradition in probabilistic choice models both in psychology and economics, anyone trying to get acquainted with the literature is hindered by a surprising lack of unanimity in the terminology, especially when it comes to the use of words as “theory” or “model”. As a consequence, it will seldom be immediately clear what is meant when we hear that a “model is tested”. We were confronted with this difficulty in a research project within which we were to compare a numerous subclass of probabilistic choice theories and models known as “probabilistic unfolding”. In order to define what was actually at stake in this comparison, we were forced to construct a conceptual framework with a clear, distinct meaning for “model” and “test of a model”, and within which the relation between theory and observations could be adequately described and evaluated. A sketch of this framework is presented in this paper. We do not present an overview of probabilistic unfolding, for which the reader is referred to Croon (in press) and Bossuyt and Roskam (1985). The probabilistic choice theories mentioned in this paper only serve an exemplary purpose. They are discussed within their axiomatized form, with a variable set and a variable binary choice probability form, the familiar set-theoretical and logical constants, and the usual predicates and operations on the reals. Within the class of probabilistic choice theories, we will make a rather uncommon distinction between Type I and Type 11 theories. Any probabilistic choice theory assumes some form of regularity in the choices people make. In Type I theories these regularities are defined through ordinal or equality constraints on the choice probabilities. In Type XI theories the regularity resides in relations between the choice probabilities and a representation by means of a real-valued utility function, defined over the set of alternatives. Close to its meaning in logic, a possible realization of a theory will be a system of the appropriate set-theoretical structure: for a theory on probabilistic choice, this will be an ordered couple consisting of a set of alternatives and a set of choice probabilities (a choice probability function). A possible realization of a theory will be called a model of this theory if it

Testing Probabilistic Choice Models

209

satisfies all its valid sentences: its axioms and all sentences that can be logically derived from them. Within our view, a model of a theory cannot be a model of the data, for the simple reason that probabilities are in general not part of the recorded data, which means that they cannot be derived from the observations through a coding or classification procedure. At most they can be estimated, but never observed. Corresponding to the idea of a possible realization of a theory, we therefore use, following Suppes (1962), the notion of a possible realization of the data. This is again a set-theoretical structure of the appropriate type, containing all the information needed to test the theory in question. A possible realization of the data will be called a model of the data if the information it contains is valid. Along this exposition, we will defend the thesis that the relation between assumptions on probabilistic choice and observations should more frequently be evaluated through Type I theories, and with a statistical approach similar to the one that has up till now been followed for Type I1 theories. The difference between the use of Type I theories and Type II theories for evaluating the theory-observations relation parallels an existing distinction between “scaling” and the evaluation of necessary and sufficient axioms for a homomorphic mapping of an algebraical relational system on a numerical relational system in axiomatic measurement theory (Krantz, Luce, Suppes, & Tversky, 1971, pp. 32-33). 2. Models of Theory To ease the exposition, we will restrict ourselves to (forced) binary choices, or paired comparisons, and a single choosing entity, either a single subject, a group or a population. In the following we will refer to this entity as “the subject”. In a binary choice situation we have a nonempty set of alternatives from which option sets of two elements are constructed. Out of each option set, the subject has to choose one, and only one, element, the no-choice option being eliminated. A probabilistic choice theory describes a choice from an option set probabilistically. This means, as the result of an independent Bernoulli mal, with a particular choice probability which is supposed to remain constant over repeated presentations of the same option set. A possible realization of a theory of probabilistic binary choice is then an ordered couple

Bossuyt & Roskam

210

4 , p > satisfying the following: BCP.l S is a nonempty set BCP.2 p is a real-valued function defined on S x S as follows : v x , y E s, x # y : a. 0 I p ( x , y ) I1 b. P @ , Y ) + p b , x ) = 1 c. p (x,x) = 112. We will call such a couple 4 , p > a binary choice probability (BCP) system, and read p ( x , y ) as “the probability that the alternative x is chosen out of an option set consisting of x and y only”. In addition to the fundamental probabilistic assumptions, all probabilistic choice theories assume some form of consistency or regularity in the choice probabilities. We will distinguish between two classes or theories, which we will introduce with some simple examples.

2.1 Theories of Probabilistic Choice - Type I In a first class of theories the consistency is imposed through a number of constraints on the choice probabilities. A simple theory capturing this consistency assumption is the “weak stochastic transitivity” theory T I , defined through the following axiom A. 1: A.l V x , y , z E S : P ( X , Y ) 2 p ( Y , x ) & p ( Y , z ) ~ P ( z , Y )imply p ( x , z ) 2 p ( z , x ) . Loosely interpreted, this axiom expresses the notion that if a subject is inclined to choose x out of { x , y } and y out of { y , z ) , this subject will also be inclined to choose x out of {x,z}. In a way, weak stochastic transitivity is the simplest probabilistic analogue of the rational transitivity prescription in algebraic utility theory. A second theory, which we will call T 2 , can be defined through the “substitutability” axiom A.2: A.2 V x , y , z E S : P(X,Y) 2 p ( Y , x > ifs p ( x , z ) 2 p b z ) . This axiom states that whenever a subject is inclined to choose x rather than y out of { x , y } , the probability of choosing x out of an option set with a third element will exceed that of choosing y , and vice versa. To

Testing Probabilistic Choice Models

21 1

complete our set of examples of Type I theories, we present T 3 . It is defined through axiom A.3 which states that the probability of two intransitive choice cycles is the same in any subset of three alternatives:

A.3 V x , y , z E S : P(X,Y)P(YJ)P(ZJ)=P(X,Z)P(Z,Y)P(YJ). So far we have three examples of Type I theories of probabilistic choice. In each theory a distinct form of consistency is imposed on the choice probabilities, either through ordinal constraints (A.1, A.2) or through an equality (A.3). For any of the theories on probabilistic choice, a possible realization will be called a model of theory Ti if it satisfies the corresponding axiom A.i. If a BCP system (as defined through BCP.1, BCP.2) is given, testing whether or not it is a model of a particular theory then becomes a straightforward task: one simply checks the relevant axioms. As an example, consider the BCP system <E,p‘>: E = { e , f , g ) p ’ ( e , f ) = 3/5 p ’ ( f , g ) = 2 / 3 p‘(e,g) = 7/10.

(1)

It is easy to see that this BCP system satisfies weak stochastic transitivity (A. 1) and is therefore a model of T 1. It satisfies substitutability (A.2) and is a model of T 2 , but it fails to satisfy A.3, so it fails to be model of theory T 3 . The three theories we presented as an example are hierarchically related to one another. Theory T1 is a subtheory of T 2 : substitutability implies weak stochastic transitivity and, hence, any model of T 2 will also be a model of T I . The same holds for T z and T 3 : T z is a subtheory of T 3 . As a consequence, if a BCP system fails to be a model of T 2 , it cannot be a model of T 3 . In the following subsection we turn to the Type I1 t heories.

2.2 Theories on Probabilistic Choice - Type I1 Whereas in the Type I theories the consistency was imposed through constraints on the choice probabilities themselves, in Type I1 theories the regularity in choices is defined through a representation by means of a real-valued function defined over the set of alternatives S (or, alternatively, its product set). Usually the latter function is interpreted as a utility function. Here also we will present three simple examples.

Bossuyt & Roskam

212

-

The first example is the weak utility theory T 4 : A.4 V x , y

E

S : p(x,y) 2 p(y,x)

u ( x )2

uQ).

This theory states that there exists a utility function u defined over S, with a tendency to choose x rather y if the utility of x , u (x), exceeds that of y . A second example is a Fechnerian theory T g: A S For a real-valuedfunction H : V X ,y E S: p ( ~ , y=) H [ u ( x ) ~ ( y ) ] .

-

As a third example we present the strict utility theory T g :

How can we test whether or not a BCP system is a model of any of these Type I1 theories? If we succeed in finding an adequate representation, this means, a utility function u for which the choice probabilities satisfy the relevant axiom(s), we have demonstrated the model relation. Take again the BCP system <E,p’> (1). It is a model of the weak utility theory T4 (or, alternatively, “it is a weak utility model”) since the choice probabilities satisfy axiom A.4 with the following function u‘ on E : u‘(e) = 3, u’cf) = 2, u’(g) = 1. We were not able to find a similar function to satisfy either A S or A.6, so we cannot make a decision on the model relation between <E,p’> and the theories T 5 and T 6. Fortunately, there exists another way to test these model relations. Ever since the late fifties, a number of authors have been studying the equivalency relations between what we have labeled Type I theories and Type II theories (see for example Block & Marschak, 1960; Luce & Suppes, 1965). Simultaneously, the study of the formal foundations of measurement led to the formulation of sets of necessary and sufficient conditions for homomorphic mappings of certain algebraic relational systems to numerical relational systems (Krantz et al., 1971; Suppes & Zinnes, 1963). The study of probabilistic choice has also taken advantage of these results. Some examples might illustrate these relations. (We refer to the authors mentioned earlier for proofs of the results to follow.) If a BCP system 4 , p > is a weak utility model, then it is also a model of T 1, but not conversely. So axiom A . l is a necessary, but not a sufficient

Testing Probabilistic Choice Models

213

condition for a BCP system to be a model of the weak utility theory T 4 . If 4 , p > is a model of T 1, then it is also a model of T4 if S is finite. A second example: if a BCP system 4 , p > satisfies A.6, and, hence, is a model of theory T 6 , then it will also satisfy A.3 and be a model of T 3 , but not conversely. The converse relation holds if the set S is finite, and vx,y E

s: 0 < p ( x , y ) < 1.

A third example: a BCP system satisfying A S will also satisfy A.2, but no sufficient conditions are known for a BCP system satisfying A.2 to be a model of T S. These relations can help us a great deal in evaluating the model relation between a BCP system and a Type I1 theory. If the system fails to satisfy a necessary condition, the model relation has to be rejected; if it satisfies a set of sufficient conditions, it is a model of the corresponding theory. In other cases, no decision can be made. Take again the BCP system <E,p‘>. It failed to satisfy A.3, so it cannot be a strict utility model. However, it satisfies A.2, so it still may be a model of a Fechnerian theory T 5 . A number of authors share the opinion that only Type I1 theories can be called probabilistic choice theories; a view that reveals itself when the Type I theories are discussed as “observable properties” (as in Luce & Suppes, 1965). We think that this distinction is unwarranted, and incorrect. Type I and Type 11 theories differ in the way the regularity in (probabilistic) choice is defined: in the first class of theories the constraints are defined on the choice probabilities themselves, whereas in the second class of theories they are related to a representation by a utility function. In a way, both classes of theories can be seen as defining “properties” on the choice probabilities, but neither the first set of axioms nor the second is “observable”. It will be obvious that the utility functions u are unobservable, but the same holds for the binary choice probabilities. Since we never observe choice probabilities a BCP system cannot be a possible realization of empirical data. How a researcher interested in the relation between a probabilistic choice theory and a body of observations can proceed is discussed in the following section.

Bossuyt & Roskam

214

3. Models of Data Corresponding to possible realizations of theory, we will use the notion of possible realizations of data as a help in evaluating the theory-data relation. As far as we know, Suppes (1962) has been the first to make this distinction. Due to the background of probabilistic choice modeling, our conception of “models of data” differs slightly from his. A possible realization (or valid interpretation) of the data will be a set-theoretical structure of the appropriate type, designed to incorporate all the information about the experiment which can be used in tests of the adequacy of the theory. In our view, “data” refers to “recorded data”: everything that can be obtained from empirical observations through a coding or classification procedure. The “data” as data are never observed: “behavior does not yield data by parthogenesis” (Coombs, 1964). It may be obvious that not all observations will be coded. All details that appear inessential for the intended use will be omitted. Ultimately, what is to be coded will depend on the conceptual framework used. For the probabilistic choice theories introduced earlier, an ordered couple d,C>will be a convenient realization of the data if it satisfies the following: BCS.l S is a nonempty set BCS.2 C is a set of indexed binary-valued functions q , defined on S x S as follows: either one of the following holds a. c&,y) = 1 & c l ( r , x ) = 0 b. c~(x,Y)= 0 & c ~ @ , x = ) 1 c. q ( x , y ) = 0 & q ( r , x ) = 0. If these conditions are fulfilled, we call d , C > a binary choice (BC) system. We read cl(x,y) = 1 as “the alternative x has been chosen out of the option set { x , y } on occasion 1” and q ( x , y ) = 0 as “the alternative x has not been chosen out of the option set {x,y} on occasion 1”. Condition BCS.2.c implies that the option set {x,y} has not been presented on occasion 1. A BC system will be called a model of the data if its components are valid, both within the sense of the empirical observations and the

Testing Probabilistic Choice Models

215

fundamental probabilistic assumption. Here “validity” has to be interpreted as “truth preserving”. One side of this quality is easily understood: if a subject chooses a out of { a , b } on occasion h, and a BC system contains ch(a,b) = 0, it obviously cannot be a model of the data. However, a complete evaluation of this validity relation is not easily made, since it involves checking a plentitude of assumptions, related to the data collection procedure used, the experimental design, and what Suppes (1962) has called “ceteris paribus conditions”: disturbing environmental conditions, such as unwanted noise, bad lighting, and so on. To mention one, our basic probabilistic assumptions imply that every presentation of an option set of two alternatives can be treated as equivalent to any other presentation of this option set. This assumption will not be automatically met and will require a careful design of the choice environment. Though this subject certainly deserves more-elaboration, we will not expand on it in this paper. 4. Theory Versus Data: Evaluating Goodness-of-Fit Suppose we have a theory on probabilistic choice and a set of observations. We will be interested in the relation between observations and theory. This means that we are interested in the question whether out theoretical assumptions can be maintained in the light of the empirical observations made. This does not means that we will try to evaluate whether or not the model or the data - the binary choice system - is a model of the theory. This cannot be done, for it would imply checking the relevant axioms: a senseless job, since these are expressed in terms of choice probabilities, and the model of the data only contains choices. For Type I1 theories the following strategy is usually adopted. Given a model of the data, one mes to construct a model of the theory corresponding maximally to it. Most frequently, the likelihood will be the correspondence criterion. The likelihood function expresses the joint probability of a BC system, given the estimated BCP system. Because of the fundamental probabilistic assumption, each choice (i.e., each q ( x , y ) ) is a realization of an independent Bernoulli trial, governed by the choice probability p ( x , y ) . To give an example, to construct a strict utility model of theory T g , the utility function u is sought maximizing the likelihood, where the choice probabilities are calculated through A.6. To construct a

Bossuyr & Roskam

216

Type I theories

Type I1 theories

1 model of theory goodness-of-fit relation

I

model of data

I

empirical observations

Figure 1. The theory-observations relation in probabilistic choice.

Fechnerian model of theory T 5 , a particular distribution function H is selected and the maximum likelihood utilities u are sought. The likelihood is calculated using AS. Suppose all this has been done. At this stage, a number of authors present some statistical decision procedure, calling it a “test of the model”. No matter what this procedure consists of, this terminology itself cannot be correct. Obviously, the BCP system obtained through the maximum likelihood strategy will be a model of the theory, by construction. On the other hand, the BC system had to be a model of the data, for, if not, the whole strategy could not have been adopted. What will actually be done then is an evaluation of the correspondence relation between the model of the theory and the model of the data, a more common word for “correspondence” in this context being “goodness-of-fit” (Figure 1). If the maximum likelihood method has been used, the likelihood function is the indicated device for this purpose. Within the Neyman-Pearson approach to statistical testing, the generalized likelihood-ratio test can be used to test the null hypothesis that the model of the data constitutes a set outcomes from a BCP system that is a model of the data, versus the alternative hypothesis that it is not.

Testing Probabilistic Choice Models

217

It is worth reemphasizing that this is not a test of the model relation, but an evaluation of the statistical correspondence between a model of the theory and a model of the data in terms of likelihood. Unfortunately, this likelihood ratio test is not very useful when it comes to evaluating why a model of the theory did not correspond to the model of the data. A failure of the likelihood ratio test tends not to be particularly instructive: it tells us that the goodness-of-fit is in fact rather bad, without indicating why. However, there exists a second way of evaluating the relation between empirical observations and Type I1 theories. Suppose we were able to construct a model of a Type I theory, fulfilling the sufficient conditions for being a model of a Type II theory also, and showing an acceptable goodness-of-fit with the model of the data. In that case we know, without having constructed a representation with the utility function(s), that the BCP system obtained will also be a model of the Type II theory. We do not claim that this idea of evaluating the goodness-of-fit of a model of the data with a Type I1 theory through a model of a Type I theory is a new one. However, the evaluation of the correspondence relation between models of the data and models of Type I theories has usually been done in an unsatisfactory way. Luce and Suppes’ (1965, p. 379) quotation still applies to a major practice in the field: Lacking satisfactory statistical methods, authors often simply report, for example the number of violations (...) and, on some intuitive basis, they conclude whether the failures are sufficiently numerous to reject the hypothesis. In olhcr situations, tables or plots of data are reported and the reader is lclt pretty much on his own to reach a conclusion. Because the rcsuIts are almost never clear cut, one is left with a distinct feeling of inconclusiveness. Usually the evaluation between observations and theory is not based on the goodness-of-fit relation between a model of the data and a model of the theory, estimated on the basis of the maximum likelihood principle. Instead a system of choice proportions is constructed and the model relation with the theory is evaluated. In general, a failure of the relevant axiom does not lead to a rejection of the model relation since, due to statistical errors, we are likely to find a number of violations. However, as Luce and Suppes remarked, the determination of the acceptable number of violations is based on rather intuitive grounds, and so is the assessment of

218

Bossuyt & Roskam

the relation between theory and observations. To give an example, take the following BC system, d ; , C > , which we take to be a model of the data, with F = { d , e , f , g , h } and C a set of 10 functions, from which the system @,k> of binary choice proportions can be obtained:

k ( d , e ) = 0.9 k ( d , f ) = 0.5 k ( d , g ) = 0.9 k ( d , h ) = 1.0 k ( e , f ) = 1.0 k ( e , g ) = 1.0 k ( e , h ) = 0.4 k ( f , g ) = 0.8 k ( f , h ) = 1.0 k ( g , h ) = 0.1. One can easily check that the system &,k> of binary choice proportions fails to be a model of the weak stochastic transitivity theory T I . There are three failures of A.l, due to the existence of an intransitive cycle in { e , f , h } . We may ask ourselves whether or not these three violations constitute enough evidence to reject the hypothesis that we can construct a model of T1 with an acceptable goodness-of-fit to the model of the data &,C>. We have no idea whether or not this is the case. Two of the intransitive triples can be made transitive simply by reversing one out of the ten choice out of the option set { d , f ) , and out of {e,g}. However, we can proceed in much the same way as we would do to evaluate the relation with a Type I1 theory. Taking the likelihood as the criterion, we can look for the “best fitting” model of T I . Using a branch and bound algorithm developed for this purpose (Bossuyt & Roskam, 1985) we found that the following BCP system &,p”> was the maximum likelihood model of the weak stochastic transitivity theory:

p”(d,e) = 0.5 p ” ( d , f ) = 0.5 p”(d,g) = 0.9 p”(d,h) = 0.5 p ” ( e , f ) = 0.5 p”(e,g) = 1.0 p”(e,h) = 0.5 p ” ( f , g ) = 0.8 p ” ( f , h ) = 0.5 p”(g,h) = 0.1. To test the null hypothesis mentioned earlier, with level 0.05, we used the log likelihood ratio. It turned out to be h = -26.676. From a series of simulations in which the behavior of the likelihood ratio under the null hypothesis was observed, we know that this value does not lie in the critical region ([-28.679,0.0], n = 1000). Therefore, we do not reject the null hypothesis. Since d;,p”> is a model of T 1 and since the set F is finite, the BCP system cF,p“> is also a weak utility model. A utility function u

Testing Probabilistic Choice Models

219

for which c F , p f f >satisfies A.4 is:

u(d)=2 u(e)=2 ucf)=2 u(g)= 1 u(h)=2 We can proceed in a similar way for other Type I theories. Contrary to evaluating the goodness-of-fit relation for each model of a Type I1 theory, the subsequent goodness-of-fit evaluation for models of hierarchically related Type I theories gives us more information on which theoretical assumptions on probabilistic choice did survive a confrontation with the data, and which did not. It is our conviction that the construction of appropriate BCP models based on the maximum likelihood principle offers both a more promising way of evaluating the relation between theoretical assumptions and empirical observations, and an approach to the representation of choice probabilities which is more in line with the principles of axiomatic measurement theory.

References Block, H. D., & Marschak, J. (1960). Random orderings and stochastic theories of response. In I. Olkin, S. Ghurye, W. Hoffding, W. Madow, & H. Mann (Eds.), Contributions to probability and statistics. Stanford, CA: Stanford University Press. Bossuyt, P. M., & Roskam, E. E. (1985). A nonparamemc test of probabilistic unfolding models. Paper presented at the 4th European Meeting of the Psychometric Society, Cambridge. Coombs, C. H. (1964). A theory of data. New York: Wiley. Croon, M. (in press). A comparison of statistical unfolding models. Psychomerrika, in press. Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971). Foundations of measurement theory (Vol. 1). New York: Academic Press. Luce, R. D., & Suppes, P. (1965). Preference, utility and subjective probability. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology (Vol. 3). New York: Wiley. Suppes, P. (1962). Models of data. In E. Nagel, P. Suppes, & A. Tarski (Eds.), Logic, methodology and philosophy of science. Stanford, CA: Stanford University Press. Suppes, P., & Zinnes, J. L. (1963). Basic measurement theory. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology (Vol. 1). New York: Wiley.

This Page Intentionally Left Blank

New Developments in PsychologicalChoice Modeling G. De Soete, H. Feger and K. C. Klauer ( 4 s . ) 0Elsevier Science Publisher B.V. (North-Holland), 1989

22 1

ON THE AXIOMATIC FOUNDATIONS OF UNFOLDING: WITH APPLICATIONS TO POLITICAL PARTY PREFERENCES OF GERMAN VOTERS Bernhard Orth University of Hamburg, FR Germany Sufficient conditions for the existence of a qualitative J-scale of the unfolding model are given in terms of an “unfolding structure”. This measurement structure is illustrated by a set of hypothetical data and then applied to an analysis of prcfcrence orderings of German political parties obtained from about 4,000 voters in 1969, 1972, and 1980. It turns out that these prefcrence orderings cannot be unfolded appropriately in either one or two dimensions. On the level of aggregated orderings according to the most preferred party, however, these exist one-dimensional unfolding solutions (representing the parties) as well as structurally simple graph theoretical representations of these groups of voters. The findings suggest the hypothesis that preferences for German parlics are determined by both the political left-right dimension and the preferred party coalition.

Earlier drafts of portions of this paper benefited very much from discussions with David H. Krantz and Clyde H. Coombs. The analyzed data are taken from the studies “Bundestagswahl 1969” and “Bundcstagswahl 1972” (stored by the Zentralarchiv fiir empirische Sozialforschung under ZA-Nr. 0426 and ZA-Nr. 06350637, respectively) and “ZUMA-Bus 1980” (Zentrum fur Umfragen, Methoden und Analysen e.V.). The 1969 and 1972 data are given in Norpoth (1970) and the 1980 data in Pappi (1983). I wish to thank Franz Urban Pappi for pointing out and correcting some errors in the 1980 data. Last but not least I thank Gesine Muller for her assistance in the data analysis. This paper is a revised version of an article published in Zeitschrifr fir Sozialpsychologie, 1987, 18, 236-249.

222

Orth

1. Introduction The measurement theoretical foundations of the method of unfolding (Coombs, 1950, 1952, 1953, 1964) are still poorly understood. Although some attempts have been made, e.g., by Suppes and Zinnes (1963), Ducamp and Falmagne (1969), and Krantz, Luce, Suppes, and Tversky (1971) necessary and/or sufficient conditions for the existence of either a qualitative or an quantitative J-scale have not yet been established. Suppes and Zinnes and Krantz et al. consider unfolding representations in terms of a single scale representing both objects (stimuli) and persons (or persons’ ideal points) whereas Ducamp and Falmagne propose a representation in terms of two different scales (one scale for the objects and the other one for the persons). Both types of representations can be said to characterize the unfolding model. But they do not correspond to the actual unfolding method which aims at the construction of a (qualitative or quantitative) stimulus scale and which provides just ordering information with respect to persons or ideal points. The approach taken in this paper closely corresponds to the actual method of unfolding. It is based on a representation suggested by Orth (1976), that takes into account a stimulus scale only. This representation, however, is not stated in terms of the primitive notions (i.e., the individual I-scales) but in terms of a defined relation on the set of objects. As a consequence, the formulation of axioms becomes fairly complex and a little bit cumbersome. For this reason and because of space limitations this paper gives sufficient conditions for the existence of a qualitative J-scale only. These axioms are stated i n terms of an unfolding structure in Section 2. The unfolding is illustrated by an example with hypothetical data (Section 3) and then applied to real data on preferences for political parties (Section 4). These data have been gathered from representative samples of Gemian voters in the years 1969, 1972, and 1980. The final Section 5 is devoted to a brief discussion of the tindings. 2. An Unfolding Structure Let A and P be sets and 2 be a binary relation on A x P . The sets A and P are conceived of as sets of empirical objects and persons, respectively, and the relation up 2 bp is to be interpreted as “ p (weakly) prefers h over a’’ or “the distance between u and p is larger than or equal to that

On the Axiomatic Foundations of Unfolding

223

between b and p.” (Note that unfolding is not concerned with up 2 b4 with p f 4; these cases will be excluded below.) A qualitative J-scale may be defined as a real-valued function @ on A, unique up to strictly increasing transformations, such that, for all a, b E A,

a

>* b

iff @ ( a )2 Q(b)

where >* is uniquely determined by 2. mere, “uniquely determined” essentially means that >* is appropriately defined in terms of 2; and exact definition will be given below.) This definition of a qualitative J-scale corresponds to the unfolding representation studied in this section. It turns out to be convenient to introduce a betweenness relation on the set A. Such a relation facilitates the formulation of sufficient conditions for the existence of a function Q satisfying the representation above as well as the construction of such a function. A betweenness relation can be defined in terms of 2 as follows: Definition I . Let A, P , and 2 be as above. For all a , b, c that b is bemeen a and c, denoted a I b I c, iff

up 2 bp, cp either a

f c

and c4

bq, aq

E

A, we say

and

or a = b = c, for some p, 4

E

P.

According to this definition, an object b is between two objects a and c (on the J-scale to be constructed) whenever there are two persons (say, p and 4 ) such that one of them prefers both b and c over a and the other one prefers both b and a over c. There are thus four combinations of two I-scales leading to a I b I c; these four cases are illustrated in Figure 1. Unfolding assumes that the individual preference orderings are singlepeaked. In terms of Definition 1, this basic property of single-peakedness can be stated as follows:

If a I b I c, then either up 2 bp or cp 2 bp (or both) for all a, b, c E A and p E P. Hence, whenever a 1 b I c, singlepeakedness is violated if there is a person preferring both a and c over b. Definition 1 also allows n-ary betweenness relations ( n > 3 ) to be defined quite naturally in terms of the ternary relation:

Orrh

224

fiil

IiJ ab

ac bc

ab

ac

bc

hill ab

ac bc

IIVI

ab

ac

bc

ap 2 b p 5 CP. c q L bq 2 aq

ap 2 b p z CP, cq L aq z b q

ap 2 CP z bp, cq a bq E aq

a p a cp z bp. cq z aq z b q

p ' cba q : abc

p: cba q: bac

p: bca q: abc

p: bca q: bac

Figure 1. Four cases of two persons' I-scalcs yielding thc the bctweenness relation a 1 b I c according to dclinition 1 .

a ( b ( c ( d iff a ) b ) c , a ) b ) dn, l c l d , a n d b l c l d ; and

a l b l c l d ) e iff a I b I c Id, a 1 bl c I e, a I b I d l e, a I c I d l e, and h I c I d l e ; and so on (for all a, b, c, d, e E A). These definitions almost directly yield a method for constructing a qualitative J-scale. Axioms for the existence of a qualitative J-scale must put suitable restrictions on the primitives A and P and especially on 2,and they have to make sure that the betweenness relation will satisfy some important properties needed for deriving a complete and consistent ordering of the objects. The conditions given in the next definition serve this purpose.

Definition 2. Let A be a set with a least two elements, let P be a nonempty set, and let 2 be a binary relation on A x P. The relational system U i , P , z > is an unfolding structure iff, for all a, h, c, d E A and p , 4 E P , the following four axioms hold: 1. Either up 2 bp or bp 2 up; and if up 2 hi/,then p = 4.

2. If ap 2 bp and hp 2 cp, then up 2 cp. 3. There exists an r

E

P such that ar 2 hr.

On the Axiotnatic Foundations of Unfolding

225

4. If a I b I c and a I c I d, then either bp 2 cp or dp 2 cp. According to Axiom 1, 2 is conditionally connected. That is, 2 holds only for pairs with a common element out of P, and for those pairs, it is connected. Thus, this axiom merely specifies the preference orderings to be individual ones and, together with Axiom 2 which asserts transitivity of these orderings, it assumes that these I-scales are weak orders. Axiom 3 ensures that there are at least two different I-scales. Note that otherwise unfolding could not be done. (As shown in the next section, this version of Axiom 3 actually is somewhat stronger than necessary.) It can be said that Axioms 1 to 3 together just described the kind of data typically used for unfolding. The crucial and empirically interesting assumption is Axiom 4. This is essentially a type of single-peakedness condition. Together with the other axioms. it implies the important property of transitivity of betweenness. 1. If a I b I c and a I c I d, then a I b I d and h I c I d; and

2. If a I b I c, b I c I d, and b

f

c, then a I h I d and a I c I d

(for all a, 6 , c, d E A). As in similar contexts (e.g., Orth, 1980), transitivity of betweenness gives rise to “unidimensionality” in the sense of a consistent ordering of the elements of the set A . Together with the following “technical” Condition C, Axioms 1 to 4 can be shown to be sufficient for the existence of a qualitative J-scale as stated in the theorem below.

Condition C. A contains a finite or countable subset A‘ such that there is E A’ with a I b’ I c, for all a, c E A.

b’

Theorem. Let 4 , P , > > be an unfolding structure satisfying Condition C. Then there exists a real-valued function @ on A, unique up to strictly increasing transformations, such that, for all cz, b E A , a 2* b

iff

@(a>2 ~ ( h )

where 2* is uniquely determined by 2.

Remarks. By “uniquely determined” it is meant that ?* is the only simple order on A with the property: if a 2* b and h 2* c, then a 1 b I c. The proof of the theorem is fairly simple and will be omitted because of space

226

Orth

limitations. Condition C is required because the unfolding structure (Definition 2) is not restricted to finite sets. If A is finite, however, Condition C can be dropped. 3. An Example with Hypothetical Data

This section gives a simple example with hypothetical data in order to illustrate how the unfolding structure can be applied empirically. Table 1 contains fictitious preference orderings of five objects a, b, c, d, and e from six persons p , q, r, s, t, and u. The first step is to determine the betweenness relation according to Definition 1. For every triple of objects, it must be checked whether a I b I c, b 1 a 1 c or a I c I b holds. (Note that a 1 b I c iff c I b I a holds.) It may happen that two or even all of these cases apply; this would be due either to ties within the I-scales or to violations of Axiom 4 of the unfolding structure. The data from Table 1 yield the betweenness relation given in Table 2. Table I . Fictitious preferences of six persons p , q, r, s t, and u for five objects a, b, c, d, and e cach.

Persons p q r S

t U

I-scales aecbd cadeb cdaeb eacbd acedb ebacd

Prefcrcncc orderings dP L bP L CP I eP L aP bq L eq L d4 I aq L cq br er 2 ar I dr cr ds bs 2 cs I as es bt dt I et I ct 2 at du cu 2 uii I bu 2 eu

Next, the axioms of the unfolding structure can be tested. The I-scales in Table 1 are connected and transitive. Thus, Axioms 1 and 2 are satisfied. Axiom 3, however, turns out to be violated because there is no person preferring d over c and no one preferring b over e. Nevertheless, it will be shown below that a qualitative J-scale does exist for the present data. Hence, this example shows that Axiom 3 is somewhat stronger than necessary. In order to test Axiom 4 one has to consider all those combinations of objects that satisfy the premise of this axiom. These cases are

On the Axiomatic Foundations of Unfolding

227

called “possible tests”, and for each possible test one has to check whether the corresponding conclusion of the axiom is satisfied for all (or perhaps for how many) persons. The present example yields the possible tests and conclusions of Axiom 4 as given in Tnble 3. The conclusion in the first line of this table is satisfied because all persons prefer either c over a or c over d (or both). Similarly, all the other conclusions turn out to hold for all persons. Thus, Axiom 4 is perfectly satisfied. Now, suppose that the I-scales in Table 1 are sufficiently distinct (in spite of the violation of Axiom 3). We can then conclude from the theorem in Section 2 that there exists a perfect qualitative J-scale for the present data. Table 2. Betweenness Relation Obtaincd From the Data in Table 1.

This scale indeed exists and it can be constructed as follows. According to Table 2, we have b I a 1 c, b I a I d, a I c I d , and b I c I d and hence b I a I c I d. Similarly, we obtain b I e I a 1 c, b I e 1 a I d, b I e I c I d, b I a I c I d, and e I a I c I d and thus the 5-ary betweenness relation b 1 e 1 u I c I d which gives the ordering of the objects on the qualitative J-scale. In case of violations of Axiom 4 for some objects or persons, this information can be used for constructing a “dominant J-scale’ ’ (Coombs, 1964). An empirical application of the unfolding structure (or at least a determination of the betweenness relation) might be useful even if strong violations of Axiom 4 are to be expected. Such an example will be studied in the next section. 4. Applications to Political Party Preferences

In this section, the unfolding structure will be applied to preference orderings of five German political parties obtained from representative samples

228

Orth

Table 3. Possible tests and conclusions of Axiom 4 when tested with the data in Table 1. (Conclusions must be tested V p E P . )

Premise

Conclusion

of voters in studies on the occasion of elections for the Federal Diet in the years 1969, 1972, and 1980. The five German parties used in these studies are given in Table 4, together with their relative position on the political left-right dimension. The 1969 and 1980 I-scales were obtained by pair comparisons and the 1972 I-scales by a rank order method. The complete and transitive preference orderings from 1969 and 1972 are given in Norpoth (1969, p. 355, Table 2) and those from 1980 in Pappi (1983, p. 432, Table 3). Since the latter ones contain some minor errors the corrected data (as well as the 1969 and 1972 data) are reproduced in Table 5. Norpoth (1979) did a multidimensional unfolding analysis (with the program MINIRSA; cf. Roskam, 1977) on the 1969 and 1972 data. He obtained one- and two-dimensional solutions and made an effort to interpret these solutions although several ones quite obviously were degenerate. Pappi (1983) examined these data and those from 1980 more carefully. He argued that the preference orderings of the majority of the voters cannot be unfolded to any dimensional structure. He then studied more closely those voters whose preferences were compatible with a Jscale corresponding to the left-right dimension. The present section gives a reanalysis of the three sets of data in terms of the axiomatic approach outlined above.

On the Axiomatic Foundations of Unfolding

229

Table 4. Five German political parties used in representative studies in 1969, 1972, and 1980. (The ordcr from top to bottom corresponds to the parties position on lhe political leftright dimension as rated by expert as we1 as by voters, e.g., Klingemann, 1972.)

K

DKP

S

SPD

F

FDP

C

CDU/CSU

N

NPD

Deutsche Kommunistische Partei (German Communist Party) (1969: Aktion DemoklratischerFortschritt) SozialdemokratischePanci Deutschlands (Social Democratic Party) Freie Demokratische Partei (Free Democratic Pany) Christliche Demokratischc Union/ Christliche Soziale Union (Christian Democratic Union/ Christian Social Union) NationaldemokratischcPartei Deutschlands (National Democratic Parly)

Table 5 contains a total score of more than 4,000 preference orderings. In determining the betweenness relation according to Definition 1 (for each set of data), it is immediately seen that all the three cases a I b I c, b I a I c, and a I c I b hold with respect to every triple (a,b,c) of the five parties K, S , F, C, and N. It follows (cf. Section 3) that Axiom 4 of the unfolding structure is violated. It is also easily seen that these violations are not accidental and thus cannot be attributed to chance. We therefore dispense with a detailed test of that axiom. For each set of data, there is clearly no (one-dimensional) qualitative J-sca1e . On the other hand, determining the betweenness relation also reveals clear differences with respect to the number of I-scales yielding either a I b I c or b I a I c or a I c I b for triples of parties. A closer look at the betweenness relation will show some systematic regularities and thereby provide some insights on why there is no qualitative J-scale. Table 6 gives the percentages of those persons whose preference ordering are not compatible with either a I b I c or b I a I c or a I c I h (for all mples of the five parties). For example, the table shows that many I-scales are

230

Orth

Table 5. I-Scales of the five German partics given in Tablc 4 together with their frequencies in representative samplcs of voters in the years 1969 (N = 907), 1972 (N = 1785), and 1980 ( N = 1316). (After Norpoth, 1979, and Pappi. 1983.) 1969

FSNCK FSNKC FSCNK FSCKN FSKNC FSKCN FKNSC

1972

1

6 16 1 1

2 1 35 33 2 3

50

1 1 1 1 7 20 1 94 1

-

FKSCN SNCFK SNCKF SNFKC SNKCF SCNFK SCFNK SCFKN SCKNF SCKFN SFNCK SFNKC SFCNK SFCKN SFKNC SFKCN SKNFC SKCNF SKCFN SKFNC SKFCN KNFCS KNFSC KCSFN KFSCN KSCFN KSFNC KSFCN

2

6 122 141 6 1 29 85 1 3 1 1

13 25 315 210 9 45

1 13 1

1 1 1 1 2

79 3 4 1 1

-

1 45 42

1

3 3

1980

9 8 160 264 8 48

13

1

-

1

-

4

2

1969

FCKSN FCSKN FCSNK FCNSK FNSCK CKSFN CSKFN CSKNF CSFKN CSFNK CSNKF CSNFK CFKSN CFKNS CFSKN CFSNK CFNKS CFNSK CNKSF CNKFS CNSFK CNFKS CNFSK NKSCF NKFSC NKCSF NKCFS NSFKC NSFCK NSCFK NFSKC NFSCK NCSFK NCFSK

2 5 19

1 2 135 138 11 2 2 56 66 1 10

1972

1980

10 31 1 2 1 2 2 97 35 1

37 40 2

6 2 40 178 17

49 119 1 3 1 74 207 2 19

1

5

3

2 7 2 9

2 2 1 10

1

4 1 1

1 2 1

1

1 1 6

1 1 1

2

On the Axiomatic Foundations of Unfolding

23 1

compatible neither with S I KI N nor with K I N I S; but almost all I-scales are compatible with KIS IN. Similarly, K l F l N and K l C l N almost uniquely hold. A somewhat different pattern holds for triples containing K (and not N) as well as for those containing N (and not K). It is very clear that neither K nor N are between two of the other three parties. It is less clear, however, whether we have, for example, K l S l F or KIFIS. Nevertheless, these nine mples (with K and/or N) are best compatible with the 5-ary betweenness relation K I S IF I C IN and thus with a J-scale corresponding to left-right dimension. However, it is the final mple (S,F,C) that severely violates this J-scale (as well as every other one). For every set of data, there is a substantial portion of I-scales being compatible with neither S l F l C nor F l S l C nor SICIF. Exactly these cases lead to substantial violations of Axiom 4. They result from the fact that every possible ordering of S, F, and C is contained in frequently observed I-scales. This has also been noted by Pappi (1983). Essentially the same result is obtained when some idiosyncratic Iscales are deleted from Table 5. The numbers in parentheses in Table 6 give the corresponding percentages after deletion of those I-scales which have not been observed at least two times in every year. This criterion excludes about 5% of the I-scales. It is interesting to see that now almost all the triples containing K andor N are uniquely determined in terms of betweenness. These are exactly those mples corresponding to the leftright dimension. The problem with the triple (S,F,C), however, still remains. The results on betweenness also provide information about possible two-dimensional unfolding representations of the five parties. We know that S, F, and C must be represented as a “triangle”. We also know that this mangle is “between” K and N which are at the extremes of such a configuration. However, these two pieces of information cannot be combined in any satisfactory manner. According to Table 6, the mples with N (and without K) do not tell us whether, for example, N is closer to F or to S (i.e., whether S IF IN or F I S 1 N). The same holds for these other triples as well as for those with K (and without N). Thus, all possible locations of K and N relative to S, F, and C are equally well (or rather poorly) supported. To put it differently, there is no “rotation” of the “triangle” built by S, F, and C that can properly be fitted “between” K and N. These problems with a two-dimensional unfolding representation

232

Orth

arise from the fact that all 12 logically possible I-scales having S, F, and C (in any order) before K and N (in any order) were frequently observed. In terms of unfolding, the three vertices of the mangle with S, F, and C yield three boundary lines separating the plane into six isotone regions (corresponding to the six orderings of S , F, and C). The boundary line between K and N, however, can cross at most four of these regions but not six. Hence, two-dimensional unfolding cannot account for all of these 12 I-scales at the same time. These results again hold for all three sets of data. Table 6, however, also reveals some interesting differences between the years 1969, 1972, and 1980, especially with respect to the triple (S,F,C). In 1969, the betweenness relation of this triple that is compatible with most I-scales is S I C I F, whereas in 1972 it is F I S I C and in 1980 it is S 1 F I C. Thus, in 1969 the largest difference is that between S and F, in 1972 it is that between F and C, and in 1980 it is that between S and C. A closer study of this type of information has been given elsewhere (Orth, 1986). It has been shown there that in 1969 F is relatively far away from both S and C which are fairly close together, in 1972 F and S are much closer to each other, and in 1980 S and C are now relatively far away from each other and F is somewhat cloJer to C than before but still closest to S. It is to be noted that these changes correspond very well with the coalition governments in Germany between 1969 and 1980. Moreover, also Pappi (1983) showed some influences of party coalitions on party preferences. It is thus tempting to conjecture that the qualitative J-scale corresponding to the political left-right dimension was not found for these data from 1969, 1972, and 1980 because it was heavily distorted by preferences for political party coalitions.

TubZe 6. Percentages of persons whose preference orderings are not compatible with either a I b I c or b I a I c for all triples of the five parties K, S, F, C, and N. Percentages in parentheses are obtained after excluding some idiosyncratic I-scales (see text).

Triples with K and N

K/S/N

WIN K/C/N

Triples with K

WSF K/S/C K/F/C

Triples with N

SF/N

S/C/N F/C/N

Triple W,C)

S/F/C

1969

1972

1980

1.1 (0.0 0.9 (0.0 1.2 (0.0

0.5 0.0 0.6 0.0 2.4 0.0

0.6 0.0) 0.3 0.0) 1.7 0.0)

S/K/N

1.5 (0.0 1.0 (0.0 1.8 (0.0

0.5 0.0 0.6 0.0 0.7 0.0

0.7 0.0) 0.5 0.0) 0.4 0.0)

K/F/S

3.2 (1.9 1.7 (0.0 1.3 (0.0

1.8 0.8 3.4 0.0 3.3 0.0

0.7 0.4) 2.5 0.0) 2.4 0.0)

WSIN

63.6 (64.6

43.4 44.6

20.1 20.6)

F/S/C

F/K/N C/K/N

KJC/S K/CF

C/S/N C/F/N

1969

1972

1980

47.2 (47.8 47.3 (47.8 47.1 (47.8

67.7 67.8 67.7 67.8 66.4 67.8

51.8 51.9) 52.0 51.9) 51.5 51.9)

K/N/S

2.4 (0.6 2.9 (0.9 2.2 (0.9

1.6 1.0 5.9 3.7 5.9 3.7

1.4 1.2) 6.8 5.0) 6.8 5.0)

S/KF

3.2 (1.5 3.1 (2.1 4.0 (1.9

2.2 0.9 2.5 1.9 2.5 0.8

3.0 2.3) 3.2 2.5) 1.8 0.4)

S/N/F

19.0 (18.6

16.5 16.8

30.2 30.6)

S/C/F

K/N/F K/N/C

SMC

F/K/C

S/N/C F/N/C

1969

1972

1980

51.7 (52.2 51.8 (52.2 51.7 (52.2

31.8 32.2 31.8 32.2 31.2 32.2

47.6 48.1) 47.7 48.1) 46.8 48.1)

96.0 (99.4 96.1 (99.1 96.0 (99.1

98.0 99.0 93.6 96.3 93.5 96.3

98.0 98.8) 92.6 95.0) 92.8 95.0)

93.6 (96.6 95.3 (97.9 94.7 (98.1

96.0 98.3 94.1 98.1 94.3 99.2

96.3 97.3) 94.3 97.5) 95.7 99.6)

17.4 (16.8

40.1 38.6

49.7 48.8)

t d

w

w

234

Orth

5. Discussion

An axiomatic approach to unfolding can facilitate a detailed analysis of structural aspects of a set of preference orderings. The concept of betweenness given in Definition 1 enables one to study not only the whole set of objects at once but also to do so for every mple or every subset of the preferential objects under study. In case of violations of axioms, this very feature of betweenness might be of use to gain some insights into what could have led to those violations. This was illustrated by applying an unfolding structure to data on political party preferences from the years 1969, 1972, and 1980. The preference orderings were shown to be not unfoldable to a qualitative J-scale of the five parties. This is mainly due to one of the ten triples of parties. Furthermore, it was argued that there is also no appropriate twodimensional unfolding representation for these data. A comparison of the three sets of data with respect to that particular triple of parties revealed differences which seem to be related to the governmental coalitions existing at the respective times. It should be noted that essentially the same differences have been found (Orth, 1986) with the data analyzed on an aggregated level where group preference orders were built according to the most preferred party. These aggregated preferences yield almost perfect J-scales for the three years differing with respect to the parties S , F, and C just as the betweenness relation does here. These findings suggest the hypothesis that political party preferences are determined by both the perceived position of the parties on the political left-right dimension (in terms of the distance from a person’s ideal party or own position on that dimension) and the preferred party coalition. Such an explanation would be consistent with Pappi’s (1983) finding that governmental coalitions have some impact on party preferences.

References Coombs, C. H. (1950). Psychological scaling without a unit of measurement. Psychological Review, 57, 145-158. Coombs, C. H. (1952). A theory of psychological scaling. Engineering Research Institute, University of Michigan, Ann Arbor. Coombs, C. H. (1953). Theory and methods of social measurement. In

On the Axiomatic Foundations of Unfolding

235

L. Festinger & D. Katz ( a s . ) , Research methods in the behavioral sciences (pp. 471-535). New York: Dryden. Coombs, C. H. (1964). A theory of data. New York: Wiley. Ducamp, A., & Falmagne, J. C. (1969). Composite measurement. Journal of Mathematical Psychology, 6 , 359-390. Klingemann, H. D. (1972). Testing the left-right continuum on a sample of German voters. Comparative Political Studies, 5, 93-106. Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971). Foundations of measurement (Vol. 1). New York: Academic Press. Norpoth, H. (1979). Dimensionen des Parteienkonflikts und Praferenzordnungen der deutschen Wahlerschaft: Eine Unfoldinganalyse. Zeitschrift fur Sozialpsychologie, 10, 350-362. Orth, B. (1976). An axiomatization of unfolding. Paper presented at the 7th European Mathematical Psychology Group Meeting, Stockholm. Orth, B. (1980). On the foundations of multidimensional scaling: An alternative to the Beals, Krantz, and Tversky approach. In E. D. Lantermann & H. Feger (Eds.), Similarity and choice (pp. 54-69). Bern: Huber. Orth, B. (1986). Grundlagen des Entfaltungsverfahrens und eine axiomatische Analyse von Priiferenzen fur politische Parteien. Unpublished manuscript, University of Hambrug. Pappi, F. U. (1983). Die Links-Rechts-Dimension des deutschen Parteiensystems und die Parteipriferenz-Profileder Wahlerschaft. In M. Kaase & H. D. Klingemann (Eds.), Wahlen und politisches System (pp. 422441). Opladen: Westdeutscher Verlag. Roskam, E. E. (1977). A survey of the Michigan-Israel-NetherlandsIntegrated series. In J. C. Lingoes (Ed.), Geometric representations of relational data (pp. 289-312). Ann Arbor, MI: Mathesis Press.

This Page Intentionally Left Blank

New Developments in Psychological Choice Modeling G. De Soete, H. Feger and K . C. Klauer (eds.) 0 Elsevier Science Publishcr B.V. (North-Holland),1989

237

UNFOLDING AND CONSENSUS RANKING: A PRESTIGE LADDER FOR TECHNICAL OCCUPATIONS Rian A . W. van Blokland-Vogelesang Free Univcrsity, Amsterdam, The Netherlands The social prestige of occupations can be measured by letting people rank order occupations according to social prestige. This has been done by Goldbcrg (1976) for technical occupations. For all judges a “consensus ranking” can be determined: the mean or median ranking. Thc rcsulting consensus ranking for these data is a ranking of these occupations according to social prestige. In unfolding the individual rankings a J scale is sought on which stimuli and individuals can be placed. This J scale forms a common reference frame for thc evaluation of stimuli. To find qualitative and quantitative J scalcs for cornplcte rankings a computer program, “UNFOLD”, has been developed. For thc Goldberg data a “nested set” of quantitative J scalcs was found: a J scalc for larger numbers of stimuli contains a smallcr J scale as a propcr subsct. This indicates a stable and rcliable solution. To cxplain departures from the perfect unfolding model, Feigin and Cohen’s (1978) error model has been used.

1. Introduction The unfolding model and technique are thoroughly discussed in Coombs (1964). The historical context in which the unfolding technique originated was the debate concerning “majority decisions”: finding a consensus ranking for a number of stimuli in a group of individuals, see Coombs (1964, Ch. 18). Under the condition that all subjects’ rankings are single peaked preference functions (SPF’s) on a common quantitative J scale, the consensus ranking proved to be a pattern of the J scale: the ranking of the This paper is a revised version of an article published in Zeifschr@ fur Sozialpsychologie, 1987, 18, 250-257.

238

vun Bloklund- Vogelesang

median individual on the J scale. This argument can be reversed. By folding back the median ranking for a group of subjects, (an approximation to) the best J scale can be found. The unfolding model is a deterministic model. It has to be rejected if departures from the perfect model occur. For a variety of reasons, however, individuals mostly do not all produce preference rankings which are consistent with one underlying J scale. In addition, much research is precisely directed at finding an underlying frame of reference in a certain domain of investigation. Therefore, we need a criterion for the “best” J scale. The best J scale will be defined as that scale for which the total number of inversions from individuals’ rankings is a minimum. The minimization of total number of inversions is an often used criterion in nonparametric statistics. In the case of ranking data this criterion also follows from the Mallows (1957) and Feigin and Cohen (1978) models. To determine scale values for stimuli and subjects on the quantitative J scale, linear programming techniques have been used. To explain departures from the perfect unfolding model, the Feigin and Cohen (1978) model has been used. The results of the unfolding procedure will be illustrated on the Goldberg (1976) data on social prestige of technical occupations. In the following, first the unidimensional unfolding model as outlined in Coombs (1964) will be shortly introduced (Section 2). Subsequently, Section 3 will concern consensus rankings, Section 4 finding the best unfolding scale. In Section 5 the Feigin and Cohen (1978) model and its adjustment to the unfolding situation will be treated. In Section 6 the unfolding procedure will be illustrated on the Goldberg (1976) data. In Section 7 a discussion follows. 2. Coombs’ Unidimensional Unfolding Model Coombs’ (1964) unidimensional unfolding model was devised for the analysis of complete orderings of preference. Suppose there are n individuals ranking k objects from most to least preferred. Each individual and each object may be represented on a single dimension, called the J scale (“Joint”). The points representing the individuals are called “ideal points”, each representing the best possible object from the point of view of the individual. Each individual’s preference ranking of objects is given by the rank order of the distances of the object points from the ideal

Unfolding and Consensus Ranking

239

point, the nearest being most preferred. In the unidimensional unfolding model possible orders of preference (“admissible patterns”) correspond to intervals of the J scale. Other orders of preference do not correspond to intervals of the J scale and, hence, are called “inadmissible patterns”. For four stimuli A, B, C and D, two different J scales (“4-scales”) are possible, depending on the order of the midpoints ad and bc. The relative magnitude of the distances d(AB) and d(CD) depends on the order of the midpoints ad and bc (see Figure 1).

I . d(AB)

>

d(CD)

(ad precedes bc)

midpoints:

ad

bc

bd

Interval : -

I-

A

11. d(AB) < d(CD)

(bc precedes ad)

mi dpoi nts:

ab

‘1

In1terval : -

,ABCD

A

C

B

1I

ac

bc

ad

I

‘2 BACD, B

D

bd

cd

I

I

1 1 1 1 1 BCAD

‘3

CBAD ‘4

CBDA

‘5

CDBA

‘7

DCBA

‘6

C

I_

0

Figure 1. The two possiblc midpoint ordcrs for 4-scale ABCD; ad preccdes bc (top) and bc prcccdcs ad (bottom).

So, without restrictions on the order relations between the midpoints, there are eight admissible patterns in 4-scale ABCD. This scale is called the qualitative J scale. With restrictions on the order relations of midpoints one of the fourth intervals 1 4 is excluded (only one order of ad and bc is possible), this scale is called a quantitative J scale. A qualitative J scale contains 2k-’ patterns, a quantitative J scale

[i]+ 1. The quantitative J

scale can be represented by a unidimensional continuum, because of the fixed midpoint order; the qualitative J scale can not.

van Bloklad-Vogelesang

240

3. Consensus Ranking The existence of common J scales is, generally speaking, a consequence of cultural homogeneity (Coombs, 1964, p. 397). The unidimensional J scale represents a reference frame for the evaluation of objects.

‘m

preference

(a)

t

A

B

preference

C

J scale

(b)

t

qualitative

I Figure 2. Single pcakcd prcfcrencc functions (lop). Black’s case: the majority decision is thc top choice of thc mcdian individual in the group (bottom). The significance of the existence of a common J scale for a group of individuals may be best understood in the context of the historical tradition in which the unfolding model originated. In the fifties many a discussion centered around the problem of “consensus rankings”: how to construct a social preference out of manifold individual preferences. Black (1948a, 1948b) proved that the majority decision (“consensus”) for a set of

Unfolding and Conscnsux Runking

24 1

options is the top choice of the median individual, given that individuals’ preferences functions are single peaked (see Figure 2). Arrow (1951) proved that the consensus ranking is a folded J scale, on which the first option is !he top choice of the median individual in the group. Goodman (1954) and Coombs (1954) proved that if individuals’ preference rankings are generated from a common underlying quantitative J scale, the consensus ranking is the ranking of the median individual on the J scale (see Figure 3). These results are very strong and one might ask whether they apply in the case where not all individuals’ ordering are single peaked. In particular one might be interested in the conditions under which the median ranking will be the consensus ranking for any set of rankings (not necessarily SPF’s). This issue was answered by Kemeny (1959) and Kemeny and Snell (1972). They showed that the mean and median rankings are consensus rankings in general, that is, without the assumption that individuals’ rankings be SPF’s. This will hold if the number of inversions between rankings is used as a distance measure and at least in the case of a substantial number of subjects. preference

t

quantitative 4

J scale

Figure 3. Coombs irnd Goodman’s case: Ihc consensus ranking is a folded J scale arid is the ranking of the median individual in thc group.

242

van Blokland-Vogelesang

A second question might be the following: under the assumption that individuals share a common frame of reference (individuals do not choose rankings at random), how sure can we be that the resulting consensus ordering is a folded J scale? By using a probability model for ranking data it can be proved that the median ranking is a folded J scale in general, at least for large numbers of subjects. Any ranking model for which probabilities of rankings decrease with increasing numbers of inversions from the median ranking can be used. The Feigin and Cohen (1978) model is an example of such a model.

social preference

collective ideal point

Figure 4. In folding back thc consensus ranking the J scale may be found.

4. Finding the Best J Scale Unidimensional unfolding is a technique for finding the latent dimension, the “J scale”, on which the preference rankings are based. The data are complete ordering of preference of n individuals for a fixed set of k stimuli. If a stable frame of reference is underlying people’s preferences, rankings will unfold into a common J scale.

Unfolding and Conscnsus Ranking

243

The best J scale is defined as that scale for which the total number of inversions from subjects’ rankings is a minimum. This can be explained briefly as follows. Each J scale has a certain number of admissible patterns (“I scales”). Every individual is supposed to have a pattern of preference in mind, the “latent pattern”, which is identical to one of the admissible patterns of the J scale. In reporting his or her latent pattern of preference the subject may make errors. So, the “manifest” pattern of preference may be different from the latent pattern. For each admissible pattern of the J scale the number of inversions from an individual’s manifest pattern of preference is assessed. That admissible pattern which has a minimum number of inversions from the individual’s pattern is taken as the latent pattern for this individual. In this way, the number of inversions needed is minimized for each individual and, i n general, for all individuals. The minimization of total numbers of inversions is an often used criterion in nonparametric statistics (cf. Lehmnnn, 1975). To find best qualitative and quantitative J scales for complete rankings of preference, a number of procedures have been devised, see Van Blokland (1988, i n press). The computer program UNFOLD has been written by Piet van Blokland on the basis of these procedures. It should be stressed that the minimization of the total number of inversions from subjects’ rankings is the only criterion used to find the best J scale. No other criteria such as quasi independence (Davison, 19759, observed versus expected numbers of errors and “uniqueness” of the found scale (Van Schuur, 1984) are used. There is no user interaction and there are no parameters which have to be set by the uscr. 4.1 The Benefits of UNFOLD -

Best qualitative and quantitative scales for subsets of 4 <: k 2 9 stimuli out of a maximum number of IS stimuli. separate analysis for each number of objects. Results for any number or subser of stimuli are never dependent on previous steps in the analysis.

-A

-

Scale values for objects, for midpoints between objects, for the patterns of the J scale and for individunls.

244

van Blokland-Vogelesang

- Options to analyze specific qualitative or quantitative J scales. - A test for goodness of fit based on a nonparametric error model for ranking data. Best qualitative J scales can be determined for subsets of a maximum of 11 stimuli. Scale values and the test for goodness of fit are determined for quantitative J scales only. All sets of data which have been analyzed to date produced bipolar scales. These scales could be interpreted in a substantially meaningful way. Moreover, the best quantitative J scales formed a “nested set” in that the items of a smaller scale were included in the larger scale in the same order. This was not deliberately sought but just followed from the data. Such a nested set of scales indicates that a stable continuum is underlying the data. The algorithms used in the program are based on: -

backtracking and branch-and-bound-methods;

- finding a good solution first; - eliminating inferior scales as quickly as possible. The computer program has four main parts: 1. finding the median ranking

2. finding the best qualitative J scale 3. finding the best quantitative J scale 4. assessing goodness of fit, scale values etc.

To get a quick estimate of the total number of inversions needed, in a first step only those qualitative J scales which arise from unfolding the median ranking are considered. Not until a later stage are all possible J scales investigated. In the first step the total number of inversions needed is (under)estimated. For the 20 qualitative scales having the best underestimate of the total number of inversions the actual minimum number of inversions is assessed. Then all possible qualitative scales are investigated, the ten best ones are retained and printed.

245

Unfolding and Consensus Ranking

The best qualitative J scales probably will include a very good quantitative J scale and are thus first investigated to assess the total number of inversions from subjects’ preference patterns. Having determined a good quantitative J scale most remaining scales can be skipped as candidates for the best quantitative J scale. The ten best quantitative J scales are retained and, on request, are printed. Scale values and the test for goodness of fit are printed too.

4.2 Scale Values by Linear Programming The procedure to assess scale values for stimuli starts from the order of the midpoints on the quantitative J scale. The distances between the successive midpoints are called the 6i’s. The distances between the midpoints must satisfy a number of equality constraints (see Van der Ven, 1977). The restrictions on the 6’s can be represented by a system of linear equations which can be solved by using linear programming techniques, under the constraints of 1.

6i > 0

2.

6i

i = 1,2,

...,

I;[

-

is at a minimum.

This last constraint is imposed on the 6’s to obtain a maximum distinction in metric relations on the J scale (see Van Blokland, in press; cf. Coombs, 1964, p. 101). To this end, SIMOPT (by E. Kalvelager, Free University, Amsterdam) was incorporated in UNFOLD. The scale value of an individual is the midpoint of the admissible pattern which corresponds to his or her pattern of preference. An inadmissible pattern is assigned the scale value of that admissible pattern which has a minimum number of inversions from it. It may happen that more admissible patterns exist which have the same number of inversions from that inadmissible pattern. In that case the inadmissible pattern will get the scale value of the admissible pattern which is closer to the median ranking (i.e., it has less numbers of inversions from it). This seems reasonable, since a unimodal and symmetric function of social preference is assumed to exist on the J scale. This rule corresponds to the principle of “regression to the mean”.

van Blokland-Vogelesang

246

5. Feigin and Cohen’s Model A nonparametric family of distributions for ranking data has been derived by Mallows (1957). A particular case of this family is the Feigin and Cohen (1978) model. The model is based on the number of inversions between rankings. Suppose there are n subjects, each of whom ranks the same set of k stimuli according to some criterion. Assume that each subject ranks the objects independent of the other subjects. The ranking of a subject is a permutation w of the numbers (1, . . . , k). The distribution of a ranking w in the Feigin and Cohen model depends on two parameters, w,, a location parameter, and 0, a non-negative dispersion parameter (0 I 0 Sl). The number of inversions between the ranking 61 and the basic ordering w, is given by X(w,,o). The probability distribution of a ranking w is

Po,.e(w)= (f (0))-1 0x ( w - w ) ,

0I051

(1)

where f (0) = C 6x(oo’0),a nonnalizing constant. Consequently, the probability distribution of X = X (o,,w)is:

where

k

[i]

= number of objects = maximum possible number of inversions

a!

= number of possible orderings with x inversions from

f (0)

=

a:

w,

W, a normalizing constant.

X

The model of Feigin and Cohen can be interpreted as stating that subjects have the same latent ordering w, in mind and make errors in reporting it. A low value of 0 corresponds to subjects making few errors. The larger the value of 0 the more errors and the more improbable the existence of one underlying ordering will be. Both parameters w, and 0 can be estimated by maximum likelihood methods. From (1) the likelihood equation given the sample of n subjects is given by

247

Unfolding and Consensus Ranking

(i = 1, ..., n ) and hence the maximum likelihood estimate for o, is given by the value 6, of o, for which C X(o,,oi)is minimal. i

This means that 6, is the median ranking (the consensus ranking of Section 1): that ordering which has a minimum number of inversions from all subjects’ rankings. The maximum likelihood estimate of 8 is found from the mean number of inversions X from the median ranking: 8 is that value of 8 for which X = E ( X I &,e) (cf. Feigin & Cohen, 1978). So, the Feigin and Cohen model fits precisely in the framework of unfolding and consensus rankings. n

5.1 Adjustment of the F&C Model to the Unfolding Situation By adjusting the Feigin and Cohen model to the unfolding situation, a probabilistic aspect is added to the deterministic unfolding model. The number of errors observed can be compared to the number of errors expected, to arrive at a measure for goodness of fit of the unfolding model to the data. The application of Fei in & Cohen’s model in the unfolding situation involves distinguishing

[!I+

1 latent classes, “latent rankings”, one for

each admissible pattern of the quantitative J scale. Analogously to the Feigin & Cohen situation, the most likely quantitative J scale is that ordered set of

[:]

+ 1 admissible

patterns for which the total number of

inversions from individuals’ rankings is minimal. The model assumptions for the application of Feigin & Cohen’s model in the unfolding situation can be formulated as follows: 1. The quantitative J scale is known or has been estimated.

2. Each individual has a latent pattern of preference identical to one of the + 1 admissible patterns of the J scale.

[i]

3. The ranking actually given by the judge has, according to Feigin & Cohen’s model, this person’s latent pattern as basic ordering. When judging objects according to their latent pattern of preference people can make mistakes.

248

van Blokland-Vogelesang

Observing an admissible pattern does not mean that the subject has made no errors. For example, a subject may have reported pattern BCAD, which is an admissible pattern of J scale ABCD. This person may have had pattern BACD in mind and made no errors on it. But she might also have had pattern ABCD in mind and made one inversion on it. With X = number of inversions made by the subject when stating her pattern of preference

Y = minimum number of inversions the researcher has to apply to an inadmissible pattern in order to change it into an admissible one the relation between X (latent number of inversions) and Y (manifest number of inversions) can be investigated. For inference about 8 from Y we have determined P (Y=y I X = x ) , the probability of needing y inversions to fit a subject’s pattern into the J scale when the subject has made x inversions on a certain admissible pattern of the J scale. P ( Y = y 1 X = x ) is independent of 8. In assessing P (Y =y I X = x ) all admissible patterns of the J scale are assumed to have equal probabilities. This seems a rather unrealistic assumption, but appears to work out very well in practice. The approach using varying probabilities for the admissible patterns of the quantitative J scale has been treated in Van Blokland et al. (1987). From P e ( X = x ) and P ( Y = y I X = x ) , P e ( Y = y ) can be determined: Pe(Y=y) =

maw)

C

P ( Y 9 I X=x)P,(X=x)

(X

2 y).

Y=O

With 7 known, 8 can be estimated analogously to the Feigin and Cohen situation: 8 is that value of 8 for which 7 equals E e ( Y ) . Instead of 8 it is recommended to use Ee(T), the mean of the dismbution of Kendall’s (1975) z. Ee(7) is unbiasedly estimated from the sample by Z (Feigin & Cohen, 1978). The advantage of using 2, rather than 8, is that it is a well-known estimate of the concordance between the subjects and their underlying rankings, and 0 I Z I 1. The number of inversions X and z are linearly related: A

A

z = 1 - 2 [;]-lx(w,,w)

and so are their mean values X and 2. Hence, 2 is a linear transformation of a sufficient statistic for 8. Once 8 is known, 2 = E e ( z ) can be found A

Unfolding and Consensus Ranking

249

from the X-distribution given the estimated value of 0. In Appendix I, E e ( X ) and Ee(T) are given for selected values of 8 and k.

5.2 Goodness of Fit of the Unfolding Model to the Data For the test of goodness of fit of the unfolding model to the data preference patterns with the same number of inversions Y from the J scale are grouped into the same category (cf. Feigin & Cohen, 1978). The Pearson X 2 test for goodness of fit has numbers of degrees of freedom which are one less than the number of categories into which the data are lumped. For each extra numerical parameter estimated one more degree of freedom should be subtracted. Since w, is not a numerical parameter but an ordering, we need not subtract an extra degree of freedom if coo is estimated. The x2 approach in this situation is then an approximate procedure. This point needs some further research. However, this is beyond the scope of this paper. The observed frequencies obs, are the frequencies of the y-values in the data, the expected frequencies exp, can be assessed via nPi(Y=y). As a test statistic Pearson's X 2 is used:

can be referred to a X*-distribution with df = maxCy) - 1 degrees of freedom. The higher values of Y may have to be grouped because of small expected frequencies.

6. The Goldberg Data on Social Prestige of Technical Occupations In this section the results of the unfolding procedure are illustrated using the Goldberg data on social prestige of technical occupations. In Goldberg (1976) the relevance of cosmopolitanfiocal orientations to professional values and behavior is discussed. Professionals are said to have a value system and behavioral patterns different from those of other occupational groups. Associated with the characteristics of professionals are the concepts of cosmopolitanism, defined as orientation to an outer reference group and localism, defined as an orientation to the inner reference group. According to Goldberg, in the literature cosmopolitanism has been confused with professionalism; also, a doubtful bipolar concept of

van Blokland-Vogelesang

250

professionals as either cosmopolitan or local has been introduced. In his article he presents theoretical arguments as well as empirical evidence which point to the conclusion that an orientation which combines both cosmopolitan and local reference groups (“cosmo-local”) may be compatible with the values and behavior considered important to professionalism.

6.1 The Goldberg Data The Goldberg data on the social prestige for technical occupations come from a sociological survey of graduates of the Faculty of Indusmal and Management Engineering (Technion, Haifa, Israel) and is referred to in Goldberg (1976), Feigin and Cohen (1978) and Cohen and Mallows (1980) who also present the data. People were asked to rank ten occupations according to the degree of social prestige associated with each one. There were 143 complete responses. The Goldberg questionnaire was in Hebrew. The translation of the ten occupations is according to A. Cohen (personal communication). The occupations are: A. B.

C. D.

E.

F. G. H. I. J.

A faculty mcmber in an acadcmic institution Mechanical Enginccr Opcrations Researchcr Technician Manager in a staff position in an industrial entcrprisc ( c g , dcaling with safcty, human rcsourccs, timc and motion study, etc.) Owner of a plant with morc than 100 workcrs Supcrvisor Industrial Enginecr Managcr of a production dcpartmcnt with more than 100 workcrs An applicd scicntist

“FAC” “MECH” “O.R.”

“TECH” “STAFF”

“OWN” “SUP” “IND”

“MAN” ‘‘APPL ’’

6.2 Results of the Unfolding Procedure In unfolding the Goldberg data consistently for each k best quantitative J scales are found which are subsets of the largest scale DBHCJAFIEG for all ten items, see Figure 5. This result is obtained after removing three outliers: subjects with 35, 38 and 39 inversions from the median ranking,

Unfolding and Consensus Ranking

25 1

whereas remaining X values are in the range 0 to 18 (see Feigin & Cohen, 1978, p. 211). The Goldberg data excluding these three outliers are called the Goldberg* data, to distinguish them from the full set of data. For both sets of data the results for the best quantitative J scales (4 Ik 2 9 ) are given in Table 1. In some cases only the best qualitative J scale is given. Table I Total numbers of inversions (Cy) and frequencies of perfect fit (y = 0) for 4 2 k I 10 stimuli from the Goldberg data (n = 143, total set) and the Goldberg* data (n = 140, without outliers). For each value of k the three best quantitative J scales are given. If they arc the same, only one is presented. Orderings with an asterisk (*) indicate that only the bcst qualitative J scale has been determined. Goldberg* data k

Ordering

4

DHFG DBAG DBFG DBHFG DBAFG DBHAG DBHAFG DBHAFEG DBHAFIEG DBHJAFIEG DBHCJAFIEG*

5

6 7 8

9 10

Goldberg data

y=o

y=o 2 3 4 29 30 30 71 143 273 428 486

138 137 136 115 111 112 88 54 27 12 16

9 9 9 38 39 40 85 169

138 137 137 115 111 112 88 54

GDBHJAFE DBHJAFIEG* GBHJAFCIED* ;DBCHJAFIEG*

From Table 1 it is clear that a stable continuum underlies the Goldberg* data. On removing three outliers from the data some disturbance is eliminated which causes slightly different results from the Goldberg data for higher values of k. Hence, further results and the underlying continuum will be discussed for the Goldberg* data only. Best J scales for 4 I k I10 are given below in Figure 5 , starting with k = 4. For each larger k one item is added to the already existing quantitative (k - 1)scale. For 4 I k I 7 the Goldberg data show results analogous to the Goldberg* data, only the number of inversions is higher due to the three

252

van Blokland-Vogelesang

Tech k (D) 4 Tech 5 Tech 6 Tech 7 Tech 8 Tech 9 Tech 10 Tech

Mech (B)

OR

Ind

Appl Fac Own Man

(C) (H)

Mech Mech Mech Mech Mech Mech O.R.

t Technical

(J)

(A)

Ind Ind

Ind Ind

Ind Ind Ind

Fac Fac Fac Appl Fac Appl Fac

Occupations

(F)

(1)

Own Own Own Own Own Man Own Man Own Man

Staff

Sup

(E)

(GI SUP

SUP Staff Staff Staff

SUP Sup

Sup Sup

Staff Sup

Managerial Occupations +

Figure 5. Quantitative J scales for the Goldberg” data for 4 I k 5 9. For k = 10 only the best qualitative J scale has been determined.

outliers. For k = 8 the best quantitative J scale is GDBHJAFE, in which the least prestigous item G has moved to the other end of the scale. This is because of the error introduced by the outlying rankings, against which the unfolding procedure is clearly not robust (but, in this case, only for as large a k as k = 8). In the unfolding of data, restrictions on possible locations of items on the scale follow from subjects’ rankings. If one object is placed in the lowest ranks by most of the people, it should get an extreme position on the J scale. If all rankings fit the J scale perfectly, the position of this least preferred item is firmly determined. However, in the situation of increasing error much information (restrictions on the positions of the items) is lost. Remaining restrictions are such that the position of the extreme item should be far from other items, hence it could be on either end of the scale. Thus, restrictions are not enough to firmly fix the item on the one or other end of the scale. Hence, the item can “flip over” to the other end of the scale. The scale found for the Goldberg* data is clearly a bipolar continuum, ranging from purely technical professions to purely managerial professions. At both ends of the scale the professions with the least prestige can be found: “Technician” as the technical profession with the least prestige and “Supervisor” as the managerial profession with the least prestige. In the center of the scale are the high status professions: “Faculty member”, “Owner of a big plant” and “Applied scientist”

Unfolding and Consensus Ranking

253

being the top status professions, in this order. If Goldberg’s (1976) conjecture is true, the cosmo-local oriented professionals are to be found here. The pure “cosmopolitans” (technical professions) or the pure “locals” (managerial professions) would be located near the ends of the scale.

6.3 Goodness of Fit of the Unfolding Model to the Data To test the goodness of fit of Feigin and Cohen’s model to the unfolded data, Pearson’s X2 statistic (eqn. (4)) will be used. Only the case k = 7 for the data with and without outliers will be considered here. The cases k = 4,5,6 gave comparable results. For k > 7 no tables for P e ( Y = y ) are as yet available. They will be available soon. Table 2 Goodness of fit for the Goldberg data with and without outliers,

case k = 7. Goldberg* data ,. 7 = 1.02, 8 = .21, T = .86

Y

obs

0 1 2 3 4 5 6 7 8 9 10

54 49 23 9 3 2 0 0 0 0 0 140

X2 55.01 46.3 1 24.46 9.73 3.25 .93 .24 .05 .o1

.02 .16 .09 .06 .06

.oo .oo x2=.37

Goldberg data 7 = 1.18, = .23, z = .84 A

e

Y

obs

0 1 2 3 4 5 6 7 8 9 10

54 49 23 9 3 2 0 0 2 0 1 143

X2 49.17 46.67 27.82 12.50 4.72 1.53 .44 .ll .02

.47 .12 .84 .98 .20

.oo .oo X 2=2.61

Case k = 7. For the Goldberg* data X 2 = .37, df = 3, .90 c p c .95. For the Goldberg data X2 = 2.61, df = 3, .25 < p c SO. If we had treated the values Y = 4 and Y 2 5 as separate categories, we would have found

254

van Blokland-Vogelesang

X 2 = 6.95, df = 4, .10 < p < .25. Hence, the Feigin and Cohen model fits in very well with the errors in the data for k = 7. See Table 2. In sum, the Goldberg data without the three outliers unfold into a quantitative J scale for nine items. The best qualitative J scale for ten items is consistent with this scale. The median ranking is a folded J scale and represents the prestige hierarchy of these technical occupations. The J scale is bipolar and can be interpreted as going from “working with techniques’’ to “working with people” (see Figure 6 ) . The Goldberg data including the outliers do not give the same results. An extreme item flips over to the other end of the scale, because of the increased level of error due to these outliers.

social prestige

‘,

\

\

i,

\ \

\

\

\\

\

‘\--”appl

\ ,

‘.---oO.R

‘. ‘\ ---\

\

I

\

‘\, ‘\\ .-41ind

\

\

\

‘

,\

‘1

rnan(r--( -41

‘...------ -

I

I

/I

.’’

’

I

I

,I

/

/

// I

I

I

/ /I /

/

/I

//’

staff+---/ 41

I

I

/

/

mech

I

I

tech

..,/

I

___C-/’

‘rconsensus ranking. fac, own,

- - - - - -, sup.

Figure 6. The unfolding scalc for the Goldbcrg* data. The median ranking is a foldcd J scalc and rcprcscnls the prcstigc ladder for thcse tcchnicd occupations.

For all k that have been investigated, deviations from the perfect unfolding model can be explained by Feigin and Cohen’s model.

Unfolding and Consensus Ranking

255

7. Discussion

In unfolding a set of data we are searching for an underlying J scale. This J scale can be seen as a reference frame for the evaluation of stimuli. If people’s rankings all unfold into the same quantitative J scale, the consensus ranking is the ranking of the median individual on the J scale. Two points come up for discussion now. The first point concerns the stability of an unfolding solution for increasing numbers of items. If the J scale is a reference frame in some domain of research, it should not be varying with the number of stimuli in the analysis. That is to say: we are looking for a J scale which grows with increasing numbers of stimuli. Also, for each k the same items must be on the J scale in a constant order. If this is the case we can conclude that an underlying reference frame has been established. However, as we have seen from the two Goldberg data sets, extreme rankings or - in general - the level of error in the data may cause stimuli not to have firm positions on the J scale. If this is the case, stimuli may start flipping over from the one end of the scale to the other end. A related problem is that of “irrelevant stimuli”. The majority decision and the consensus ranking in general are not independent of irrelevant alternatives. More specifically: by introducing irrelevant stimuli in the data, the median ranking (and the J scale) may be different for increasing k. For some k an item may crop up into the J scale and disappear again with the next larger k. If a stable continuum seems to arise from the analysis, it seems wise to ignore a J scale which contains that particular stimulus and to take a next best J scale for that value of k which is consistent with the whole set of J scales. In the previous section it was shown that the Goldberg (1976) data (excluding three outlying rankings) unfold into a nested set of J scales. Also errors from the perfect unfolding model could be explained by Feigin and Cohen’s (1978) error model (at least for k 5 7). Since the occupations were ranked according to the degree of social prestige associated with each of them, the consensus ranking is interpreted as the prestige ladder for these technical occupations.

van Blokland- Vogelesang

Appendix I E o ( X ) (first row) and Ee(T) (second row) for selected values of 6 and k.

0 .05 .10 -15 .20 .25 .30 .35 A0 .45

k 3

.loo .933 .199 367 .297 302 .392 .738 ,486 .676 .576 .616 .663 .558 .747 .502 ,828

.448 .50 .55 .60 .65 .70 .75

.80 .85 .90 .95

1.OO

.905 .397 ,978 .348 1.048 .301 1.115 .256 1.179 .214 1.239 .174 1.297 .135 1.352 ,099 1.404 .064 1.453 .03 1 1.500 0

4

5

.152 .949 .310 .897 .471 343 ,636 .788 303 .732 .972 .676 1.141 .620 1.309 .564 1.475 .508 1.638 .454 1.798 .401 1.953 ,349 2.103 .299 2.248 .251 2.388 .204 2.522 .159 2.650 ,117 2.772 ,076 2.889 .037 3.000 0

,205 .959 .421 .916 .647 ,871 .884

.823 1.132 .774 1.388 .722 1.653 .669 1.924 .615 2.199 .560 2.477 .505 2.755 .449 3.031 .394 3.304 .339 3.572 .286 3.832 ,234 4.085 ,183 4.329 .134 4.563 ,087 4.786 ,043 5.000 0

6

258 .966 .532 .929 324 .890 1.134 .849 1.464 305 1.813 .758 2.180 .709 2.566 .658 2.967 .604 3.382 .549 3.806 .492 4.238 .435 4.672 ,377 5.105 .319 5.533 ,262 5.953 ,206 6.362 .152 6.757 ,099 7.137 .048 7.500 0

7 .310 .970

.643 .939 1.Ooo ,905 1.384 368 1.797 329 2.240 .787 2.714 .741 3.221 .693 3.759 .642 4.326 .588 4.920 .53 1 5.536 ,473 6.168 .413 6.810 .351 7.455 .290 8.096 ,229 8.726 .169 9.340 ,110 9.932 ,054 10.500

0

8 .363 .974 .754 .946 1.177 .916 1.634 383 2.130 348 2.668

309 3.251 ,768 3.882 .723 4.564 .674 5.295 .622 6.075 .566 6.899 ,507 7.762 .446 8.654 .382 9.565 ,317 10.483 ,251 11.396 .186 12.293 ,122 13.164 ,060 14.000

0

9

10

.415 .977 365 .952 1.353 .925 1.884 ,895 2.463 .863 3.096 328 3.789 ,790 4.547 ,747 5.375 .701 6.277 .651 7.256 .597 8.308 ,538 9.429 .476 10.609 .411 I 1334 ,343 13.088 .273 14.350 ,203 15.601 ,133 16.823 ,065 18.000

,468 .979 .976 .957 1.530 .932 2.134 .905 2.796 376 3.525 ,843 4.327

0

,808 5.212 .768 6.190 .725 7.268 .677 8.452 .624 9.747 ,567 11.149 .504 12.651 .438 14.237 .367 15.885 ,294 17.565 .219 19.248 ,145 20.902 .07 1 22.500

0

Unfolding and Consensus Ranking

257

References Arrow, J. K. (1951). Social choice and individual values. New York: Wiley. Black, D. (1948a). On the rationale of group decision making Journal of Political Economics, 56, 23-34. Black, D. (1948b). The decisions of a committee using a special majority. Econometrica, 16, 245-261. Cohen, A., & Mallows, C. L. (1980). Analysis of ranking data. Bell Laboratories Memorandum, Murray Hill, NJ. Coombs, C. H. (1954). Social choice and strength of preference. In R. M. Thrall, C. H. Coombs, & R. L. Davis (Eds.), Decision processes. New York: Wiley. Coombs, C. H. (1964). A theory of data. New York: Wiley. Davison, M. L. (1979). Testing a unidimensional, qualitative unfolding model for attitudinal or developmental data. Psychomenika, 44, 179-194 Feigin, P. D., & Cohen, A. (1978). On a model for concordance between judges. Journal of the Royal Statistical Society, B, 40, 203-213. Goodman, L. A. (1954). On methods of amalgation. In R. M. Thrall, C. H. Coombs, & R. L. Davis (Eds.), Decision processes. New York: Wiley. Goldberg, A. I. (1976). The relevance of cosmopolitan/local orientations to professional values and behaviour. Sociology of Work and Occupation, 3 , 331-356. Kemeny, J. G. (1959). Mathematics without numbers. Daedalus, 88, 577-591. Kemeny, J. G., & Snell, J. L. (1972). Preference rankings, an axiomatic approach. Cambridge, MA: MIT Press. Kendall, M. G . (1975). Rank correlation methods. London: Griffin Lehmann, E. L. (1975). Nonparametrics. New York: McGraw-Hill. Mallows, C. L. (1957). Non null ranking models I. Biometrika, 44, 114130. van Blokland-Vogelesang, R. A. W., Verbeek, A., & Eilers, P. (1987a). Iterative estimation of pattern and error parameters in a probabilistic unfolding model. In E. E. Roskam & R. Suck (Eds.), Progress in mathematical psychology I . Amsterdam: Elsevier (North-Holland).

258

van Blokland-Vogelesang

van Blokland-Vogelesang, R. A. W. (1988). UNFOLD: A computer program for the unfolding of complete rankings of preference in one dimension. Free University, Amsterdam van Blokland-Vogelesang, R. A. W. (in press). Midpoint sequences, intransitive J scales and scale values in unidimensional unfolding. In E. E. Roskam & E. Degreef (Eds.), Progress in mathematical psychology 11 Amsterdam: Elsevier (North-Holland). Van der Ven, A. H. G. S . (1977). Inleiding in de schaaltheorie. Deventer: Van Loghum Slaterus. Van Schuur, W. H. (1984). Structure in political beliefs. Doctoral thesis, University of Groningen, The Netherlands.

New Dcvclopments in Psychological Choicc Modeling G . Dc Soctc, H. Fcgcr and K. C. Klaucr (cds.) 0Elscvicr Scicncc Publisher B.V. (North-Holland), 1989

259

UNFOLDING THE GERMAN POLITICAL PARTIES: A DESCRIPTION AND APPLICATION OF MULTIPLE UNIDIMENSIONAL UNFOLDING Wijbrandf H. van Schuur University of Groningcn, Thc Nethcrlands This paper discusses a numbcr of problems with existing unfolding modcls and proposes a strategy of analysis to overcome these probIcms. This stratcgy assumcs dichotomous or dichotomized data, and derives unfoldability critcria from information about ordcrcd triples of stimuli. A unidimcnsional unfolding scale conforming to thcse criteria can bc found for a maximal subset of stimuli. This proccdure can bc applicd to full or partial rank ordcrs of preference, which arc dichotomized to “pick kln” data, and to Likcrt-type rating scales, which arc dichotomizcd to “pick anyln” data. This procedurc is applicablc to largc data scts, such as survey data. As an cxamplc, the proccdurc is applicd to prcfcrcnccs for five German political parties in electoral survcys in 1969, 1972, and 1980. A dominant left-right unfolding dimcnsion is found, and violations of this represcntation are discusscd. 1. Introduction

Coombs’ unfolding model, first presented in 1950, is regarded by many methodologists in the social sciences as an appealing model for the analysis of preferences. Introductions to unfolding appear in many textbooks on scaling, and computer programs for unfolding analysis continue to be developed. Despite favorable attention, however, reports of successful applications of unfolding are rare, and unfolding programs have only very recently found their way into some general-purpose statistical packages. This paper is a rcviscd version of an articlc publishcd in Zcirschrifi fur Sozialpsychologie, 1987, 18, 258-273.

260

van Schuur

There are two major reasons for the relative neglect of the unfolding model by applied social researchers. One is that most techniques for unfolding operate on full rank orders of preference, which ties unfolding to a relatively unpopular form of data collection. The other is that until now we have not been able to satisfactorily unfold imperfect data (i.e., data that do not conform perfectly to the unfolding model). In this paper I propose a new strategy for unidimensional unfolding that solves both these problems. This strategy is implemented in a computer program called MUDFOLD, for Multiple UniDimensional unFOLDing. To exemplify of the technique, I present the unfolding analysis of preferences for five German political parties by German voters in 1969, 1972, and 1980. 2. Background: Problems With Existing Unfolding Models 2.1 Unfolding Analysis of Different Data Types

The tradition of using full rank orders of preference in unfolding analyses has obscured the fact that other types of data may be represented along an unfolding dimension as well. In particular, data obtained from five-point Likert-type rating scales may conform to the unfolding model. Researchers who do not realize that rating data may fit the unfolding model often subject their data to factor analysis. The use of factor analysis with data that can be unfolded gives rise to a problem, however: an extra, artificial factor will be introduced, in addition to the number of factors (i.e., dimensions) necessary for an unfolding representation (Coombs & Kao, 1960; Ross & Cliff, 1964). Such factor analysis results may then lead to interpretations of the data different from those that an unfolding analysis would suggest. However, researchers who try to unfold rating data with the currently available unfolding models often get degenerate results because their data contain many ties. The unfolding model presented below is capable of analyzing rating data without the problem of degeneracy. An essential aspect of this model is that data are dichotomized, e.g., into “preferred” and “not preferred” response alternatives. Dichotomization allows rating data and a large number of other data types, including full and partial rank orders, to be used in unfolding analysis. It is a desirable technique also for additional reasons, as will be discussed shortly.

Unfolding the German Political Parties

26 1

2.2 Unfolding Analysis of Imperfect Data When a data set is not perfectly unfoldable, its imperfections can be attributed either to random noise or to systematic deviations from the unidimensional unfolding model. Random noise can be handled by using a stochastic rather than a deterministic model. Stochastic models have been discussed by Sixtl (1973), Zinnes and Griggs (1974), Bechtel (1976), Dijkstra et al. (1980), and Jansen (1983), among others. All of these models are unsatisfactory in certain ways. Some are designed to be used only in a confirmatory way, i.e., to test whether a known order of all stimuli can be interpreted as a Jscale. Others require repeated questioning of subjects to obtain estimates of the probability with which they prefer one stimulus over the other. Still others depend on assumptions that are probably incorrect; for example, especially problematic is the assumption that if subjects are given a choice between two stimuli that lie close together on the J-scale but far away from their ideal point, they will almost deterministically prefer the one closer to their ideal point. However, they will probably prefer both to approximately the same (low) degree. Despite these difficulties with stochastic unfolding models, the strategy of relaxing the criterion for perfect representation to allow stochastic representation seems advantageous. I return to this when I propose a new strategy for unidimensional unfolding. Systematic deviations from the unidimensional unfolding model can be explained in at least four ways. According to one interpretation, respondents begin the task of picking preferred items by using the most salient common criterion, but in the course of evaluating stimuli that are less preferred, they bring other, more idiosyncratic criteria into play. According to a second explanation, the preference judgment process is multidimensional rather than unidimensional, i.e., two or more criteria for preference play an independent but simultaneous role in all preference judgments for all stimuli. Thirdly, the set of stimuli may not be homogeneous with respect to the latent unfolding dimension; that is, one or more of the stimuli are indicators of a different latent trait. Finally, the set of subjects may not be homogeneous with respect to the latent unfolding dimension: they may either use different dimensions, or they may perceive the stimuli differently on the same dimension.

262

van Schuur

Let us look at these problems in more detail and consider some possible strategies for dealing with them.

2.2.1 Analyzing Dichotomous Data: “pick anyln” and “pick kln” Analysis The unfolding model assumes that successively chosen stimuli are decreasingly good substitutes for the subject’s ideal stimulus according to the criterion used for selection. However, in the course of giving a full rank order of preference for n stimuli, a subject may begin to use other criteria for choosing that are different from the criterion with which he or she started out. Coombs (1964) talked about the “portfolio” model in this connection, and Tversky (1972) and Tversky and Sattath (1979) suggested an “Elimination by Aspects” (EBA) model, in which different criteria for preference are hierarchically ordered. To deal with this problem in such a way that we can still find the dominant criterion that is used by all subjects, we should restrict ourselves to distinguishing only the first few most preferred stimuli from the remaining ones: otherwise we risk introducing idiosyncratic noise. Distinguishing the first few most preferred stimuli from the remaining ones can be done by dichotomizing the preference responses of each subject (see Leik & Matthews, 1968; Coombs & Smith, 1973; and Davison, 1980, among others). This is accomplished by assigning the code “1” to each subject’s most preferred stimuli; and “0” to the remaining stimuli. The cutoff point between preferred and non-preferred stimuli depends on the type of data. In the case of Likert-scale items one or more response categories (e.g., “strongly in favor”) can be considered as the “preferred response” and the remaining categories as “nonpreferred responses”. Since a subject can give the “preferred response” to any number of Likert-scale items, he can pick any of the n stimuli as “most preferred”. In the case of full or partial rank orders of preference, however, the researcher generally has to decide which k ( k 2 2) most preferred stimuli will be distinguished from the remaining ones. The unfolding analysis of dichotomous data has been called “parallelogram analysis” by Coombs (1964): a data matrix of subjects and stimuli, ordered according to their position on a perfect unidimensional unfolding scale shows a parallelogram pattern of “ l ” s from top left to bottom right.

Unfolding the German Political Parties

263

Using dichotomous data has both advantages and disadvantages. An advantage is that a large number of different data types (including full and partial rank orders of preference, Likert-type rating scales, and roll call data) can all be subjected to such an analysis; all that is needed is that the most preferred stimuli can be distinguished from the others. A disadvantage is that, in contrast to the unfolding analysis of full rank orders of preference, the unfolding analysis of dichotomous data only leads to a qualitative J-scale. This means that no metric information about the relative distances between the stimuli is available and therefore that subjects cannot be discriminated as well as in a quantitative J-scale. However, Davison (1979), has argued convincingly that it is in any event unlikely that a single quantitative J-scale can be found for a large group of subjects in practical applications of unfolding analysis, because subjects often use different subjective metrics.

2.2.2 Multidimensional Unfolding Multidimensional unfolding models assume that subjects do not use a single criterion in making their preference choices, but rather use two, three, or even more independent criteria simultaneously. Multidimensional models have been proposed by Bennett and Hays (1960), Roskam (1968), Schonemann (1970), Carroll (1972), Gold (1973), Kruskal, Young, and Seery (1973) and Heiser (1981), among others. Multidimensional unfolding models are appealing in part because there are various ways for combining the different dimensions, for example, the vector model and the weighted distance model (e.g., Carroll, 1972). However, disadvantages are that technical problems of degeneracy and the representation of I-scales as points in essentially open isotonic regions are more likely to arise in doing multidimensional than unidimensional unfolding. Also problematic is the assumption that different criteria for preference (i.e., more than one dimension) are used simultaneously and independently, and that they are relevant for each stimulus and each subject. Proponents of multidimensional unfolding insist that reality is multidimensional: e.g., a chair has a color, a weight, and a size; a person has an age, a sex, and a preference for certain goods; and a political party may be large, religious, and right wing. Still, subjects often do not evaluate items on the basis of all possible attributes at once. Often they compare them with respect to one attribute only, e.g., sizes of chairs, ages of

264

van Schuur

subjects, and ideological positions of political parties. There may be instances in which a multidimensional model is indeed the best one. But the relative merit of multidimensional versus unidimensional unfolding models in particular situations should be determined empirically.

2.2.3 Selecting a Maximal Subset: of Stimuli or of Subjects? It is an established practice in (multidimensional) unfolding analysis to assign stress values to subjects. This practice reflects the assumption that difficulty in finding a representation can be explained by reference to subjects who apparently used criteria other than the overall dominant one(s), or who perhaps even behaved randomly. A possible procedure for reducing imperfection in one’s data is thus to delete subjects whose stress values are too high. However, high stress values may arise because one or more stimuli do not belong in the same universe of content along with the other stimuli, and therefore cannot be adequately incorporated into the same representation. For unfolding to apply, subjects should differ in their preferences for the stimuli, but they should agree about the cognitive aspects of the stimuli: whether gentlemen prefer blondes or brunettes is a different matter from establishing whether Marilyn is blond or brown. If there is disagreement among the subjects about the characteristics of a stimulus, differences in preference will be difficult to represent; such a stimulus can better be deleted from an unfolding scale. There are reasons for preferring the deletion of stimuli to the deletion of subjects from an unfolding scale. Subjects are often selected as representatives of a larger population. Deleting subjects therefore lowers the likelihood that the result will generalize successfully from a sample to a population. Stimuli, in contrast, are rarely a random sample from a population of stimuli, but rather are intended to serve as the best and most prototypical indicators of a latent trait. In other words, we are often less interested in the actual stimuli than in their potential for allowing us to measure subjects along a latent trait. This means that the deletion of stimuli can generally be defended more easily than the deletion of subjects. Regardless of whether stimuli or subjects are deleted, an explanation for their nonscalability is called for. For stimuli this is especially true in the case that they constitute an entire population, such as all political

Unfolding the German Political Parties

265

parties of a country. The nonscalability of certain stimuli in one dimension may mean that a less parsimonious spatial (multidimensional), or a discrete (cluster, or tree) representation is needed instead of a unidimensional unfolding representation. Alternatively, different well-specified groups of subjects may consistently use different criteria in judging a set of stimuli. Explaining why certain subjects are difficult to represent on an unfolding dimension or in an unfolding space is generally even more difficult than explaining why certain stimuli do not fit. Such explanations are virtually nonexistent in the applied unfolding literature.

3. A Proposed Strategy for Unidimensional Unfolding The unidimensional unfolding model proposed in this paper is based on a combination of three of the strategies discussed above for dealing with data that are not perfectly unfoldable. It allows for a stochastic representation of a maximal subset of stimuli and all subjects in one dimension, using only the highest preference judgments of each subject. Subjects’ most preferred stimuli are distinguished from the remaining ones in a dichotomous way. The approach used here to find an unfolding scale is a form of hierarchical cluster analysis. The optimal smallest unfolding scale is first found and this is then extended with additional stimuli, for as long as the stimuli jointly continue to satisfy the criteria for an unfolding scale. If no more stimuli can be added to the p-stimulus unfolding scale the procedure begins again by selecting the optimal smallest unfolding scale among the remaining n - p stimuli. The process by which more than one maximal subset of unidimensionally unfoldable stimuli can be found in a given pool of stimuli is called “multiple scaling”.

3.1 The Concept of “Error” We generally do not know in advance which stimuli are representable in an unfolding scale, much less the order in which they are representable. The smallest unfolding scale consists of three stimuli, since it takes at least three stimuli to falsify a proposed proximity relation. For the unfolding scale of the ordered triple ABC, the response pattern in which A and C are preferred but B is not is defined as the “error pattern” for that uiple of stimuli. Since part of our analysis is to establish the order in which the stimuli form an unfolding scale, we must consider all three

266

van Schuur

permutations in which each of the three stimuli is the middle one: BAC, ABC, and ACB (a reflection of the scale is an admissible transformation). For the triple consisting of the stimuli A, B, and C in this order, the response pattern 101 would be the error pattern for an unfolding scale ABC, 110 for the scale ACB, and 01 1 for the scale BAC. The amount of error in an individual response pattern to more than three stimuli in a proposed scale order is defined as the number of proximity relations in that pattern that violate the unfolding model, i.e., the number of triples that contain the error pattern. For example, the pattern (ABCD, 0101) contains one error, namely in the triple BCD; the pattern (ABCD, 1011) contains two errors: in the triples ABC and ABD, the pattern (ABCDEFG, 1110111) contains nine errors: in the triples ADE, ADF, ADG, BDE, BDF, BDG, CDE, CDF, and CDG, and the pattern (ABCDEFG, 1011111) contains five errors: in the triples ABC, ABD, ABE, ABF, and ABG. The amount of error in a data set being evaluated for its fit with a candidate unfolding scale is defined as the sum of errors over the response patterns of all subjects. This figure can be calculated by summing the number of errors in each triple of stimuli fist over all subjects, and then over all triples of stimuli. The number of errors in a data set can be calculated for each candidate unfolding scale, i.e., for each set of three or more stimuli in each of their possible permutations.

3.2 Calculating the Expected Number of Errors The number of errors found in a candidate unfolding scale must be compared with the number of errors that would be expected under statistical independence, i.e., under the assumption that a subject’s preferences for the stimuli are completely unrelated. In this “null model” it is assumed that subjects do not differ systematically from each other in their probability of giving a positive preference response to the stimuli. When subjects are free to select as many stimuli as they wish as their most preferred (the “pick anyln” situation), the expected frequency with which a given set of stimuli is preferred is the product of the relative frequencies with which each of the stimuli is preferred times the number of subjects:

Unfolding the German Political Parties

267

Exp.Frey.(ijk, 101) =p;(l

-pJ)*pk.N

where p 1 is the relative frequency with which stimulus i is “picked” and N is the number of subjects. In the case of “pick kln” data, calculating the expected number of errors under statistical independence is a two-step procedure. This is first explained for “pick 3/n” data, and then generalized. a. The expected frequency of the “111”-response patterns is first determined by applying the n-way quasi-independence model (e.g., Bishop, Fienberg, & Holland, 1975). b. From the expected frequency of the “111”-response to each triple the expected frequency of other response patterns (e.g., 01 1, 101, or 110) is deduced. Ad a. In a data matrix in which each of the N subjects picks three of the n stimuli as most preferred, we can find the relative frequency p i with which each stimulus is picked. Under the statistical independence model these pi's arise from the addition of the expected frequency of triples ( i , j , k ) for all combinations of j and k with a fixed i . The expected frequency of triple i, j , k (is., alJk) is the product of the item parameters f,), f J , and fk times a general scaling factor f,without interaction effects: aijk

=

f f, ‘4 *fk. ‘

The values for f and each f, are found iteratively. This procedure, first described by Davison (1980), was developed by Van Schuur and Molenaar (1982). The details of this procedure are given in Van Schuur (1984). Ad b. Once the expected frequency of the “1 1 1”-pattern of all triples is known, the expected frequency of the other response patterns can be found, assuming that each subject picked exactly three stimuli as most preferred. For example: consider the situation in which subjects pick three out of five stimuli A, B, C, D, and E. For the unfolding scale ABC, the error pattern is the one in which stimuli A and C are picked, but B is not. Since exactly three stimuli were picked, either D or E must have been picked in addition to A and C. We can therefore calculate the expected frequency across all subjects of the error response pattern for the triple (ABC,lOl) by summing the frequency of the expected “ 1 11”-patterns for

van Schuur

268

the mples ACD and ACE. In general: Exp.Freq (ijk, 101) =

f e f i

*fk

*

C

fs.

s+i,j,k

This procedure can be easily generalized to the “pick kln” case, where k = 2 or where k > 3. First we find the expected frequency of each k-tuple, ranking between 1 and Second, we calculate the expected

[i].

frequency of the error response pattern of an unfolding scale of three stimuli, as follows: Exp.Freq. (ijk, 101) = f * f i*fk .Q where Q is the sum over all

[;I;] k - 2 tuples of the product of their

fi’s, where s is not equal to i, j , or k. 3.3 A Coefficient of Scalability Once we know for a triple of stimuli in a particular permutation both the frequency of the error response observed, 0bs.Freq. (ijk, lOl), and the frequency expected under statistical independence, Exp.Freq.(ijk, 101), a coefficient of scalability can be defined analogous to Loevinger’s H (cf. Mokken, 1971, who uses Loevinger’s H for multiple unidimensional cumulative scale analysis): H(ijk) = 1 -

0bs.Freq.(ijk, 101) Exp.Freq. (ijk,101) *

For each triple of stimuli (i, j , and k), three coefficients of scalability can be found: H (jik),H (ijk), and H (ikj). Perfect scalability is defined as H = 1. This means that no error is observed. When H = 0 the amount of error observed is equal to the amount of error expected under statistical independence. The scalability of a (candidate) unfolding scale of more than three stimuli can also be evaluated. In this case we simply calculate the sum of the error responses to all relevant triples of the scale for both the observed and expected error frequency, and then compare them, using the coefficient of scalability H :

Unfolding the German Political Parties

269

el 0bs.Freq. (ijk, 101)

C H=l-

ijk=l

P

3

Exp.Freq. (ijk, 101) ijk=l

The scalability of individual stimuli in the scale can also be evaluated. This is done for each stimulus separately by adding up the frequencies of the error patterns, observed and expected, in only those mples that contain the stimulus, and then comparing these frequencies by using the coefficient of scalability H .

3.4 The Search Procedure for an Unfolding Scale After obtaining all the information needed for calculating the coefficients of scalability of each triple of stimuli in each of its three essentially different permutations (i.e., 0bs.Freq (ijk, 101), Exp.Freq (ijk, lOl), and H ( i j k ) , we can begin to construct an unfolding scale. This is done in two steps. First the best elementary scale (the “best mple”) is found, and second, new stimuli are added in one by one to the existing scale. The best elementary scale is defined as the triple of stimuli that conforms best to the following criteria: Its scalability value should be positive in only one of its three permutations. This guarantees that the best triple has a unique order of representation. Its scalability value is higher than some user-specified lower boundary. This helps to ensure that the scale will be interpretable in a substantively relevant way. In practical applications a lower boundary of H > 0.30 is suggested as a rule of thumb; this value is modeled on Mokken’s (1971) approach to cumulative scaling.

If more than one triple satisfies the first two criteria, we select that mple with the highest absolute frequency of the sum of the perfect patterns that contain at least two of the three stimuli. Each mple contains eight response patterns, one of which (101) is the error pattern, and four of which (000, 100, 010, and 001) are not very informative about preferences for sets of stimuli. The high frequency of Occurrence of the patterns 111, 011, and 110 guarantees the

van Schuur

270

representability of the largest group of respondents for the elementary scale. Once the best elementary scale is found, each of the remaining n - 3 stimuli is investigated to see whether it might make the best fourth stimulus. The best fourth stimulus (e.g., D) may be added to the best triple (e.g., ABC) in any of four positions: DABC, ADBC, ABDC, or ABCD. These places are denoted as place 1 to place 4. The best fourth or, more generally, p + l’th - stimulus has to meet the following criteria to be included in a p-stimulus unfolding scale: 1. All new

k]

triples that include the p

+ i’th

stimulus and two

stimuli from the existing p-stimulus scale have to have a positive

H (ijk)-value. This guarantees that all stimuli are homogeneous with respect to the latent dimension. 2. The p + l’th stimulus should be uniquely representable, i.e., it can be positioned in only one of the p possible places in the p-stimulus scale. This helps to ensure the later usefulness and interpretability of the order of the stimuli in the scale.

+ l’th stimulus, as well as the H-value of the scale as a whole, have to be higher than some user-specified lower boundary (see second criterion for the best triple). Actually, adding a stimulus to a scale may even increase the H-value of the scale as a whole, depending on the scalability quality of the triples that are added to the scale.

3. The Hi-value of the p

4.

If more than one stimulus conforms to the criteria mentioned above, that stimulus will be selected that leads to the highest scalability for the scale as a whole.

This procedure of extending a scale with an additional stimulus is repeated as long as the criteria mentioned above are satisfied. When no further stimulus conforms to the criteria, the p-stimulus scale is taken as a maximal subset of scalable stimuli. This maximal subset can then be further evaluated as an unfolding scale with additional goodness-of-fit criteria.

Unfolding the German Political Parties

27 1

3.5 Maximizing Perfection or Minimizing Error? The search procedure of finding an unfolding scale is based on identifying a maximal subset of stimuli that contains the smallest proportion of errors in its mples. An alternative procedure might be to find a maximal subset of stimuli that contains the largest proportion of perfect patterns among all of its patterns (e.g., Davison, 1980). If we had applied this procedure we would have been interested in the extent to which the number of perfect patterns found exceeds the frequency of perfect patterns to be expected under statistical independence. We should not accept a set of stimuli as a scale if the observed frequency of perfect patterns is no more than can be explained by assuming statistically independent responses. Observed and expected frequencies of perfect patterns can also be compared by applying Loevinger’s coefficient of homogeneity. For a “pick 3/n” analysis this becomes: H=l-

0hs.Frey. ( i j k , 111) Exp.Freq.(ijk, 11 I ) ’

where 0bs.Freq.(ijk, 11 1) and Exp.Frey.( i j k , 11 1) are counted and calculated, respectively, in the same way as the error response patterns. Perfect response patterns - especially the “ 111”-responses to adjacent stimuli - should occur more frequently than expected under statistical independence, and should have a negative H-value, whereas imperfect patterns, that is, “1 11 ”-responses to non-adjacent stimuli, should occur less often than expected under statistical independence and should have a positive H-value. There are at least two problems in using the frequency of perfect patterns and the H (ijk, 11 1)-coefficients to find an unfoldable subset of stimuli. One problem is the difficulty in finding a “best” or even unique ordering of stimuli. Whereas the unique ordering of a “best” triple of stimuli follows from the (non)occurrence of errors in the three permutations, no unique ordering of the stimuli is implied in the H (ijk, 11 1)coefficients. A more important problem is that with this procedure the evaluation of a set of stimuli as an unfolding scale cannot be based on the error patterns of all triples, but only on the set of (perfect) patterns of adjacent triples. In the “pick kln” case this involves only the evaluation of n - k

+ 1 patterns,

whereas i n the procedure I am advocating all

212

van Schuur

triples are considered. Evaluating the frequencies of the perfect patterns will therefore only be used heuristically at the end of the scaling procedure to help in considering other possible start sets for the search procedure described above, and in evaluating an hypothesized unfolding scale. 3.6 The Dominance Matrix and the Adjacency Matrix The use of the coefficient of scalability as a test for the goodness-of-fit of a candidate unfolding scale can be criticized on grounds that the coefficient is not specifically tuned to the unfolding model: a good fit can be obtained for data that conform either to the unfolding model or to Guttman’s cumulative scaling model. Although this criticism is justified for the “pick anyln” case, its force can be reduced by subjecting the dominance matrix and the adjacency mamx of the unfoldable stimuli, in their scale order, to visual inspection. The dominance matrix is a square asymmetric matrix whose cells ( i J ) display the percentage of subjects who prefer stimulus i but not stimulus j . When the stimuli are ordered in their scale order along the J-scale, then for each stimulus i the percentage pi, should decrease from the first column toward the diagonal and increase from the diagonal toward the last column. The adjacency matrix is a lower triangle whose cells ( i , j ) show the percentage of subjects who “picked” both i and j . When the stimuli are ordered in their scale order along the J-scale, then for each stimulus i the percentage pi, should increase from the first column toward the diagonal and decrease from the diagonal toward the last row. The procedure for detecting stimuli that disturb the expected pattern of characteristic monotonicity is analogous to the procedure Mokken (197 1) used in multiple unidimensional cumulative scaling. Table 1 shows the dominance matrix and the adjacency matrix for a perfect unidimensional unfolding data set. Note that in the dominance mamx no column-wise monotonicity is expected. If stimuli form a cumulative scale rather than an unfolding scale, the monotonicity patterns of the dominance matrix of stimuli will differ from those just described, in that they will not reverse around the diagonal. An important difference between the use of the coefficients of scalability and the use of the dominance and adjacency matrices must be

Unfolding the German Political Parties

213

Table 1. Dominance Matrix and Adjacency Matrix

for a Perfect Four Stimulus Unfolding Scale. ~~

Data matrix A

B

C

D

Frequency

1

0 1

0 0 1

0 0

P

0 0 0

0

r

0

1

S

0

0 0 1 0

t

1

X

C p + t q+t

D p+t+w q+t+u+w

1

0 0 1

0

0 0 1 1

0 1 1

1 1 I 1

4

U V W

Dominance matrix A A B C

D

B

P q+u+x

r+u+v+x s+v+x

r+v s+v

r+u+w S

Adjacency matrix A

B

B C

t+W W

u+w+x

D

0

X

C

v+x

mentioned here. The coefficients of scalability reflect the relative number of errors, whereas the matrices reflect the absolute number of errors. Dijkstra et al. (1980) have shown already that the characteristic monotonicity requirement is not a sufficient condition for a set of stimuli to be interpreted as an unfolding scale. They give a counter example in which a perfect characteristically monotone dominance matrix was derived from Iscales that did not belong to the same J-scale. Looking at the pattern of absolute frequencies only and disregarding the information from the H coefficients may therefore lead to unjustified acceptance of an unfolding

214

van Schuur

scale.

3.7 Scale Values Once an unfolding scale of a maximal subset of stimuli has been found, scale values for stimuli and subjects can be determined. The scale value of a stimulus is defined as its rank number in the unfolding scale. The scale value of a subject is defined as the mean of the scale values of the stimuli the subject “picked” as most preferred. Subjects who did not pick any stimulus from the scale cannot be given a scale value, and have to be treated as missing data.

4. An Application to Preference for German Political Parties Transitive rank orders of preference were derived from pairwise preference comparisons for five German parties by German voters in 1969 (N = 907) and in 1980 ( N = 1316). Full rank orders of preferences given in 1972 were obtained directly from a random sample of 1785 German voters (the data are published in Pappi (1983); Norpoth (1979a, 1979b) also discusses the 1969 and 1972 data). The MUDFOLD model will be applied to a “pick 2/5” and a “pick 315” analysis of these three data sets. The parties are denoted by the capital letters A-E as follows: A: CDU/CSU (Christian democrats); B: SPB (social democrats); C: FDP (liberals); D: NPD (neo-national socialists); E: DKP (communists). The scalability values of each triple of all stimuli in each permutation, as well as the dominance and adjacency matrices, are given in the Appendix in Table 2 through Table 7. For each permutation of a triple i,j,k (e.g., jik, ijk, and ikj) the observed and expected frequency of the error patterns are given (i.e., the patterns ijk,Oll, ijk,lOl, and ijk,llO, which are the error patterns of the scales jik, ijk, and ikj, respectively), as wel! as their appropriate H-value. Expected frequencies are rounded to the nearest integer. On the basis of this information an unfolding scale of a maximum subset of stimuli is constructed. In the “pick 3/5” analyses, the observed and expected frequencies of the “ijk,lll”-patterns are also given, together with the matching H-value. The dominance matrix contains the percentage of subjects who “pick” the row party but not the column party among the most preferred. The adjacency matrix contains the percentage of subjects who “pick” both the row party and the column

Unfolding the German Political Parties

215

party among the most preferred.

4.1 A “Pick 2/5” Analysis of the 1969 Data Five of the ten triples have a positive and high enough coefficient of scalability (i.e., > 0.30) in only one of the three possible permutations, that is, they are “unique” triples:

BAD : SPD - CDU - NPD EBA : DKP - SPD - CDU CAD : FDP - CDU - NPD ADE : CDU - NPD - DKP BED : SPD - DKP - NPD

(H = 0.71) ( H = 0.80) (H = 0.66) ( H = 0.80) (H = 0.71).

However, it is impossible to construct a scale of more than three stimuli. The three major parties (ABC, or: CDU, SPD and FDP) have a negative H-value in all three permutations, which means that they cannot be represented together i n one unidimensional unfolding scale. Moreover, the scale value of a larger scale containing all three major parties would be very low. Finally, the position of the DKP (stimulus E) - either to the left of the SPD (stimulus B), or close to the NPD (stimulus D) - cannot be uniquely determined. The five three-item scales can be interpreted either in terms of a left-right dimension (the first three), or in terms of a government-opposition dimension (the last two). On the basis of the dominance and adjacency matrices an unfolding order of the stimuli SPD CDU - FDP - NPD - DKP might have been expected. However, the “unique” triple CAD (FDP - CDU - NPD), which we have already seen has an acceptably high coefficient of scalability, violates this order. The fact that the triple with the three major parties cannot be unfolded suggests that the German voters did not all use the same criterion for preference for the five political parties.

4.2 A “Pick 3/5” Analysis of the 1969 Data.

The best candidate starting triple is the only “unique” ordered triple, BCE, that has a H-value larger than 0.30, namely 0.49. Since the triple BCD has negative H-values in all three permutations, the scale BCE cannot be extended with stimulus D. If we follow the strict procedure, the best triple cannot be extended with stimulus A either, since A can be represented in two places in the scale: position 1 (giving scale ABCE), or

van Schuur

276

position 2 (giving scale BACE). Moreover, in both positions the scalability value of stimulus A, H(A), falls slightly below the user-specified lower boundary of 0.30. And even if we are willing to accept stimulus A in the unfolding scale, it is difficult to choose between these two positions on the basis of the monotonicity patterns in the dominance and adjacency matrices. However, if we relax the criterion of unique representability to allow stimulus A to be represented in the position that gives the highest overall H-value, then it will be represented in scale BACE (SPD - CDU FDP - DKP). This scale can probably best be interpreted in terms of a “government-opposition” dimension: the “Great Coalition” between SPD and CDU governed the Federal Republic until 1969. The scale is rather weak, however. Scale ABCE A: CDU B: SPD C: FDP E: DKP

Scale BACE

Pi

Hi

0.98 0.97 0.94 0.02

0.28 0.32 0.32 0.34 H = 0.32

B: SPD A: CDU C: FDP E: DKP

Pi

Hi

0.97 0.98 0.94 0.02

0.35 0.29 0.32 0.37 H = 0.33

where pi is the proportion of subjects who “pick” stimulus i as most preferred, and Hi coefficient of scalability for item i.

4.3 A “Pick 2/5” Analysis of the 1972 Data The best triple among the “unique” triples ADE, CBE, and BED is CBE (or reflected as EBC: DKP - SPD - FDP): its frequency of admissible patterns (011 and 110) is highest, and its H-value is 1.00. Unfortunately, as in the analysis of the 1969 data, this triple cannot be extended to form a scale that comprises all three major parties (CDU, SPD, and FDP). This is because each of the three pairs that can be made of these three parties (CDU + SPD; CDU + FDP; and SPD + FDP) are mentioned approximately as often as would be expected under statistical independence. There are 1730 respondents, or 97%, who “picked” two of the three major parties as most preferred. The representation of the two small parties DKP (stimulus E) and NPD (stimulus D) is also problematic. The unique triples ADE and BED suggest that D and E are relatively close together, whereas information

Unfolding the German Political Parties

277

from BAD and CAD suggests that D is relatively close to A (CDU), and information from ABE suggests that E is relatively close to B (SPD), which is in accordance with the suggestion from other analyses that D and E are the end points of the scale. The two other criteria for evaluating data as an unfolding scale (the Occurrence of perfect patterns and the characteristic monotonicity patterns of the dominance and adjacency mamces) do not suggest the same solution: the four pairs of parties that are mentioned together more often than expected under statistical independence, (AD, DE, BE, and BC) might lead us to expect a scale ADEBC (i.e., CDU - NPD - DKP - SPD - FDP). However, the dominance and adjacency matrices suggest an unfolding scale ECBAD (DKP - FDP - SPD - CDU - NPD), in which the only deviations of the characteristic monotonicity patterns involve the item pairs BE (SPD-DKP) and CE (FDP-DKP). In fact, for the three major parties this last scale conforms to the order that Norpoth (1979a, 1979b) suggested on the basis of his own analyses: FDP - SPD - CDU, which he interpreted in terms of a “religious-secular” dimension. Still, this scale has an H-value of only 0.08, which makes the null hypothesis of statistical independence very plausible. The position of the two smaller parties, DKP and NPD, is based on the responses of a small number of subjects and therefore highly unstable: of the 1785 subjects only 22 mentioned the DKP, 35 mentioned the NPD, and 2 mention both the DKP and NPD among their two most preferred parties.

4.4 A “Pick 3/5” Analysis of the 1972 Data The best elementary unfolding scale is the triple ACE, since the sum of the patterns 011, 110, and 111 is higher than for the other “unique” triples (the ordered triples ABE, ADE, BED, and CED). Its H-value is 0.38. (In this example the triple ABE could also have been considered: the sum of its patterns 011, 110 and 111 is only marginally less, and its H ( i j k ) value is larger (0.65). The search procedure would lead to the same final conclusion, however.) This best triple cannot be extended with stimulus D, since there is at least one negative scalability value in the triples ACD, ADE, and CDE for each of the four possible places. Stimulus B can be represented in more than one position: position 2 (scale ABCE) and position 3 (scale ACBE). The position that gives the highest overall H-value is position 3, giving as

278

van Schuur

a final scale: CDU - FDP - SPD - DKP. The two perfect response patterns of this scale (ABC and BCE) are the two most preferred patterns, and they occur more often than expected under statistical independence, so these results do not violate the unfolding interpretation. This order of the stimuli conforms to the reflected order of the parties on the ideological left-right continuum. Since it is customary to represent political parties from left to right, I have reversed the order to EBCA in the final scale, as well as in the dominance matrix and adjacency mamx. Scale EBCA

E: DKP B: SPD C: FDP A: CDU

Pi 0.05 0.98 0.98 0.93

Scale ECBA

Hi 0.63 0.43 0.37 0.32 H = 0.42

DKP FDP SPD CDU

Pi

Hi

0.05 0.98 0.98 0.93

0.54 0.29 0.36 0.30 H = 0.36

Let us return for a moment to the nonrepresented party, the NPD (stimulus D). The triples incorporating stimulus D that have the highest scalability values are BAD, CAD, and ADE. The triples BAD and CAD are mentioned relatively frequently (24 and 30 times, respectively), more frequently than would be expected under statistical independence. This suggests that the NPD should be represented to the right of the CDU. The scale in this case would be EBCAD. However, triple ADE also has a high scalability value ( H = 0.76), and these three stimuli are also mentioned together more often than expected under statistical independence. In fact, all triples including both stimuli D and E (NPD and DKP) occur more often than expected under statistical independence. This relatively frequent co-occurrence of NPD and DKP - which are at opposite end on the ideological left-right continuum - suggests that at least some subjects used another dimension in establishing their preference order (e.g., a “protest”, “anti-system”, or “government-opposition” dimension). 4.5 A “Pick 215” Analysis of the 1980 Data

Among the unique mples (ACB, ADE, BED, CAD and CBE) mple CBE is the best one, according to the criteria given in 3.4. It can be extended with stimulus D in the fourth place, giving scale CBED. The best triple

Unfolding the German Political Parties

279

cannot be extended with stimulus A: although all H(ijk)-values for the scale ACBE are positive, the scalability value of the scale as a whole drops below 0.30. The representation of stimulus D (NPD) next to E (DKP) rather than at the other end of the scale, next to the FDP, depends on a single person who mentions stimuli D and E together. The expected number of subjects who would mention D and E together under statistical independence is 0.089. Because of this one subject, the values for H(EBD) and H(ECD) become negative, which precludes the scale EBCD, according to our criteria. Moreover, the scalability of scale DEBC is higher then that of the scale EBCD. Scale EBCD

E: DKP B: SPD C: FDP D: NPD

Scale DEBC

Pi

Hi

0.01 0.69 0.77 0.01

0.46 0.67 0.67 0.52 H = 0.61

NPD DKP SPD FPD

Pi

Hi

0.01 0.01 0.69 0.77

0.96 0.83 0.83 0.89 H = 0.88

4.6 A “Pick 3/5” Analysis of the 1980 Data

The best triple has to be sought among the three unique triples with a H(ijk)-value of over 0.30: ABE, ACE, and BED. Triple ACE has the highest sum of the frequencies of the 011, 110 and 111 patterns, and is therefore chosen as the best triple (H = 6.40).Stimulus D cannot be added to the scale: according to the H-values of triple CDE, E should be represented between C and D, but due to the negative H-value of the ordered triple AED this representation is not possible. Stimulus B can be represented in two places: position 2 (scale ABCE) and position 3 (scale ACBE). Representation of stimulus B in position 3 gives the highest overall H-value, with no violations of the characteristic monotonicity patterns of the dominance matrix and the adjacency matrix. This scale (in reflected order: DKP - SPD - FDP - CDU) can be interpreted in terms of a left-right dimension.

van Schuur

280

Scale ECBA

Scale EBCA

E: DKP B: SPD C: FDP A: CDU

Pi

Hi

0.07 0.97 0.99 0.92

0.81 0.74 0.64 0.58 H = .70

E: DKP C: FDP B: SPD A: CDU

Pi

Hi

0.07 0.99 0.97 0.92

0.75 0.21 0.39 0.35 H = .39

5. Discussion

Applications of MUDFOLD, a computer program for the unidimensional unfolding analysis of dichotomous data, to preferences for five political parties by West German voters in 1969, 1972, and 1980, lead to unfolding scales for four of the five parties. It is not possible to represent all five German parties in a unidimensional unfolding scale. The difficulty in unfolding the preference rankings of the five German parties has already been pointed out by Norpoth (1979a, 1979b) and Pappi (1983). The detailed information, obtained through MUDFOLD analyses, suggests two major reasons for this difficulty: first, that there is very little structure in preferences for the three major parties, and second, that the two smallest parties can be represented in two conflicting ways. In the three years for which data have been analyzed, he number of subjects who mentioned one of the three pairs of the three major parties together as most preferred in the “pick 2/5” analysis, or who mentioned all three parties together as most preferred in the “pick 3/5” analysis, hardly deviates from the number that would be expected under statistical independence. Two possible explanations may be given for this finding, both of which are compatible with an unfolding representation. According to the first, the three parties are very close together on the unfolding scale, and are therefore difficult for subjects to distinguish. Second, subjects may differ in their interpretation of the position of the three major parties along the underlying dimension. For instance, for some people the FDP may be representable to the right of the CDU, whereas for others the FDP should be placed between SPD and CDU, or even to the left of the SPD. Such cognitive difference would make the unidimensional representation of differences in preferences impossible. Klingemann (1972) and Pappi (1980), among others, present some evidence supporting this phenomenon.

Unfolding the German Political Parties

28 1

The conflicting possible representation of the DKP and NPD as either close together or each at opposite ends of the unfolding scale is found in all three data sets. This also suggests that different subjects may base their preference judgments on different criteria. However, only a small number of subjects mentioned these parties together among their two or three most preferred ones, and it is difficult to make valid inferences on the basis of a comparison between small numbers of observed and expected errors. An alternative explanation for the results for these two parties is that some respondents or some coders may have inadvertently reversed the appropriate pairwise preference judgments or the preference I-scales. The least-preferred parties would then have been interpreted as the most preferred, and vice versa. This reversed order is more in agreement with the dominant unfolding interpretation. However, there is no way to validate this suggestion on the basis of the published data. Despite the difficulty of representing all parties along one unidimensional unfolding scale, we still find some easily interpreted structure in subsets of the data. In 1969 the preference effects of the “Great Coalition’’ are clearly visible in a “government-opposition” dimension. Most of the additional structure found among unfoldable triples or four-tuples of parties can be interpreted in terms of the “left-right” dimension, which Klingemann (1972) also identified as important on the basis of other evidence. These results do not conform to the interpretation given by Norpoth (1979a, 1979b) for the same data. Norpoth analyzed these data by constructing an unfolding scale for a maximal subset of subjects rather than a maximal subset of stimuli, and concluded that the three major parties would form the best unfolding scale in the order FDP - SPD - CDU, which he interpreted as a “religious-nonreligious” dimension. However, he did not find this interpretation very plausible: “... the overwhelming share [of subjects] claimed by this dimension strains credulity. Religious issues have rarely if ever topped the priority list of the public in recent years” (1979b, p. 729). By insisting on keeping all stimuli in the scale, which forced him to throw out at least 20% of his subjects without any substantive explanation, he found it impossible to obtain the left-right results that he also had expected on the basis of Klingemann’s previous studies.

282

van Schuur

The major reason for the difference between Norpoth’s findings and those presented here, lies in Norpoth’s emphasis on absolute numbers of errors, compared to my emphasis on the number of errors relative to the number of errors that would be expected under the null hypothesis, i.e., the hypothesis that subjects’ responses are statistically independent of each other. It is true that the permutation FDP-SPD-CDU, that Norpoth accepts as an unfolding scale among the three major parties, is the order that leads to the smallest absolute number of errors. However, this number of errors does not differ significantly from the number of errors that would be expected under the null hypothesis of statistical independence.

Appendix. Detailed Information on MUDFOLD Analyses Table 2. 1969 Data, Pick 215, N = 907 Error paitcrns

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE

Obs

Exp

Huik)

Obs

Exp

143 3 11 1 2 6 1 2 6 6

141 10 9 3 2 .2 3 2

-.01 .71 -.29 .66 .20 -32.54

163 15 2 15 2 2 3 11 11 2

162 12 10 12 10 10 10 9 9 2

.2 .2

.66 -20 -32.54 -32.54

H(iJk) Obs -.01 -.29 .80 -.29 .80 30

.71 -.29 -.29 .20

Dominance matrix

B: SPD A:CDU C:FDP D:NPD EDKP

561 561 561 163 163 15 143 143 3 1

Exp

H(ikJ)

558 558 558 162 162 12 142 142 10 3

-.01 -.01

-.01 -.01 -.01 -.29 -.01 -.01

.71

.66

Adjaccncy matrix

SPD

CDU

FDP

NDP

DKP

-

17 -

63 64

18

16

-

2

1

1

2

3 2

79 80 34 2

78

20

SI’D

SI’D 81 CDU 62 FDP 16 34 2 NPD 0 - D K P 1

CDU

FDP

NDP

2 0

1

18 0 0

DKP

Table 3. 1969 Data, Pick 315, N = 907

C

E m r patterns

2s

111 patterns

a

00

Obs ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE

Exp

14 12 2

20 17 3

18 8 6 23 7 8 2

17 3 .2 22 4 .3 .4

H(jik)

Obs

Exp

H(ijk)

Obs

Exp

H(ikj)

Obs

Exp

.29 .29 .37

18 19 3

26 22 4

.30 .14 .27

49 822 863

51 818 853

.03 -.01 -.01

818

-.07 -1.60 -24.08 -.05 -.76 -26.87 -4.48

47 6 5 45 4 6 3

43 8 12 43 8 11 7

-.09 .24 .57

819 835 62 820 830 57 29

814 832 65 813 827 60 39

-.01

-.04

-49 .44 .56

Dominance matrix SPD B: SPD A:CDU C:FDP D:NPD E:DKP

2 3 3 1

FDP

NDP

DKP

2

5 6

91 91 91

96 97 93 8

I

-

5 I

.04

-.01 -.OO .05 .25

B

810

-.01

Q

45 4

43 8

17 1 2 12 2 0 6

22 4

-.05 .48 .22 .74 -8.69 .28 .33 1.oo -73.47

.2 17 3 .2 .I

2

s 3

%

% =. r,

Q

e%

3.

2

Adjacency matrix

CDU

2 1 1

-.OO

H(111)

SPD CDU FDP NPD DKP

SPD

CDU

FDP

NDP

96 92 6 1

92 7 1

4 1

1

DKP

h)

01

w

van Schuur

284

Table 4. 1972 Data, Pick 215, N = 1785 Error patterns

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE

Obs

Exp

Huik)

Obs

Exp

690 7 19 3 0 2 3 0 2 2

677 20 12 7 5 .1 7 5 .1 .1

-.02 .64 -.55 .58 1.oo -14.38

279 23 1 23 1 1 7 19 19 0

283 8 5 8 5 5 20 12 12 5

.58

1.00 -14.38 -14.38

H(iJk) Obs .01 -1.82 .80 -1.82 .80 .80 .64 -55 -.55

1.00

Dominance matrix DKP

E: DKP C:FDP B: SPD A:CDU D:NPD

-

SPD

CUD

NPD

1

0 16

1 39 40 -

1 54

-

82 60 2

44 44 2

17 2

H(ikj)

.o 1 .o1 .o 1

768 768 768 283 283 8 677 677 20 7

.01 .01 -1.82 -.02 -.02

.64 .58

Adjacency matrix

FDP

54

761 761 761 279 279 23 690 690 7 3

Exp

1

82 58

-

DKP DKP FDP SPD CDU NPD

FDP

SPD

39 16 0

43 0

CDU

0 1

0 0

1

NPD

Table 5. 1972 Data, Pick 315, N = 1785 Error patterns

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE

Obs

Exp

122 45 79 45 79 2 31 3

123 66 58 66 58 1 20 18 .9 .8

5

5

111 patterns

Hulk)

Obs

Exp

H(ijk)

Obs

Exp

H(ikj)

Obs

Exp

.oo

32 34 6 28 10 8 25 7 84 80

36 19 17 18 16 33 19 17 74 74

.ll -.76 .65 -.53 .38 .76 -.33 .58 -.14 -.08

30 1601 1619 1597 1625 54 1673 1639 68 74

34 1607 1609 1607 1610 37 1648 1656 83 84

.12

1595 24 6 30 2 4

1590 18 16 19 17 .2 65 58 .7 .7

.32 -.36 .32 -.36 -.48 -.56 .83

-4.64 -4.89

Dominance matrix DKP E: DKP B: SPD C: FDP A: CDU D: NPD

93 94 92 5

FDP

CDU

NPD

0

1 2

4 7 7

5 94 94 90

3

-

2 2

-.01 .01 -.01 -.47 -.02 .01 .18 .12

44 78 1 1

-.oo -.32 .63 -.56 .88 -19.72 .33 -.36 -.66 -.44

Adjacency matrix

SPD

2 2 2

.oo

H(111)

DKP DKP SPD FDP CDU NPD

SPD

FDF

CDU

96 91 4

91 4

3

5 5 1 0

NPD

van Schuur

286

Table 6. 1980 Data, Pick 215, N = 1316 Error pattcrns

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE

Obs

Exp

636 1 15 0 3 1 0 3 1 1

626 6 6 9 9 .1 9 9 .1 .I

Huik)

Obs

Exp

-.02 .84 - 1.36

382 17 0 17 0 0 1 15 15 3

378

-.01

4 4 4

-3.44 1.00 -3.44 1.00 1.00 .84 -1.36 -1.36 .66

1.00

0.66 -10.29 2.00

.66 -10.29 -10.29

4 4

6 6 6 9

H(ijk)

D.NPD E: DKP B: SPD C: FDP A: CDU

-

DKP

FDP

1

1

1

68 77 50

0 29 30

1 21 21

1

69 78 49

SPD

Exp

26 1 26 1 26 1 382 382 17 636 636

275 275 275 378 378 4 626 626 6 9

1

0

H(ikj) .05 .05

.05 -.01 -.01

-3.44 -.02 -.02 .84 1.00

Adjnccncy matrix

Dominance matrix NDP

Obs

NDI’

CDU 0 1 50 49

-

NPD DKP SI’D

FDI’ CDU

DKI’

SPD

FDP

1 0 0

48 20

29

CDU

0 0 0 1

-

Table 7. 1980 Data, Pick 3/5, N = 1316

C

Error patterns

ABC ABD ABE ACD ACE

ADE BCD BCE BDE CDE

Obs

Exp

100 20 80 21 81 1 37 2 4 3

100 45 56 45 57 1 18 23 1 .3

Huik)

Obs

-.MI

37 39

.55 -.43 .53 -.43 .09 -1.00 .91 -3.22 -7.68

4

11 4 2 8 1 81 81

Exp

H(iJk)

Obs

40 18 23 5 7 29 5 7 62 78

.08 -1.20 .82 -1.10 .40 .93 -.49 .85 -.30 -.04

9 1167 1174 1167 1202 44 1246 1186 28 56

3 @a

Exp

H(ikj)

Obs

Exp

H(111)

5

12 1169 1168 1186 1180 23 1219 1207 49 62

.23

1166 8 1 36 1 3 20 80 0 1

1163 5 7 18 22 .1

-.00 -.56 .85 -1.04 .96 -29.33 .55 -.43 1.oo -.18

9 3Q

Dominance matrix

.OO -.01

.02 -.02 -.93 -.02 .02

.43 .09

44 56 .2 .8

m

3

% 2

s 0

E

e3 2.

8

Adjacency matrix

~

~~

E:DKP B: SPD C: FDP A: CDU D: NPD

3t?:

111 patterns

DKP

SPD

FDP

CDU

NPD

-

0

0 1

6 8 8

6 95 95 89

2

-

91 93 92 5

3 3 3

1 1

DKP DKF' SPD FDP CDU NF'D

SPD

FDP

CDU

96 89 2

91 4

4

NDP

-

6

6 0 0

N

2

288

van Schuur

References Bechtel, G . G . (1976). Multidimensional preference scaling. The Hague: Mouton. Bennett, J. F., 8z Hays, W. L. (1960). Multidimensional unfolding: Determining the dimensionality of ranked preference data. Pyschometrika, 25, 27-43. Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (1975). Discrete multivariate analysis: Theory and practice. Cambridge, MA: MIT Press. Carroll, J. D. (1972). Individual differences and multidimensional scaling. In R. N. Shepard, A. K. Romney, & S. Nerlove (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences (Vol. 1, pp. 105-155). New York: Seminar Press. Coombs, C. H. (1950). Psychological scaling without a unit of measurement. Psychological Review, 57, 148-158. Coombs, C. H. (1964). A theory of data. New York: Wiley. Coombs, C. H., & Kao, R. C. (1960). On a connection between factor analysis and multidimensional scaling. Psychometrika, 25, 219-231. Coombs, C. H., & Smith, J. E. K. (1973). On the detection of structure in attitudes and developmental processes. Psychological Review, 80, 337-351. Davison, M. L. (1979). Testing a unidimensional, qualitative unfolding model for attitudinal or developmental data. Psychometrika, 44, 179194. Davison, M. L. (1980). A psychological scaling model for testing order hypotheses. British Journal of Mathematical and Statistical Psychology, 33, 123-141. Dijkstra, L., van der Eijk, C., Molenaar, I. W., van Schuur, W. H., Stokman, F. N., & Verhelst, N. (1980). A discussion on stochastic unfolding. Methoden en Data Nieuwsbrief, 5 , 158-175. Gold, E. M. (1973). Metric unfolding: Data requirements for unique solutions and clarification of Schonemann’s algorithm. Psychometrika, 38, 44 1-448. Heiser, W. J. (1981). Unfolding analysis of proximity data. Leiden: University of Leiden. Jansen, P. G. W. (1983). Rasch analysis of uttitindinul data. Nijmegen:

Unfolding the German Political Parties

289

Catholic UniversityRhe Hague: Rijks Psychologische Dienst. Klingemann, H. D. (1972). Testing the left-right continuum in a sample of German voters. Comparative Political Studies, 5, 93-106. Kruskal, J. B., Young, F. W., & Seery, J. B. (1973). How to use KYST. Bell Laboratories, Murray Hill, NJ. Leik, R. K., & Matthews, M. (1968). A scale for developmental processes. American Sociological Review, 54, 62-75. Mokken, R. J. (1971). A theory and procedure of scale analysis. The Hague: Mouton. Dimensionen des Parteikonflikts und Norpoth, H. (1979a). Praferenzordnungen der deutschen Wahlerschaft: Eine Unfoldinganalyse. Zeitschrijt fur Sozialpsychologie, 10, 350-362. Norpoth, H. (1979b). The parties come to order! Dimensions of preferential choice in the West German electorate, 1961-1976. American Political Science Review, 73, 724-736. Pappi, F. U. (1983). Die Links-Rechts Dimension des deutschen Parteiensystems und die Parteipraferenz-Profile der Wahlerschaft. In M. Kaase & H. D. Klingemann (Eds.), Wahlen und politisches System, Analysen aus Anlass der Bundestagswahl 1980 (pp. 422-441). Opladen: Westdeutscher Verlag. Roskam, E. E. (1968). Metric analysis of ordinal data. Voorschoten: VAM. Ross, J., & Cliff, N. (1964). A generalization of the interpoint distance model. Psychometrika, 29, 167-176. Schonemann, P. H. (1970). On metric multidimensional unfolding. Psychometrika, 35,349-366. Sixtl, F. (1973). Probabilistic unfolding. Psychometrika, 38, 235-248. Tversky, A. (1972). Elimination by aspects: A theory of choice. Psychological Review, 86, 542-573. Tversky, A., & Sattath, S. (1979). Preference trees. Psychological Review, 86, 542-573. van Schuur, W. H., & Molenaar, I. W. (1982). MUDFOLD: multiple stochastic unidimensional unfolding. In H. Caussinus, P. Ettinger, & R. Tomassone (Us.), COMPSTAT 1982 (Part I, pp. 419-426). Vienna: Ph y sica-Verlag. van Schuur, H. (1984). Structure in political beliefs, a new model for stochastic unfolding with application to European party activists.

290

van Schuur

Amsterdam: CT Press. Zinnes, J. L., & Griggs, R. A. (1974). Probabilistic multidimensional unfolding analysis. Psychometrika, 39, 327-350.

New Developrncnts in Psychological Choice Modeling G. De Soete, H. Fcger and K. C. Klauer (eds.) 0 Elsevicr Scicncc Publisher R.V. (North-Holland), 1989

29 1

PROBABILISTIC MULTIDIMENSIONAL SCALING MODELS FOR ANALYZING CONSUMER CHOICE BEHAVIOR Wayne S. DeSarbo University of Michigan

Geert De Soete University of Ghcnt, Belgium

Kame1 Jedidi University of Pennsylvania We review the development of two new stochastic multidimensional scaling (MDS) methodologies that operate on paired comparisons choice data and render a spatial representation of subjects and stimuli. In the probabilistic vector MDS model, subjects are represented as vectors and stimuli as points in a T-dimensional space, where the scalar products or projections of the stimulus points onto the subject vectors provide information about the utility of the stimuli to the subjects. In the probabilistic unfolding MDS model, subjects are represented as ideal points and stimuli as points in a T-dimensional space, where the Euclidcan distance between the stimulus points and the subject ideal points provides information as to the respective utility of the stimuli to the subjects. To illustrate the versatility of the two models, a marketing application measuring consumer choice for fourteen actual brands of over-the-counter analgesics, utilizing optional reparameterizations, is dcscribcd. Finally, other applications are identified.

The second author is supported as “Bcvoegdverklaard Navorser” of the Belgian “Nationad Fonds voor Weknschappelijk Onderzock”. This paper is a revised version of an article published in Communication & Cognition, 1987, 20, 17-43.

DeSarbo, De Soete, & Jedidi

1. Introduction

The method of paired comparisons involves presenting a subject two stimuli at a time. The subject is then required to choose one of the two presented stimuli (cf. e.g., David, 1963; Thurstone, 1927). Since this paper is concerned with understanding consumer behavior, we will be using the terminology of consumers (for subjects) and productshrands (for stimuli). The method of paired comparisons can be gainfully applied in consumer behavior research whenever it is not possible or feasible to make continuous measurements of the utilities of a set of products or brands. With J products, each of the I consumers typically makes

(”21

judgments. However, if this number is too large, incomplete designs may be utilized (cf. Bock & Jones, 1968; Box, Hunter, & Hunter, 1978) in order to reduce the number of judgments a consumer must make. Since consumers are often inconsistent when making judgments, probabilistic models are needed for analyzing such paired comparisons data. To display the structure in paired comparisons data, several models have been presented in the psychometric literature which represent the consumers and the products in a joint uni- or multidimensional space. There have been an number of unidimensional scaling procedures proposed to obtain scale values for products from such (aggregated) paired comparisons data (for a survey, see Bock & Jones, 1968; Torgerson, 1958). More recently, multidimensional scaling models have been devised to account for the multidimensional nature of the products. Here, two general classes of models have been typically utilized to represent such preferencehhoice data: vector and unfolding models. A vector or scalar products multidimensional scaling model (Slater, 1960; Tucker, 1960) represents the consumers as vectors and the products are points in a 7’-dimensional space. Figure 1 represents a hypothetical two-dimensional portrayal of such a representation where there are two consumers (represented by two vectors I and 11) and five products (represented by the letters A-E). Here, utility or preference order for a given consumer is assumed to be given by the orthogonal projection of the products onto the vector representing that consumer. For example, for consumer I, product B has the highest utility, then E, then A, then D, and finally C. For consumer 11, the order of utility (from highest to lowest) is A, B, C, D, and E. The goal of the analysis here is to estimate the “optimal” vector directions and product

Analyzing Consumer Choice Behavior

293

coordinates in a prescribed dimensionality. An intuitively unattractive property of the vector model is that it assumes preference or utility to change monotonically with all dimensions. That is, it assumes that if a certain amount of a thing is good, more must be even better. (The isoutility contours therefore are parallel straight lines perpendicular to a consumer’s vector.) According to Carroll (1980). this is not an accurate representation for most quantities or attributes in the real world (perhaps with the exception of money, happiness, and health).

Figure I . Two-dimensional illustration of the vector model (taken from Carroll & DeSarbo, 1985).

There has been some work done concerning analyzing paired comparisons via such vector or scalar products models. Bechtel, Tucker, and Chang (1971) have developed a scalar products model for examining

294

DeSarbo, De Soete, & Jedidi

graded paired comparisons responses (i.e., where consumers indicate which of two products are preferred and to what extent). Cooper and Nakanishi (1983) have devised two logit models (vector and ideal point) for the external analysis of paired comparisons data. Carroll (1980) has proposed the wandering vector model for the analysis of such paired comparisons data. According to this vector model, it is assumed that each consumer can be represented by a vector and that individual consumers will prefer that brand from a pair having the largest projection on that vector. The direction cosines of this vector specify the relative weights the consumer attaches to the underlying dimensions. The wandering vector model assumes that a consumer’s vector wanders or fluctuates from a central vector in such a way that the distribution of the vector termini is multivariate normal. De Soete and Carroll (1983, 1986) have developed a maximum likelihood method for fitting this model and have proposed various extensions of the original model to accommodate additional sources of error as well as graded paired comparisons. Unfortunately, the De Soete and Carroll (1983, 1986) model requires replicated paired comparisons per subject (or group of subjects) to estimate more than one vector. This turns out to be a rather difficult data collection task in consumer behavior research. Without such replications, a group of subjects must be considered as replications of each other. Assuming considerable heterogeneity within the group of subjects, the centroid vector for the group may be estimated with considerably high variances on the terminus. In addition, no provision is available to explore individual differences (with replications) as a function of specified subject differences (such as demographic characteristics). DeSarbo, Oliver, and De Soete (1986) propose an alternative probabilistic vector MDS model which operates on paired comparisons. This model can estimate separate subject vectors without requiring withinsubject replications. A variety of possible model specifications are provided where vectors andor stimuli can be reparameterized as a function of specified background variables. We will describe its model structure as well as its program options, and provide a marketing application. The other major type psychometric model to represent such preference/choice data is the unfolding model (Coombs, 1964). We will discuss only the simple unfolding model of Coombs (1964). In the simple unfolding model, both consumers and products are represented as

Analyzing Consumer Choice Behavior

295

x2

t

@ III (DABEC)

@ I (BACDE)

Figure 2. Two-dimensional illustration of the simple ideal point model (taken from Carroll & DeSarbo, 1985).

points in a T-dimensional space. The points for the consumers represent ideal points, or optimal sets of dimension values. The farther a given product point is from a consumer’s ideal point, the less utility that product has for the consumer. This notion of relative distance implies a Euclidean metric on the space which implies that, in T = 2 dimensions, iso-utility contours are families of concentric circles centered at a consumer’s ideal point. Carroll (1980) demonstrates that the vector model is a special case of this unfolding model where the ideal point goes off to infinity. Figure 2 illustrates a hypothetical two-dimensional space from an unfolding perspective. Here there are three consumers represented by ideal points labeled I, 11, and 111, and five products labeled A-E. The figure specifies the preferencehtility order for each consumer as a function of distance

296

DeSarbo, De Soete, & Jedidi

away from the respective ideal point. The objective in unfolding analysis is to estimate the “optimal” set of ideal points and product coordinates in a prescribed dimensionality. Although several unidimensional stochastic unfolding models have been proposed in the literature (Bechtel, 1968, 1976; Coombs, Greenberg, & Zinnes, 1961; Sixtl, 1973; Zinnes & Griggs, 1974), only three multidimensional unfolding models have been developed to accommodate paired comparisons data. The first one by Schonemann and Wang (1972) and Wang, Schonemann, and Rusk (1975) is based on the well-known Bradley-Terry-Luce model and consequently assumes strong stochastic transitivity. In the multidimensional unfolding model proposed by Zinnes and Griggs (1974), it is assumed that the coordinates of both the consumer and the product points are independently normally distributed with a common variance. Zinnes and Griggs (1974) assume that for each element of the product pair, a consumer independently samples a point from his or her ideal point distribution. In the Zinnes-Griggs model, the probability that consumer i prefers product j to k is defined by

where F ”(v1,v2,h1,h2) denotes the doubly non-central F distribution with degrees of freedom v1 and v2 and noncentrality parameters hl and hz, and di; (respectively &) the Euclidean distance between the mean point of consumer i and the mean point of product j (respectively k). More recently, De Soete, Carroll, and DeSarbo (1986) and De Soete and Carroll (1986) have proposed the wandering ideal point model for the analysis of such paired comparisons data as an unfolding analogue of the wandering vector model. According to this model, it is assumed that each consumer can be represented by an ideal point and that he or she will prefer that product from a pair which has the smallest Euclidean distance from that ideal point. This model assumes that a consumer’s ideal point wanders or fluctuates from a central ideal point in such a way that the distribution of the ideal point coordinates is multivariate normal. De Soete, Carroll, and DeSarbo (1986) have developed a maximum likelihood method for fitting this model and show that it is the only existing probabilistic multidimensional unfolding model requiring only moderate stochastic transitivity.

Analyzing Consumer Choice Behavior

297

Unfortunately, as in the case of the wandering vector model, the De Soete, Carroll, and DeSarbo (1986) model also requires replications of paired comparison matrices per consumer to estimate more than one ideal point. Again, this turns out to be a rather difficult task in terms of data collection. Without such replications, only one centroid ideal point can be estimated for a sample of I consumers. Assuming considerable heterogeneity in the sample, the single centroid ideal point me be estimated with considerably high variances. In addition, no provision is currently available to explore individual differences (with replications) as a function of specified consumer differences (such as demographic characteristics), or have similar reparametrizations on products (vis a' vis attributes or features). DeSarbo, De Soete, and Eliashberg (1987) propose an alternative probabilistic MDS unfolding model which also operates on paired comparisons. This model can estimate separate consumer ideal points without requiring within-consumer replications. A variety of possible model specifications are provided where ideal points and/or product coordinates can be reparameterized as a function of specified background variables which aids in the understanding of consumer choice behavior. We will describe its model and structure as well as its program options, and provide a marketing example.

2. Methodologies

2.1 Research objectives As stated, the objective of this paper is to review the two probabilistic MDS models proposed by DeSarbo, Oliver, and De Soete (1986) and DeSarbo, De Soete, and Eliashberg (1987) for representing paired comparison judgments so that consumers and products can be displayed in a joint space, thus permitting inferences concerning the nature of the consumer choice under investigation. In doing so, two sub-objectives will be addressed. The first concerns the ability to investigate the nature of individual (consumer) differences on preference/choice and its measurement, while the second involves modeling the effect of specific product features on the measurement of preferencehhoice. The discussion section will suggest further potential applications to the investigation of still other latent constructs.

DeSarbo, De Soete, & Jedidi

298

2.2 Notation

Let

i = 1, . . . , I consumers, j,k = 1, . . . ,J brands/products, f = 1, . . . , T dimensions, 1 = 1, . . . , L brand features, n = 1, . . . ,N consumer variables,

b

1 if consumer i finds product j more satisfying than k, = 0 else,

{

= the I-feature/attribute value for the j-th brand, Yi, = the n-th background variable value for the i-th consumer, = the f-th coordinate for consumer i, bit = the f-th coordinate for brand j , aa = the impact coefficient of the n-th consumer variable on the r-th

Hjl

dimension, ylt = the impact coefficient of the I-th brand variable on the f-th dimension. 2.3 The Vector Model DeSarbo, Oliver, and De Soete (1986) define a latent consumer preference or utility construct: Vi, = Ui,

+ ei,,

(1)

where V , = the (latent) utility of brand j to consumer i, T

aitbjt,

uij = t=1

ei, = error. Here, U, refers to a “true” utility or latent preference score for consumer i concerning brand j . It is modeled as equal to the scalar product of the brand coordinates (bjt)and the consumer vector (ad). The order of utility or preference for a given consumer is thus assumed to be given by the projection of the brand into the vector representing that consumer. As is characteristic for a vector MDS model, it also assumes that utility or

299

Analyzing Consumer Choice Behavior

preference changes monotonically with all dimensions. Assume now that:

eij

-N(o,~?)

(2a)

(where 0: is the variance parameter for the i-th consumer),

Cov(eij,eik)= 0, V i , j z k ,

(2b)

Cov(eij,eisk) = 0, V i

(2c)

#

i ‘,j,k.

Suppose that consumer i is presented two brands j and k and is asked to select the one that is “more preferred”. Then

P(6ijk = 1) = P(Vij > Vik)

where
DeSarbo, De Soete, & Jedidi

300

The general form of the likelihood function, assuming independence over subscripts i , j , and k , is

where:

a(’)= P(6ijk = 1). Taking logs, one obtains the log likelihood function: I

In L =

C

J

[6i;k

i=l jd

In

(@(a))

+ (1 - ti,,) In (1 -

@(.)I

(6)

DeSarbo, Oliver, and De Soete (1986) developed a procedure for estimating A = ((ai,)) and B = ((b;[)),given A = ((6,)) and T by maximizing (6). Unlike Carroll’s (1980) original formulation of the wandering vector model, the model here posits no explicit distribution on the consumer vectors. Rather, it is a type of random utility model (Thurstone, 1927) where a normally distributed error term is added to a latent construct which is derived from the vector model. 2.4 The Unfolding Model

DeSarbo, De Soete, and Eliashberg (1987) define a similar latent consumer ‘‘dispreference” (inversely related to preference) or disutility construct:

where V ; = the (latent) disutility of stimulus j to consumer i, m

ei; = error. It is assumed that the three assumptions expressed in ( 2 ) concerning ei, also hold here. Suppose that consumer i is presented two brands j and k and is asked to select the one that is “more satisfying”. Then:

Analyzing Consumer Choice Behavior

301

Similarly,

P(6ijk = 0) = 1 - Q,

I

-

-

The general form of the likelihood function, assuming independence over

Taking logs, one obtains the same log likelihood functions as ..I expression (6). DeSarbo, De Soete, and Eliashberg (1987) similarly use maximum likelihood procedures to estimate A = ((ail)), B = ((bjt)),and ( C i ) given A = ( ( 6 i j k ) ) and T . Contrary to De Soete et al.’s (1986) original formulation of the wandering ideal point model, the model here posits no explicit distribution on the consumer’s ideal point. Rather, it too is a type of random utility model (Thurstone, 1927) where an error term is added to a latent construct derived from the unfolding model. Both the wandering vector and the wandering ideal point models, again, require replications of paired comparisons data for each consumer in order to estimate more than a single vectodideal point, since the consumer vector/ideal point is modeled as being explicitly normally distributed. However, both the wandering vector and ideal point models have the advantage of implying only moderate stochastic transitivity, whereas the present two models, like Thurstone’s

DeSarbo, De Soete, & Jedidi

302

(1927) Law of Comparative Judgment Case V, imply strong stochastic transitivity. 2.5 Program Options

These two probabilistic choice models can accommodate a number of different model specifications and options. One can perform either an internal analysis (where the user estimates both brand points and vectorshdeal points) or an external analysis (where the user can f i x one or more sets of coordinates throughout the analysis). The user can also select among a number of methods for generating starting estimates, including a userdefined option. Also, since the scalar products model is invariant under nonsingular linear transformations, and Euclidean distances are invariant under orthogonal rotations, options are provided to rotate either A or B to principal axes for possible enhancement of interpretation. Perhaps the most valuable program options concern the possibility of reparameterizing consumer vectorhdeal point coordinates and/or stimulus coordinates as functions of prespecified background features and attributes. That is, one may reparameterize consumer vectorshdeal points via: N ail

=

Yinaatlr

(1 1)

n =1

and/or brand coordinates via:

As in CANDELINC (Carroll, Pruzansky, & Kruskal, 1980), Three-Way Multivariate Conjoint Analysis (DeSarbo, Carroll, Lehmann, & O’Shaughnessy, 1982), and GENFOLD2 (DeSarbo & Rao, 1984, 1986), one can use these reparameterization options to examine what impact such featuredatmbutes have on the derived solution. This can often aid in interpreting the resulting solution. Note that, because of potential problems associated with placing such restrictions on consumer vectors as discussed in Carroll et al. (1980) and DeSarbo et al. (1982), an option exists to estimate a multiplicative stretching/shrinking parameter, ~ i on , the righthand side of expression (11) for the vector model. These zi parameters are multiplied by the ai, coordinates (the vectors) and act as stretching or shrinking factors for the vectors.

303

Analyzing Consumer Choice Behavior

As mentioned, these reparametrizations can aid in the interpretation of the derived dimensions (cf., Bentler & Weeks, 1978; Bloxom, 1978; de Leeuw & Heiser, 1980), and can replace the post-analysis property-fitting method often used in an attempt to interpret the results. In addition, as shall be discussed, the imposition of these sets of reparameterizations can provide an effective tool for understanding the nature of preference or choice. It should be noted that when a linear function replaces a product or subject coordinate, the number of background variables in the linear function cannot exceed the number of entities that exist for those variables. For example, if J brands have L attributes, J 2 L since one can only identify at most JT coordinates (excluding rotational indeterminacy). Similarly, if I consumers have N background variables, I 2 N since one can only identify at most IT coordinates (excluding rotational indeterminacy). Thus, in most applications, such reparameterizations actually improve the degrees of freedom of the model by reducing the number of parameters to be estimated. In all cases, the degrees of freedom of any of the models are equal to or greater than those in competing joint space models, as shown below.

2.6 Degrees of Freedom One typically collects I

[ J(J; I)]

independent paired comparison

responses in an application. Defining the degrees of freedom of the model as the effective number of parameters, one can calculate the degrees of freedom for the various models accommodated by the vector model methodology. These are shown in Table 1, where it is assumed one is interested in estimating ~i where appropriate. Note that in all these vector models and adjustment of T 2 is required due to the well-known invariance of such bilinear models to nonsingular transformations (Kruskal, 1978). Also, in vector models where bjl is not reparameterized, one can add a constant c to all stimulus vectors b, and not affect the choice probabilities in equation (3). This necessitates a subtraction of T degrees of freedom. Finally, in estimating zi, one can set the overall scale by fixing one zi = 1. The degrees of freedom in the unfolding models are shown in Table 2, where it is assumed on is interested in estimating cfj for all consumers i.

DeSarbo, De Soete, & Jedidi

304

Tubfe I . Degrees of freedom for the various vector models Model

Effective number of model parameters T(I +J ) - T 2- T

T(I + L ) - T2

bjt

T ( N + J ) + (I - 1) - T 2 -T

Tuble 2. Degrees of freedom for the various unfolding models

Model

Effective number of model parameters

T(Z+J)+(Z- 1 ) - T ( T - 1 ) 2

-*

T(I+L)+(Z-1 ) -

T(T- 1) 2

T(N+J)+(I- 1)-

T(T-1) 2

T(N+L)+(I- 1)-

T(T-1) 2

Note that in all these unfolding models, an adjustment of T(T - 1)/2 is

Analyzing Consumer Choice Behavior

305

required due to the well-known invariance of such unfolding models to orthogonal rotations. Also, in unfolding models where b,, is not reparameterized, one can add a constant c to all brand points b, and not affect the choice probabilities in equation (8). This necessitates a subtraction of T from the degrees of freedom. Finally, in estimating q,one can set the overall scale by fixing one q = 1. 2.7 The Algorithms

Maximum likelihood estimates of the set of specified parameters can be obtained by maximizing In L (or minimizing -In L ) in expression (6) for each of the models. The method of conjugate gradients (Fletcher & Reeves, 1964) is utilized to solve these nonlinear, unconstrained optimization problems. Automatic restarts have been implemented to enhance its convergence properties. This conjugate gradient method is particularly useful for optimizing functions of several parameters since it does not require the storage of any matrices (as is necessary in Quasi-Newton and second derivative methods.) A number of goodness-of-fit measures are computed for these models: 1. The log likelihood function: In L.

2. A deviance measure (McCullagh & Nelder, 1983; Nelder & Wedderburn, 1972): I

D = -2

J

C C 6,k

In (&jk)

+ (1 - 6,)

In (1 - i i j k )

(13)

i=l jck

where ;i,k is the estimated probability that consumer i finds brand j more preferable than brand k as expressed in equations (3) and (8) for the two models. Note that one can test nested models within each model type (vector or unfolding) as the difference between respective deviance measures. This difference is (asymptotically) x2 dismbuted with the difference in model degrees of freedom providing the appropriate x2 test degrees of freedom. This test is appropriate in testing dimensionality as well as the various models described in Tables 1 and 2 because of the nested terms. Recall, this is an asymptotic test, however. One obvious problem with this approach concerns the presence of incidental parameters in the likelihood function ( k . , parameters whose order vary according to the

DeSarho, De Soete, & Jedidi

306

order of A, such as the ail’s) as there are no within-subject replications. In such cases, maximum likelihood estimators may not be consistent. However, preliminary Monte Carlo analysis by DeSarbo, Oliver, and De Soete (1986) and DeSarbo, De Soete, and Eliashberg (1986) suggest that the asymptotic likelihood ratio test might still be useful in this situation.

3. The proportion of correct predictions in A. Here, the proportion of times the solution correctly predicts 6,k is calculated for the total sample as well as for each subject. 3. An Illustration

3.1 Study Design A sample of I = 30 undergraduate business students at the University of Pennsylvania were asked to take part in a small study designed to measure preferences for various brands of existing over-the-counter (OTC) analgesic pain relievers. These respondents were initially questioned as to the brand(s) they currently use (as well as frequency of use) and their personal motivations for why they chose such brand(s) (e.g., ingredients, price, availability, etc.). They were then presented fourteen existing OTC analgesic brands: Advil (A), Anacin (B), Anacin-3 (C), Ascriptin (D), Bayer (E), Bufferin (F), Cope (G), CVS Buffered Aspirin (a generic) (H), Datril (I), Excedrin (J), Nuprin (K), Panadol (L), Tylenol (M), and Vanquish (N). The letters in parentheses are plotting codes that will be used furtheron. Initially, they were presented colored photographs of each brand and its packaging, together with price per 100 tables, ingredients, package claims, and manufacturer (cf. DeSarbo & Carroll, 1985). Table 3 presents summaries of selected portion of the descriptions for each of the brands. Each subjecthonsumer was requested to read this information and return to it at anytime during the experiment if he/she so wished. After a period of time, they were asked to make paired comparison preference judgments for all possible 91 pairs of brands. They were told that they had to choose one from each pair (i.e., no ties were allowed). The presented pairs were randomized for each subject.

Analyzing Consumer Choice Behavior

307

Table 3. OTC Analgesic Brand Descriptions

Brand

Advil Anacin Anacin-3

Ascriptin Bayer Bufferin COP

cvs

Dauil Excedrin Nuprin Panadol Tylenol Vanquish

Mg.of Aspirin 0 400 0 325 325 324 421 500

Mg. of Acetaminaphon 0

0

Mg. of Ibuprofen

Mg. of Caffeine

200 0 0 0

0 32 0 0

0 0 0

0 0

0

500 0 0 0 0 0 500

250

250

0 0 0

0

0

200

0

500 3 25 194

0 0 0

0 221

32 0 0

65 0 0 0

33

Mg. of Buffercd Compounds 0

0 0 150 0 100 75 100 0 0 0 0 0 15

Price in US. dollars

Max. Dosage

6.99 3.91 5.29 3.29 2.69 3.89 5.31 1.99

6 10 8 12 12 10 8 12

5.15

8

4.99 1.59 4.99 3.69 4.99

8 6 8 8 12

3.2 Analysis We conducted an analysis of A in T = 1, 2, and 3 dimensions for both models with oj = 1, V i, and the reparameterization option via B = Hy, where H is defined via the feature variables (standardized to zero mean and unit variance) presented in Table 3. This specification was preferred since H contains features that consumers stated (in a pretest) were important in their choice of a specific OTC analgesic brand. All consumers were encouraged to read this information contained on the color photographs of the brand and packaging prior to the paired comparison task. Vector model results. To determine how well, if at all, preference could be defined in terms of the hypothesized manifest variables, an analysis of the 30 x 14 x 14 A array was conducted in T = 1, 2, and 3 dimensions with the probabilistic vector model using the reparameterization option for the b;"s as a function of the feature matrix presented in Table 3. Reparameterization options for ail were not utilized since the students were fairly homogeneous with respect to demographics (except for sex). Table 4 presents the results for the three analysis. If one assumes that the asymptotic x2 test is valid, it is apparent that the twodimensional solution is the most appropriate for the present data. Even if one were to ignore such a test, the differences in the proportion of correct predictions between the three solutions clearly indicate that the two-

DeSarbo, De Soete, & Jedidi

308

Table 4. Vector model results

Dimensionality T=l T=2 T=3

1nL

D

-1421.82 -1262.73 -1255.89

2843.64 2525.46 2511.78

Difference of deviance measures

Proportion of correct predictions

318.18' 12.68

.532 .781 303

* p I.01

dimensional solution is the most appropriate one for this application. In order to assist in interpreting the two dimensions, Table 5 presents the correlations between each of the feature variables (H) and the brand coordinates (B). Here, the first dimension is dominated by buffered ingredients and the aspirin vs. acetaminophen contrast. The second dimension is most highly associated with price, maximum dosage, and the aspirin vs. ibuprofen contrast. (Note that these correlations will of course vary according to the type of rotation used for interpreting A and/or B). Table 5. Correlations between design variables (H) and derived stimulus coordinates (B) for the vector model

Dimension Feature variable

I

I1 ~~

1 2 3 4 5 6 7

.436 -.588

.249 .3 17 .637 .248 .118

~

~~

~

-.910 .397 .7 19 -.284 -.614 369 -.878

Analyzing Consumer Choice Behavior

309

Figure 3 presents the two-dimensional joint space where the 'yb and the ai, coordinates are both represented as vectors (normalized to equal length for convenience) and the brands as points (A-N). The numbers 1-7 represent the 'yft coordinates for the columdvariables of the features mamx H, while subject vector are represented by an arrow. Figure 3 aptly shows how brands in the second and third quadrants are most preferred given the preponderance of subject vectors in these quadrants. Brands C, I, L, and M, all acetaminophen based brands located in quadrant 2, are highly preferred. The two ibuprofen brands, A and K, are located together in quadrant 1. There seems to be some preference for the simpler types of aspirin like E, B, and H. However, those brands with complex aspirin compounds (e.g., with buffered ingredients), located in quadrant 4, are away from the major concentration of subject vectors. It is interesting to note that the first dimension separates the simple vs. complex ingredient brands, while the second dimension separates the aspirin brands vs. the acetaminophen and ibuprofen brands. Unfolding model results. Table 6 presents the statistical summary results for the probabilistic unfolding model for T = 1, 2, and 3 dimensions and the same reparameterization option for the brand coordinates. Here, too, T = 2 dimensions appear to render a parsimonious solution to the structure in the paired comparisons data. In comparison with the vector model results in Table 4, the unfolding model performs slightly better as should be expected due to the fact that the vector model can be formulated as a special case of the unfolding model. Table 7 presents the correlations between the features mamx (H) and the brand coordinates (B). The first dimension is positively related to aspirin ingredients with caffeine and high maximum dosages, as contrasted with the more expensive ibuprofen brands. The second dimension correlates highly with buffered ingredients and high maximum dosages. Figure 4 presents the resulting joint space of ideal points (*), brands (A-N), and feature vectors (1-7). As shown, dimension one separates the ibuprofen brands (A, K) from the heavy aspirin brands with caffeine (B, J, and N). Note how the acetaminophen brands cluster just to the left of the origin. Dimension 2 separates brands with buffered ingredients @, F, H, N) from those with none. Note the distribution of ideal points throughout the space indicating somewhat diverse preferences, although most ideal points are near the acetaminophen brands as presented in the vector model

DeSarbo, De Soete, & Jedidi

310

-2

-1

0

1

2

Figure 3. Two-dimensional solution for the vector model.

Table 6. Unfolding model results

Dimensionality T=l T=2 T=3 * p I .01

In L

D

-1395.11 -1203.65 -1188.43

2790.22 2407.30 2376.86

Difference of deviance measures

Proportion of correct predictions

382.92* 30.44

.682 .831 .846

Analyzing Consumer Choice Behavior

31 1

Table 7. Correlations between design variables (H) and derived stimulus coordinates (B) for the unfolding model Dimension Feature variable

I

i1

.583 -.023 -.761 .519 .098 -.716 .711

.298 -.151 -.340 -.380 -.882 -.477 -.608

results.

3.1

1.i

b

. . . . . . ... . . . . . . . ... .

1

w

..

"--;--I

"" i . . . . . . . . . . . . . . . . .

c."*

e

..... ..

-2.5

. . . . . . . . . .. . .

-1.5

-0.5

0.5

. .' . . . .w . . . dB

1.5

.: . . . . . . .

2.5

Figure 4. Two-dimensional solution for Lhc unfolding model.

3.5

312

DeSarbo, De Soete, & Jedidi

To compare the two MDS procedures, a canonical correlation analysis was performed on the two sets of brand coordinates as an approximate configuration matching technique given the specific types of rotational indeterminacies in each of the two solutions. Here the canonical correlations were: hl = 0.919 and h2 = 0.359, indicating that a least one of the dimensions extracted from each analysis appears to be similar. These different findings are not uncommon in MDS given the diffeerenr utility models underlying each procedure. 4. Discussion

Two new MDS methodologies for the spatial analysis of paired comparisons data have been presented and contrasted to existing methodologies in terms of model structure, stochastic assumptions, input requirements, and model specification options. The models, their assumptions, and the variety of different reparameterization options available for various analyses have been described. A marketing application of the methodologies to a measurement problem in consumer preference was described in some detail for OTC analgesics where seven hypothesized attribute determinants were measured via a features mamx. Two analysis were performed where brand coordinates were directly reparameterized in terms of the features matrix. The procedures each produced a two-dimensional joint space with at least one common brand dimension. These methodologies should prove equally viable for various other applications where paired comparisons are collected. They can aid in similar measurement problems concerning latent, unobservable constructs such as utility, similarity, risk, intention/attitude, etc. With the various reparameterization options for ai, and bjt, additional flexibility is provided for investigating determinants of both individual differences ( e g , demographic information) and stimulus differences. Such reparameterization options would also be valuable in utilizing these methodologies for an external type of preference MDS analysis generally referred to as conjoint analysis. Here, a design matrix is presented defining object stimulus profiles, and a dominance judgment such as preference or intention to buy is asked in paired comparison form. The methodology then derives the contribution of each (orthogonal) object design variable to the resulting derived dimensions. This has proven to be

Analyzing Consumer Choice Behavior

313

of substantial interest to the marketing profession for product design applications (see DeSarbo et al., 1982).

References Bechtel, G. G. (1968). Folded and unfolded scaling from preferential paired comparisons. Journal of Mathematical Psychology, 5 , 333-357. Bechtel, G. G. (1976). Multidimensional preference scaling. The Hague: Mouton. Bechtel, G. G., Tucker, L. R., & Chang, W. (1971). A scalar product model for the multidimensional scaling of choice. Psychometrika, 36, 369-388. Bentler, P. M., & Weeks, D. G. (1978). Resmcted multidimensional scaling models. Journal of Mathematical Psychology, 17, 138-151. Bloxom, B. (1978). Constrained multidimensional scaling in N spaces. Psychometrika, 43, 397-408. Bock, R. D., & Jones, L. V. (1968). The measurement and prediction of judgment and choice. San Francisco: Holden-Day. Box, B. E. P., Hunter, W. G., & Hunter, J. S. (1978). Statistics for experimenters. New York: Wiley. Carroll, J. D. (1980). Models and methods for multidimensional analysis of preferential choice (or other dominance data). In E. D. Lantermann & H. Feger (Eds.), Similarity and choice. Bern: Hans Huber. Carroll, J. D., Pruzansky, S . , & Kruskal, J. B. (1980). CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters. Psychometrika, 45, 3-24. Carroll, J. D., & DeSarbo, W. S. (1985). Two-way spatial models for modeling individual differences in preference. In E. C. Hirschman & M. B. Holbrook (Eds.), Advances in consumer research (Vol. 12). Association for Consumer Research. Coombs, C. H. (1964). A theory of data. New York: Wiley. Coombs, C. H., Greenberg, M., & Zinnes, J. L. (1961). A double law of comparative judgment for the analysis of preferential choice and similarities data. Psychometrika, 26, 165-171. Cooper, L. G., & Nakanishi, M. (1983). Two logit models for external analysis of preference. Psychometrika, 48, 607-620. David, H. A. (1963). The method of paired comparisons. New York:

314

DeSarbo, De Soete, & Jedidi

Hafner. De Leeuw, J., & Heiser, W. (1980). Multidimensional scaling with restrictions on the configuration. In P. R. Krishnaiah (Ed.), Multivariate Analysis V. Amsterdam: North-Holland. DeSarbo, W. S., Carroll, J. D., Lehmann, D. R., & O’Shaughnessy, J. ( 1982). Three-way multivariate conjoint analysis. Marketing Science, I, 323-350. DeSarbo, W. S., De Soete, G., & Eliashberg, J. (1987). A new stochastic multidimensional unfolding model for the investigation of paired comparisons in consumer preference/choice data. Journal of Economic Psychology, 8, 357-384. DeSarbo, W. S., Oliver, R. L., & De Soete, G. (1986). A probabilistic multidimensional scaling vector model. Applied Psychological Measurement, 10, 79-98. DeSarbo, W. S., & Rao, V. R. (1984). GENFOLD2: A set of models and algorithms for the GENeral unFOLDing analysis of preferenceldominance data. Journal of ClassGcation, I , 147-186. DeSarbo, W. S., & Rao, V. R. (1986). A constrained unfolding model for product positioning. Marketing Science, 5 1-19. De Soete, G., & Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model. Psychometrika, 48, 553-566. De Soete, G., & Carroll, J. D. (1986). Probabilistic multidimensional choice models for representing paired comparisons data. In E. Diday, Y. Escouffier, L. Lebart, 3. Pages, Y. Schektman, & R. Tomassone (Eds.), Data analysis and informatics IV. Amsterdam: North-Holland. De Soete, G . , Carroll, J. D., & DeSarbo, W. S. (1986). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data. Journal of Mathematical Psychology, 30, 28-41. Fletcher, R., & Reeves, C. M. (1964). Function minimization by conjugate gradients. Computer Journal, 7, 149-154. Kruskal, J. B. (1978). Factor analysis and principal components. I. Bilinear models. In W. H. Kruskal & J. M. Tanur (Eds.), International encyclopedia of statistics (Vol. 1). New York: The Free Press. McCullagh, P., & Nelder, J. A. (1983). Generalized linear models. New York: Chapman & Hall. Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized linear

Analyzing Consumer Choice Behavior

315

models. Journal of the Royal Statistical Society, A, 135, 370-384. Schonemann, P. H., & Wang, M.-M. (1972). An individual difference model for the multidimensional analysis of preference data. Psychometrika, 37, 275-305. Sixtl, F. (1973). Probabilistic unfolding. Psychometrika, 38, 235-248. Slater, P. (1960). The analysis of personal preference. British Journal of Statistical Psychology, 13, 119-135. Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273-286. Torgerson, W. S. (1958). Theory and methods of scaling. New York: Wiley. Tucker, L. R. (1960). Intra-individual and inter-individual multidimensionality. In H. Gulliksen & S . Messick (Eds.), Psychological scaling: Theory and applications. New York: Wiley. Wang, M.-M., Schonemann, P. H., & Rusk, J. G. (1975). A conjugate gradient algorithm for the multidimensional analysis of preference data. Multivariate Behavioral Research, 10, 45-99. Zinnes, J. L., & Griggs, R. A. (1974). Probabilistic multidimensional unfolding analysis. Psychometrika, 39, 327-350.

This Page Intentionally Left Blank

New Developments in Psychological Choice Modeling G.De Soete, H. Feger and K. C. Klauer (eds.) 0 Elsevier Science Publisher B.V. (North-Holland), 1989

317

PROBABILISTIC CHOICE BEHAVIOR MODELS AND THEIR COMBINATION WITH ADDITIONAL TOOLS NEEDED FOR APPLICATIONS TO MARKETING Worfgang Gaul University of Karlsruhe, FR Germany The paper emphasizes the fact that for applications of choice behavior models based on the Thurstonian scaling approach additional tools for the interpretation of the results obtained can improve the readiness of adoption of such techniques. Among others multidimensional generalizations, a combination of external and internal analyses, and the use of stochastic ideal points and stochastic preference directions are discussed. Examples within the framework of the recently developed factorial model, probabilistic ideal point model and probabilistic vector model approaches are given. For marketing applications the need of interpretation-supporting tools within these probabilistic choice behavior models is demonstrated.

1. Introduction In this paper the discussion of probabilistic choice behavior models is resmcted to the recently developed so-called generalized Thurstonian scaling approaches originating from generalizations of the LCJ (Law of Comparative Judgment, Thurstone, 1927). Examples are the factorial model (Takane, 1980), the wandering vector model (De Soete & Carroll, 1983), and the wandering ideal point model (Blickenholt & Gaul, 1984b, 1986; De Soete, Carroll, & De Sarbo, 1986). The underlying procedure for gathering choice behavior data is the method of paired comparisons. A bibliography concerning references on the paired comparisons literature up

This research has been supported by the Deutsche Forschungsgerneinschafi. This paper is a revised version of an article publishcd in Communication & Cognition, 1987, 20, 77-92.

318

Gaul

to the middle of the seventies is given in Davidson and Farquhar (1976), for further details about the historical development of the ideal point model and vector model approaches, see e.g., Bijckenholt and Gaul (1986). Thus, no attempt of a chronological review of the here described approaches will be made. Instead, it will be shown how the interpretation of the results obtained by different models is affected by the consideration of multidimensional generalizations, a combination of external and internal analyses, the use of stochastic ideal points and stochastic preference directions or segmentation aspects which some of these models allow for. 2. Some Paired Comparisons Based Choice Behavior Models

The Law of Comparative Judgment is the basis for the following generalizations. Thurstone (1927) associated with every choice object 01 from a set of L interesting objects a discriminal process of the form

u, = U[ + El,

1 = 1, . . * , L

(1)

where ul is the (unknown) utility scale value of object 01 and the stochastic vector E’ = (el, . . . , EI, . . , , EL) describes random error which obeys a multivariate normal distribution according to E

- N(O,x) with c = (Ojk), O1[ = Of.

(2)

Z denotes the covariance matrix of

E and 0: the variance of EL. If it is assumed that within a paired comparisons experiment object 0, is preferred to object ok whenever the difference of the corresponding discriminal processes

ujk = uj

- uk

(3)

is positive then - using standard statistical knowledge - the probability of prefemng object 0, to object ok is given by

where 0 denotes the standard normal distribution function and d7k = Oj2 -tOk 2 - 2Ojk = Gj’ + 02 - 2rjkOjOk

(5)

Probabilistic Choice Models for Marketing

319

describes the variance of u,k. Following Gulliksen (1958) the di; are called comparatal dispersions whereas the 01 are called discriminal dispersions; the r,k are correlation coefficients for the corresponding discriminal processes. If S is the total number of subjects judging the pairs of objects the observed choice proportions j j k =

njk 7 can be used to estimate p;k

where

njk is the number of subjects who have preferred object oj to object ok. One gets L ( L - 1)/2 equations of the form Uj

- Uk

= @)-lG;k)d;k

(6)

where @-' is the inverse transformation of the normal distribution function. Equation (6) should be solved with respect to the unknown parameters of the Thurstone model. However, as the number ( L + 4) ( L - 1)/2 of unknown ul-, (31- and r;k-parameters exceeds the number of equations further restrictions such as

ol I CJ > 0, i.e., all comparatal disper-

LCJ case V (r;k sions are equal),

I 0,

LCJ case I11 ;(k' correlations),

= 0, i t . , all discriminal processes have zero

have to be imposed to ensure the solution of (6) unless sufficient estimates for the d;k are available. Of course, many discussions have been published whether or not the LCJ case 111, case V, etc., assumptions reflect reality well enough. These arguments will not be repeated here, instead, the following recently developed generalizations of the Thurstonian approach by Takane (1980), De Soete and Carroll (1983), De Soete et al. (1986), Bkkenholt and Gaul (1984b, 1986) will be described and it will be shown how e.g. the use of stochastic ideal points and stochastic preference directions can help with the interpretation of the results obtained. The classical LCJ allows a one-dimensional scaling of the objects via the corresponding ul-values. A link between multidimensional scaling techniques - which work up measures of (dis-)similarity - and the classical one-dimensional LCJ can be given by the comparatal dispersions (5). This fact has been emphasized e.g., by Sjoberg (1975, 1980) who carried

Gaul

320

out various investigations to find empirical evidence for the conjecture that r,k is in agreement with the rated similarity between the objects 0, and Ok. Indeed, if similarity and, hence, correlation between discriminal processes increase, comparatal dispersion decreases according to equation (5). Additionally, comparatal dispersions satisfy distance properties as was pointed out by Halff (1976). Takane (1980) proposed what is called the factorial model of comparative judgment. Assumed that with an L x M matrix X the covariance mamx C allows the following decomposition

Z=Xx’ then, instead of the o j k (or rjk) the coefficients of the X matrix are now the unknown parameters. Given this reparametrization the comparatal dispersions can be rewritten in the form

m=l

m=l

m =1

m=l

and may be interpreted as distances between the objects 0, and ok in an M-dimensional space in which object 01 is represented by the I-th row XI’ = ( x l l , . . . , x h , . . . , x I ~of) the matrix X . If X is an L x L diagonal matrix one gets the known LCJ case I11 assumptions. The factorial model is an interesting multidimensional generalization of the LCJ, see also Figure 1 and the arguments given there, but it does not allow a simultaneous representation of the subjects and objects in a joint space. To incorporate subjects explicitly in further model developments assume that a subject (or a homogeneous group of subjects) s (s = 1, . . . , S) who is asked to select that object 01 from a pair of objects that is of the greatest utility to him decides according to his utility imaginations Us[where Usl is described by a random variable, then, with the utility-difference

the probability of prefemng

0,

to o k is given by

Up to now the description follows the discussion which we already know from Thurstone’s LCJ. The only thing which remains to be done is to

32 1

Probabilistic Choice Models for Marketing

specify

the

utility terms in an appropriate manner. Let X I ’ = ( ~ 1 1 ., . . , X ~ M )denote the point in the joint M-dimensional space describing object 01 and R, = (R,1,

. . . , R,M) with R,

- N(e,,l)

(10)

denote the random point or vector taking into account the inter- and intra-individual irregularities within the paired comparisons choice behavior of subject s where I describes the identity matrix (of course, the approach allows for RZ - N ( e J , c , l ) , c, > 0, via the transformation R , = l/&R: with e, = l / K e J in analogy with De Soete and Carroll, 1983). In the PVM (Probabilistic Vector Model) (De Soete & Carroll, 1983) R, describes a stochastic preference vector for which it is assumed that subject s samples a vector realization r, from the multivariate normal distribution each time when a pair of objects oj, ok is presented and prefers 0, to ok whenever the relation

holds which corresponds to the fact that x, has a greater projection on the sampled preference vector direction than xk. In a similar manner, in the PIPM (Probabilistic Ideal Point Model) (Bockenholt & Gaul, 1984b, 1986; De Soete et al., 1986) R, describes a stochastic ideal point where, again, it is assumed that subject s selects an ideal point realization r, from the multivariate normal distribution each time when a pair of objects o,, ok is presented and chooses that object which is closest to the actually sampled ideal point realization, i.e. prefers 0, to ok whenever the relation ( X j - rs)’ws(xj

- rs) < (xk - rs>’ws(xk - rs>

(12)

holds where W , is an M x M diagonal matrix of weights. With

Us[= Xl‘R,

(134

or

us1 = - ( X I - R,)‘W,(Xl

- R,)

as corresponding utility values the probability of prefemng object object ok is given by

(13b) 0,

to

Gaul

322 usj

with us/ = xl‘e, and (7)) for PVM or

d!k j

- usk

= <xi- xk)‘(xj - xk) (see also the factorial model

us[ = -(XI - es)’ws(xi - e s ) and d:jk = 4 ( ~-j xk)’W:(~j - x k ) for PIPM where d& are, again, the corresponding variance terms of Of course, the dimensionality M will have to be chosen in such a way that a “best” fit to the data under consideration is possible but variation within the data not accounted for by the chosen dimensionality can be taken into consideration via

uy

= USf+ Es[

where

describes additional random disturbances. With ‘El = u(c)- U ( E ) USJ SJ sk

one, then, gets

Now, notice that in the classical Thurstonian LCJ one was interested in the uf-, of- and rjk-parameters to solve a one-dimensional scaling problem. Here, the coordinates of the es- and xl-vectors - together with the main diagonal of W, (in case of PIPM) and the o-parameter if one uses (14b) - are needed to solve a multidimensional scaling problem in which the different dimensions together with the expected ideal points or expected preference directions may provide the researcher with additional

Probabilistic Choice Models for Marketing

323

support for the interpretation of choice behavior phenomena. The question is whether these attempts to improve interpretation possibilities are sufficient or whether further interpretation-supporting tools are needed within the just described probabilistic choice behavior approaches as will be demonstrated within the following examples.

3. Examples Three examples known from the literature will be used to describe how evaluations of solutions achieved by the different probabilistic choice behavior models can be interpreted and how interpretation possibilities can be improved.

3.1 Rumelhart and Greeno (1971) Study The first example is based on the Rumelhart and Greeno (1971) data. This example is not typical for the marketing area but it has already been used by several other researchers (Bkkenholt & Gaul, 1984; De Soete, 1983; De Soete et al., 1986; Takane, 1980) so that an easy illustration and comparison of known results is possible. In the Rumelhart and Greeno (1971) study 234 college students had to judge the names of well-known personalities with respect to the question “With whom would you prefer to spend an hour of conversation?” on a paired comparisons basis. Three groups of personalities were selected for this experiment. The first group consisted of three politicians (Harold Wilson [HW], Charles De Gaulle [CD], Lyndon B. Johnson Lq), the second group of three athletes (Johny Unitas [JU],Carl Yastrzemski [CY], A. J. Foyt [An), and the third group of three movie stars (Brigitte Bardot [BB], Elisabeth Talyor [Em, Sophia Loren [SL]). Table 1 contains a summary of selected analyses of the Rumelhart and Greeno (1971) data obtained by our computer routines. Similar although not as comprehensive results can be found in De Soete (1983), De Soete et al. (1986) and Takane (1980). Already a cursory glance at Table 1 shows the following: -

The LCJ case V model is not appropriate.

-

The LCJ case In results have a bad fit.

324

Gaul

Tablc 1. Summary of selected analyses on the Rumelhart and Greeno (1971) data (see Bockenholt & Gaul, 1984b). Test against null model

Model specification dimensionality

hL

effective no. of parameters

Null model

-5310.65

36

LCJ Case V LCJ Case I I I

-5351.76 -5327.24 1

Wandering ideal point model (w, = 1. V m )

2 2 3 3 Weighted wandering ideal point model (wltlw12 = 1.23)

a2

add."

add. add.

2

xz

d.f.

p-value

AIC (-1oaOO)

8 16

82.22 33.18

28 20

4.001 0.032

719.52 686.48

-5339.14

9

56.98

27

4001

696.28

-5315.69 -5315.38 -5312.32 -5312.02

16 17 23 24

10.08 9.46 3.34 2.74

20 19 13 12

0.967 0.965 0.996 0.997

663.38 664.76 670.64 672.04

-5315.55

19

9.8

19

0.957

669.10

693.30

-5315.12

20

8.94

18

0.961

670.24

Wandering vector model

2

-5317.68

16

14.06

20

0.827

667.36

Factorial model

2

-5312.99

22

4.68

14

0.989

669.98

2

a

add.

Additional oz parameter.

- Multidimensional versions of the generalized Thurstonian scaling approaches increase the fit significantly. A comparison of the factorial and the wandering (probabilistic, stochastic) vector model was already done by De Soete (1983) who showed that in the underlying example the wandering vector model outperforms the factonal model in terms of the AIC what can also be checked by Table 1. In Bockenholt and Gaul (1984b) (see Tab. 1) and in De Soete et al. (1986) the wandering ideal point model was shown to have the best AIC-value of all considered model versions. Notice, that incorporation of the additional random disturbances parameter o2 could not add much to the interpretation of the data. These arguments, however, establish only part of the mathematical justification for this class of probabilistic choice behavior models. For interpretation purposes the advantages of the simultaneous representation of expected ideal points or expected preference directions together with the choice objects become clear from a look at Figures 1, 2,

Probabilistic Choice Models for Marketing

LJ

CY

JU

**

+9

*?

*S

.a ”

325

AF

BB

SL

Figure I . Two-dimensionalsolution of the Rumelhart and Greeno data according to the factorial model.

3. In Figure 1 one recognizes that the reparamemzation of the factorial model allows a representation of the data in a two-dimensional space which according to Table 1 and in comparison to the poor results of the LCJ case I11 and LCJ case V solutions has a very good fit. The object points cluster in the way they should do and allow to recognize separable groups of politicians and athletes and movie stars where the latter two groups seem to be more similar to each other than to the remaining group of politicians as could be expected. However, an obvious question within a choice behavior experiment “What persons have been mostly preferred” can not be answered from Figure 1. The projections of the object points on the expected preference vector in Figure 2 as well as the distances of the object points from the expected ideal point in Figure 3 allow an answer to the above question.

Gaul

326

!

.s

CD

CY *>

* J

HW

LJ .a

ET

.S

SL

Figure 2. Two-dimensional solution of the Rumelhart and Grecno data according to the wandering vector model. The students in the sample put a conversation with the politician Lyndon B. Johnson at the head of the rank whereas the athlete Carl Yastrzemski takes the last position in this ranking. Of course, once such a ranking is known another obvious question would be “Why?”. Up to now nothing is known about the dimensions in which the solutions are presented. Are there connections between the dimensions in the used joint space and salient aspects of the choice behavior approaches or is the dimensionality just something needed for fitting the mathematical model? One of the many attempts to use the dimensions of the chosen space for interpretation purposes of the solutions obtained is contained in Heiser and de Leeuw (1981) who reanalyzed data originally collected by Sjoberg (1967) from studies by Ekman (1962). In the underlying paired comparisons experiment offenses had to be judged

Probabilistic Choice Models for Marketing

321

with respect to “immorality” and in the two-dimensional space directions concerning reckless vs. intentional causes and a graduation of damage caused by the offenses could be distinguished. While these interpretations are derived without collecting additional information about objects (and subjects) additional tools which could be combined with the up to now described evaluation possibilities should be of interest.

CY

-* JU *a

AF

.z

HW CD

.I

.expected ideal point

Figure 3. Two-dimensional solution of the Rumelhart and Grecno data according to the wandering ideal point model.

3.2 Kaas (1977) Study The second data set is taken from Kaas (1977) who collected paired comparisons data for 10 stimuli consisting of seven hair spray brands and three amounts of money (see Table 2). Hundred customers of a supermarket were asked to judge each brand-brand combination and each brand-money combination with respect to the question “Which stimulus

Gaul

328

possesses a higher worth?” Concerning the money-money combinations it was assumed that a higher amount of money will be preferred, see Table 3 for the aggregated paired comparisons data. Table 2 . Ten choice objects from Kaas (1977).

Amounts of moncy

Brands 1 2 3 4 5 6 7

Elidor Gard Poly Pretty hair Riar Shamtu Taft

8 9 10

2.00DM 2.50 DM 3.00 DM

Table 3. Aggregated paired comparisons matrix for the choice objects in Table 2. 0 72 46 29 37 56

64 31 37 52

28 0 18 21 23 34 28 14

19 41

54 52 0 34 41 64 66 44 43 67

71 79 66 0 53 72 68 44 59 68

63 77 59 47 0 63 71 43 57 69

44 66 36 28 37 0 48 31 35 50

36 72 34 32 29 52 0 29 35 47

69 86 56 56 57 69 71 0 100

100

63 81 57 41

43 65 65 0 0 100

48 59 33 32 31 50 53 0 0 0

Already a cursory glance at Table 4 shows the following:

- The LCJ case V model is not appropriate. -

The LCJ case 111 results have already a non-rejectable fit.

-

All other one-dimensional model versions and also the twodimensional wandering vector model (for which a comparison with the two-dimensional factorial model is interesting) have a bad fit.

Probabilistic Choice Models for Marketing

329

Thus, the attempt to incorporate price as dominant dimension to support the interpretation of the paired comparisons choice behavior data within one-dimensional Thurstonian scaling models - as was done in the original study - was not fully successful. Again, the additional random disturbances parameters 02 could not increase the fit of the models significantly, and was omitted (except for the one-dimensional wandering ideal point and wandering vector model approaches) in Table 4. Tab& 4. Summary of selected analyses on the Kaas (1977) data.

Model specification dimensionality

OZ

C8.W m

Weighted wandering ideal pint model Wandering vector model

Factorial model

1

add.'

2 3 2 1 2 3 2

I~L.

effedive no. of

xz

d.f.

p-value

AIC (-5000)

parameters

Null model LCJ Case V Wandering ideal pin1 model

Test against null model

add.

-2653.68 -2773.30 -2669.02

45 9 18

239.24 30.68

36 27

<0.001 0.284

397.36 564.60 374.04

-2755.44

10

203.52

35

~0.001

530.88

-2664.87 -2659.26 -2662.08

18 26 21

22.38 11.16 15.80

27 19 24

0.717 0.918 0.895

365.74 370.52 366.16

-2772.57

10

237.78

35

-2674.22 -2659.19 -2659.73

18

26 25

41.08 11.02 12.10

27 19 20

~0.001 565.14

0.040 0.923 0.912

384.44 370.38 369.46

Additional uz parameter.

However, Kaas (1977) study demonstrates one of the attempts to use interpretation supporting tools within probabilistic choice behavior models. Figure 4 shows the version of the wandering ideal point model for which the value of the AIC measure indicated the best fit. The graphical representation reveals a perceived similarity of Poly [3], Pretty hair [4] and Riar [5] and of Elidor [l], Shamtu [6] and Taft [7]. Additionally, the amounts of money [8, 9, 101 form a separate cluster as it should be. Concerning the given data one might wish to incorporate three isopreference price lines (circles, ellipsoids,...) which could help with the interpretation of different price levels and/or ideal prices for the group of products under study. Here, the position of Card [2] suggests a lower bound of 3.00 DM for a possible price of this brand, whereas a lower price bound of 2.00

330

Gaul

DM and an upper price bound of 2.50 DM is indicated for e.g., Pretty hair [4] and Riar [ 5 ] .

Figure 4. Two-dimcnsional solution of the Kaas data according to the wandering ideal point model.

This price argumentation also seems to reveal that a low price segment (consisting of Poly [3], Pretty hair [4] and Riar [5]) and a high price segment (consisting of Elidor [l], Gard [2], Shamtu [6], and Taft 171 where Gard [2] has a somewhat more exceptional position) can be separated via the wandering ideal point approach. Of course, if price is the only dominant dimension of the perceptual map a one-dimensional approach would be sufficient but the results of the one-dimensional models in Table 4 do not support this hypothesis. Thus, in a further stage it is meaningful to interpret the dimensions or interesting directions in a yielded multidimensional configuration. For the Kaas (1977) data, unfortunately, no

Probabilistic Choice Models for Marketing

331

additional external stimulus data were at our disposal. Thus, no further interpretations will be given here.

3.3 B’kkenholt and Gaul (1984a) Study The third example is taken from Bijckenholt and Gaul (1984a). Paired comparisons data from 36 subjects (15 participants of continued education at the Chamber of Industry and Commerce of Karlsruhe and 21 students of an introductory course of marketing at Karlsruhe University) who had to judge 10 cognac print ads were collected with respect to the question “Which advertisement do you prefer?”. A short break was made before starting the final part of the experiment in which the subjects were asked to respond to the print ads - this time shown one by one - for a second time. In this second part of the experiment rating scales concerning dimensions - which had been pretested in Gaul (1984) - like credibility, sympathy, extravagance, preciousness, insignificance, etc., had to be filled in. Table 5 contains a summary of selected analyses of the Bijckenholt and Gaul (1984a) data obtained by our computer routines. Already a cursory glance shows the following: the LCJ case V and case I11 results have already a non-rejectable fit. Thus, is it really necessary to look for more sophisticated models? Indeed, the one-dimensional utility scale values (properly normalized) of different models do not show dramatical changes as can be seen from Figure 5. Now, to answer the question “Why?” it is time to remember that a distinction is useful between internal analysis of paired comparisons data in which only the pairwise choice data are taken for further evaluation, and external analysis where a priori given object dimensions are used to explain the pairwise preference judgments in terms of these object dimensions and that object dimensions such as credibility, sympathy, extravagance, preciousness, insignificance, etc., are available from the second part of the experiment. Thus, besides external analyses based on Cooper and Nakanishi (1983) a factor analysis was conducted revealing that two factors which were easy

332

Gaul

Table 5. Summary of selected analyses on the Bkkenholt and Gaul (1984a) data.

Test against null model

Model specification dimensiondity Null model LCJ Case V LCJ Case III Factorial model Wandering vector model

a2

2

InL

effective no. of parameters

-1024.89 -1043.72 -1036.56 -1028.53

x’

d.f.

p-value

AIC (-2000)

45 9 18 25

37.66 23.34 7.28

36 27 20

0.393 0.667 0.995

139.78 105.44 109.12 107.06

1

add.”

-1043.54

10

37.30

35

0.364

107.08

2 2 3 3

add. 0 add. 0

-1031.41 -1033.51 -1027.42 -1027.83

19 18 27 26

13.02 11.24 5.06 5.88

26 27 18 19

0.984 0.925 0.999 0.998

100.80 103.02 108.84 107.66

’ Additional oz parameter to interpret and accounted for 91% of the variance could be extracted, see Table 6. The first factor accounts for 58% and indicates a more emotional-erotic dimension while the second factor presents an image dimension which is characterized by attributes such as extravagance and preciousness. Of course, it should be of interest to perform a multidimensional generalized Thurstonian scaling approach and check whether such a model will reveal similar results. In Figure 6 the solution of a twodimensional probabilistic vector model is shown together with the directions of the object dimensions derived from the external data of the second part of the experiment via simple regression. The expected preference vector direction is situated in the angle between the first and second factor directions and in close agreement with the first factor direction. Thus, it seems that within the used dimensions the dominant dimension has been found. The projection of the object points on the expected preference vector reproduces the one-dimensional solution of Figure 5 but the combination of external and internal analyses together with appropriate multidimensional versions of generalized Thurstonian scaling approaches gives a better understanding of the choice behavior phenomena of the underlying experiment. Within the used object dimensions attributes like “sympathetic”, “credible” and “stimulating” seem to make up such a

Probabilistic Choice Models for Marketing

333

dominant dimension that even one-dimensional approaches give an appropriate fit. For further explanation the reader is referred to Biickenholt and Gaul (1984a). least preferred

7

most preferred

8

4

1

3

10 5

Figure 5. Utility scale values.

Table 6. Factor analysis results.

Dimension

Factor 1

Factor 2

sympathetic credible extravagant stimulating precious Variance explained by each factor

0.986 0.969 -0.163 0.967 0.092 2.88

0.050 0.003 0.911 0.012 0.922 1.68

Up to now no segmentation aspects (with respects to subjects) have been discussed although the ideal point and vector model approaches allow such a proceeding. Of course, the obvious segmentation into the two groups of students and participants of courses at the Chamber of Industry and Commerce was analyzed but did not show results worthwhile to be reported here. In general, it should be of interest to analyze

334

Gaul

probabilistic ideal point and probabilistic vector model approaches with several expected ideal points and expected preference vector directions. Here, different expected ideal points and expected preference vectors could correspond to clusters of subjects and it would be of interest how the fixed positions of the object points in the joint space have to be interpreted with respect to the possibly different viewpoints of the simple clusters. A first description of such segmentation aspects within the mentioned generalized Thurstonian scaling approaches is given in Gaul and Bkkenholt (1986).

Bisquit

+9

Courvoisier

+a

+7 Hennessv

R emv + 6

native

+

'+

mean preference vector

+5

Martell 10 Martell

Courvoirier

+3 I

R emy ++ II

Figure 6. Two-dimensional solution of the Bockenholt and Gaul (1984a) data according to the wandering vector modcl with addi-

tional explanatory object dimensions.

Probabilistic Choice Models for Marketing

335

4. Conclusions

Some advantages of recently developed probabilistic choice behavior models are illustrated. Of course, ideal point and vector models are wellknown in the area of choice behavior. One advantage of stochastic versions of these models which are here described in the framework of generalized Thurstonian scaling is their ability to allow statistical estimation and testing and to provide goodness-of-fit criteria. Sometimes, however, additional external information is needed for interpretation purposes of the solutions obtained. Whereas an internal analysis uses the paired comparisons data, only, an external analysis is based on further object atmbutes by which the choice behavior phenomena can be explained. Marketing examples indicate how internal and external analyses can be successfully combined. Efforts to incorporate segmentation aspects in these models are under consideration.

References Biickenholt, I., & Gaul, W. (1984a). A multidimensional analysis of consumer preference judgments related to print ads. In Methodological advances in marketing research in theory and practise. Copenhagen: EMACESOMAR. Biickenholt, I., & Gaul, W. (1984b). A wandering ideal point approach to the analysis of choice behavior. Discussion paper 68, Institute of Decision Theory and Operations Research, University of Karlsruhe

(TH). Biickenholt, I., & Gaul, W. (1986). Analysis of choice behavior via probabilistic ideal point and vector models. Applied Stochastic Models and Data Analysis, 2 , 209-226. Cooper, L. G., & Nakanishi, M. (1984). Two logit models for external analysis of preferences. Psychometrika, 48, 607-620. Davidson, R. R., & Farquhar, P. H. (1976). A bibliography on the method of paired comparisons. Biomerrics, 32, 241-252. De Soete, G. (1983). On the relation between two generalized cases of Thurstone’s Law of Comparative Judgment. Marhematiques et

Gaul

336

Sciences humaines, 81, 41-51. De Soete, G., & Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model. Psychometrika, 48, 553-566. De Soete, G., Carroll, J. D., & DeSarbo, W. S. (1986). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data. Journal of Mathematical Psychology, 30, 28-41.

Ekman, G . (1962). Measurement of moral judgment: A comparison of scaling methods. Perceptual and Motor Skills, 15, 3-9. Gaul, W, (1978). Zur Methode der paarweisen Vergleiche und ihrer Anwendung im Marketingbereich. Methods of Operations Research, 35, 123-139. Gaul, W. (1984). Datenanalyse auf der Basis von Ordinalurteilen. Studien zur Klassifrkation, 15, 142-152. Gaul, W., & Btickenholt, I. (1986). Generalized Thurstonian scaling of advertising messages. Discussion paper, Institute of Decision Theory and Operations Research, University of Karlsruhe (TH). Gulliksen, H. (1958). Comparatal dispersion, a measure of accuracy of judgment. Psychometrika, 23, 137-150. Halff, H. M. (1976). Choice theories for differentially comparable alternatives. Journal of Mathematical Psychology, 14, 244-246. Heiser, W. J., & De Leeuw, J. (1981). Multidimensional mapping of preference data. Mathematiques et Sciences humaines, 73, 39-96. Kaas, K. P. (1977). Empirische Preisabsatzfunktionen bei Konsumgutern. Berlin: Springer. Rumelhart, D. L., & Greeno, J. G. (1971). Similarity between stimuli: An experimental test of the Luce and Restle choice models. Journal of Mathematical Psychology, 8, 370-381. Sjoberg, L. (1967). Successive intervals scaling of paired comparisons. Psychometrika, 32, 297-308. Sjoberg, L. (1975). Uncertainty of comparative judgments and multidimensional structure. Multivariate Behavioral Research, I I , 207-218. Sjoberg, L. (1980). Similarity and correlation. In E. D. Lantermann & H. Feger (Eds.), Similarity and choice. Bern: Hans Huber. Takane, Y. (1980). Maximum likelihood estimation in the generalized case of Thurstone’s model of comparative judgment. Japanese Psychological Research, 22, 188-196.

Probabilistic Choice Models for Marketing

337

Thurstone, L. (1927). A law of comparative judgment. Psychological Review, 34, 273-286.

This Page Intentionally Left Blank

339

AUTHOR INDEX

A Akaike, H., 133, 135 Anderson, N. H., 178,203 Arabie, P., 58, 73 Arbuckle, L., 145, 157 Aronson, E., 30 Arrow, J. K., 241, 257 Ashby, F. G., 180, 203

B Bagman, R. E., 141, 157 Barlow, R. E., 9, 28 Barnes, S. H., 111, 119 Bartholomew, D. J., 9, 28 Beaver, R. J., 101, 119 Bechtel, G. G., 78, 80, 95-96, 124, 135, 147, 151, 157, 179, 189, 203, 261, 288, 293, 296, 313 Becker, G. M., 125, 135 Bennett, J. F., 59, 72, 263, 288 Bentler, P. M., 63, 72, 110, 119, 148, 150, 157, 303, 313 Beyle, H. C., 4, 28 Birnbaum, M. H., 179, 203 Bishop, Y . M. M., 267, 288 Black, D., 240, 257 Block, H. D., 100, 119, 212, 219 Bloxom, B., 63, 73, 141, 145, 151, 157, 303, 313 Bock, R. D., 141, 145, 148, 151,

157-158, 292, 313 Biickenholt, I., 180, 204, 317-319, 321, 323-324, 331, 333-336 Bonett, D. G., 110, 119 Borg, I., 11, 27-28, 30, 73 Bossuyt, P. M., 77, 80, 89, 95-97, 207-208,218-219 Box, B. E. P., 292, 313 Bradley, R. A., 101-105, 120, 124, 135, 139, 145, 158 Brernner, J. M., 9, 28 Browne, M. W., 59, 74, 148, 153, 158 Brunk, H. D., 9, 28 Bulgren, W. G., 183-184, 204 Bush, R. R., 120, 205, 219-220

C Capozza, D., 125, 137 Carroll, J. D., 4, 10-11, 26, 28, 30, 58-60, 62, 73-74, 123-125, 131, 136, 141, 144-145, 158, 161-163, 166-167, 171, 174176, 180, 189, 204, 263, 288, 293-297, 300, 302, 306, 313314, 317, 319, 321, 336 Caussinus, H., 289 Chandler, J. P., 188, 204 Chang, J. J., 163, 175 Chang, W., 147, 157, 293, 313

340

Chave, E. J., 3, 5, 31 Chernikova, N. V., 93, 97 Christoffersson, A., 149, 152, 158-159 Clark, L. A., 73, 166, 175 Cliff, N., 260, 289 Cohen, A., 237-238, 242, 246-251, 255, 257 Coombs, C. H., 1-2, 6, 8, 14, 28, 33, 38, 42, 45, 47, 51-52, 5455, 58-59, 73, 77-78, 80, 84, 95, 97, 116, 119-120, 123-125, 136, 162, 175, 180-181, 204, 214, 219, 222, 227, 234-235, 237-238, 240-241, 245, 251, 260, 262, 288, 294, 296, 313 Cooper, L. G., 294, 313, 331, 335 Coxon, A., 116 Cronbach, L. J., 27-28 Croon, M., 78, 80, 95, 97, 99, 180, 204, 208, 219 Cunningham, J. P., 164, 175

David, H. A., 292, 313 Davidon, W. C., 69, 73 Davidson, J. A., 59, 73 Davidson, R. R., 318, 335 Davis, R. L., 257 Davison, M. L., 60, 73, 243, 257, 262-263, 267, 271, 288 Debreu, G., 143, 158 Degreef, E., 176, 258 DeGroot, M. H., 125, 135 De Leeuw, J., 8, 19, 27-29, 59, 63, 73, 75, 141, 144, 158, 303, 314, 326, 336

Author index

Dempster, A. P., 108, 120 DeSarbo, W. S., 57-58, 60, 62-63, 68-69, 73-74, 78, 97, 123-124, 136, 145, 158, 161-162, 166167, 175-176, 180, 204, 291, 294, 296-298, 300-302, 306, 313-314, 317, 336 De Soete, G., 123-125, 128, 131132, 134-136, 141, 144-145, 149, 158, 161-162, 165-167, 171, 175-176, 180, 204, 291, 294, 296-298, 300-301, 306, 314, 317, 319, 321, 323-324, 335-336 Diday, E., 136, 176, 314 Dijkstra, L., 83, 97, 261, 273, 288 Ducamp, A., 222, 235 Dykstra, R. L., 89, 97

E Edwards, A. L., 4, 28 Efron, B., 27-28 Eilers, P., 257 Eisler, H., 178, 204 Ekman, G., 326, 336 Eliashberg, J., 161, 175, 297, 300-301, 306, 314 Ellis, M., 180, 205 Enslein, K., 30 Escouffier, Y., 176, 314 Ettinger, P., 289

F Falmagne, J. C., 222, 235 Farquhar, P. H., 318, 335 Feger, H., 33, 73, 137, 158, 160, 175, 235, 313, 336

Author Index

Feigin, P. D., 237-238, 242, 246251, 255, 257 Ferguson, L. W., 4, 28 Festinger, L., 235 Fienberg, S. E., 101, 105, 120, 267, 288 Fishbein, M., 98 Fletcher, R., 69, 74, 305, 314 Forlano, G., 4, 30 Furnas, G. W., 162, 166-167, 176

G Galanter, E., 120, 205, 219-220 Gaul, W., 180, 204, 317-319, 321, 323-324, 331, 333-336 Ghurye, S., 119-120, 219 Gill, P. E., 7 1, 74 Gleser, G. C., 27-28 Gold, E. M., 263, 288 Gold, M., 21, 23, 28 Goldberg, A. I., 237-238, 249-250, 253, 257 Golledge, R., 205 Goodman, L. A., 241, 257 Green, P. E., 58, 74 Greenacre, M. J., 59, 74 Greenberg, M., 78, 95, 97, 124, 136, 296, 313 Greenberg, M. G., 78, 92, 94, 97 Greeno, J. G., 125, 134, 136, 143, 151, 159, 323 Griggs, R. A., 78, 95, 98, 124, 137, 161, 176, 180, 183-184, 205, 261, 290, 296, 315 Gulliksen, H., 75, 315, 319, 336 Guttman. L., 5-6, 27, 29, 148, 158

34 1

H Halff, H. M., 125, 136, 142-143, 158, 170, 176, 320, 336 Hays, W. L., 59, 72, 263, 288 Hefner, R. A., 180-181, 183, 204 Heiser, W. J., 3, 8-9, 12, 14, 1719, 21, 26-29, 59-60, 63, 7374, 141, 144, 158, 263, 288, 303, 314, 326, 336 Henry, N. W., 101, 120 Himmelblau, D. M., 71, 74 Hinckley, E. D., 4, 29 Hoeffding, W., 119-120 Hoffding, W., 219 Hoffman, D. L., 78, 97 Hohle, R. H., 145, 158 Holbrook, M. B., 313 Holland, P. W., 267, 288 Horst, P., 29 Hovland, C. I., 4, 29-30 Hunter, J. S., 292, 313 Hunter, W. G., 292, 313

I Indow, T., 146, 159 Inglehart, R., 111, 113, 120

J Jansen, P. G. W., 80, 95, 97, 261, 288 Jech, T., 100 Jedidi, K., 161, 175, 291 Johnson, J., 63, 75 Johnson, S. C., 163, 176 Jones, L. V., 141, 145, 148, 151, 158, 292, 313

342

Author Index

Joreskog, K. G., 110, 120, 141, 146, 148, 150, 152, 154, 159-160

Kaas, K. P., 327, 329-330, 336 Kaase, M., 235, 289 Kao, R. C., 260, 288 Katz, D., 235 Kemeny, J. G., 241, 257 Kendall, M. G., 184, 190, 204, 248, 257 Klingemann, H. D., 235, 280-281, 289 Krantz, D. H., 125, 136, 143, 159, 209, 212, 219, 222 Krishnaiah, P. R., 30, 73-74, 135, 3 14 Kruskal, J. B., 9-11, 29-30, 59, 62, 73-74, 198, 204, 263, 289, 302-303, 313-314 Kruskal, W. H., 314 Kurtz, R., 180,205

L Laird, N. M., 108, 120 Lantermann, E. D., 73, 137, 158, 160, 175,235,313, 336 Larntz, K., 101, 105, 120 Lazarsfeld, P. F., 101, 120 Lebart, L., 176, 314 Lehmann, D., 62, 74, 302, 314 Lehmann, E. L., 243, 257 Leik, R. K., 262, 289 Levine, M. V., 8, 30 Liken, R., 5 , 30 Lindzey, G., 30

Lingoes, J. C., 28, 30, 59, 63, 73, 75, 235 Looman. C. W. N., 27, 31 Luce, R. D., 100, 104, 120, 124, 136, 145-146, 159, 162, 176, 205, 209, 212-213, 217, 219220,222

M MacKay, D. B., 177, 180, 182183, 190, 197, 204-205 Madow, W., 119-120, 219 Mallows, C. L., 238, 246, 250, 257 Mann, H., 119-120, 219 Marschak, J., 100, 119, 125, 135, 212, 219 Mattenklott, A., 106, 120 Matthews, M., 262, 289 McArdle, J. J., 154, 159 Mckullagh, P., 305, 314 McDonald, R. P., 148, 154, 159 McFadden, D., 162, 176 McGuire, W. J., 7, 30 Messick, S., 75, 315 Meulman, J., 17, 26-29 Mieschke, K, J., 106, 120 Mokken, R. J., 268-269, 272, 289 Molenaar, I. W., 83, 97, 267, 288-289 Muller, M. W., 17, 30 Murray, W.,71, 74 Muthen, B., 149, 152, 159

N Nagel, E., 219 Nakanishi, M., 294, 313, 331, 335

343

Author Index

Nanda, H., 27-28 Nebergall, R. E., 31 Nelder, J. A., 305, 314 Nerlove, S., 28, 73, 75, 174, 204, 288 Noma, E., 63, 75 Norpoth, H., 221, 228, 235, 274, 277, 280-281, 289 Nugent, J. H., 145, 157

0 Oliver, R. L., 161, 175, 294, 297298, 300, 306, 314 Olkin, I., 119-120, 219 Orth, B., 221-222, 225, 232, 234-235 O’Shaughnessy, J., 62, 74, 302, 3 14

P Pages, J., 176, 314 Pappi, F. U., 221, 228, 232, 234235, 274, 280, 289 Patnaik, P. B., 184, 205 Pendergrass, R. N., 101-102, 104105, 120 Pintner, R., 4, 30 Plackett, R. L., 101, 120 Powell, M. J. D., 69, 74 Pruzansky, S., 62, 73, 166, 175, 302, 313

R Rajaratnam, N., 27-28 Ralston, A., 30 Ramaswamy, V., 167, 175 Ramsay, J. O., 78, 95, 97, 133,

136 Rao, V. R., 57-58, 60, 63, 68-69, 74, 302, 314 Redner, R. A., 119-120 Reeves, C. M., 305, 314 Restle, F., 143, 145, 159 Robertson, T., 89, 97 Romney, A. K., 28, 73, 75, 174, 204, 288 Roskam, E. E., 10, 30, 45, 59, 75, 77, 80, 93, 95-97, 116, 207208, 218-219, 228, 235, 257258, 263, 289 Ross, J., 260, 289 Rubin, D. M., 108, 120 Rumelhart, D. L., 125, 134, 136, 143, 151, 159, 323 Rusk, J. G . , 124, 161, 176, 296, 315 Russo, J. E., 125, 137, 143, 160

S Saaty, T. L., 179, 205 Sattath, S., 137, 151, 160, 262, 289 Schefft!, H., 179, 205 Schektman, Y., 176, 314 Schonemann, P. H., 59, 75, 78, 95, 97, 124, 136, 161, 176, 189, 205, 263, 289, 296, 315 Seery, J. B., 10, 30, 59, 74, 198, 204, 263, 289 Sehr, J., 106, 120 Shepard, R. N., 28, 73, 75, 174, 204, 288 Sherif, C. W., 4-5, 19-20, 31, 96, 98

344

Author Index

Sherif, M., 4, 29-31, 96, 98 Shocker, A. D., 60, 75 Sixtl, F., 78, 80, 95-96, 98, 124, 136, 261,289, 296, 315

Sjoberg, L., 125, 137, 143, 151, 159-160, 179, 205, 319, 336 Slater, P., 292, 315 Smith, J. E. K., 262, 288 Snell, J. L., 241 Sorbom, D., 148, 150, 159-160 Spence, I., 59-60, 75 Srinivasan, V., 60, 75 Stephenson, W., 27, 31 Stokman, F. N., 83, 97 Stouffer, S. A., 29 Strauss, D., 145, 160 Stuart, A., 184, 190, 204 Suck, R., 257 Suppes, P., 100, 120, 180, 205, 209, 212-215, 217, 222

326,

154,

302, 315, 317-318, 337

Timmerman, H., 205 Tomassone, R., 289, 314 Torgerson, W. S., 5, 15, 31, 59, 75, 292, 315

Townsend, J. T., 180, 203 Tucker, L. R., 58, 75, 147, 157, 292-293, 313, 315 Tversky, A,, 125, 137, 143, 145, 151, 160, 209, 219, 222, 262, 289

V Van Blokland-Vogelesang, R. A. W., 237, 243, 245, 248, 257-258

Van Buggenhaut, J., 176 Van der Eijk, C., 83, 97, 288 Van der Ven, A. H. G. S., 245, 258 183, 219,

T Takane, Y., 59, 75, 139, 141, 144-145, 151, 160, 317, 319320, 323, 336 Tanur, J. M., 314 Tarski, A., 219 Ter Braak, C. J. F., 27, 31 Terry, M. E., 120, 124, 135, 145, 158 Thrall, R. M., 257 Thurstone, L. L., 1-3, 5, 29, 31, 140-141, 144, 160, 167-168, 176, 180-181, 205, 292, 300-

Van Schuur, W. H., 83, 97, 243, 258-259, 267, 288-289 Verbeek, A., 257 Verhelst, N., 83, 97

Walker, H. F., 119-120 Wang, M.-M., 78, 95, 97, 124, 136, 161, 176, 296, 315

Wedderburn, R. W. M., 305, 314 Weeks, D. G., 63, 72, 303, 313 Wegener, E., 204 Wilf, H. S., 30 Wold, H., 160 Wolff, R. P., 180, 205 Wright, F. T., 89, 97 Wright, M. H., 71, 74

Author Index

Y Yanai, H., 152, 160 Yellott, J. I., 102-103, 120-121 Young, F. W., 10, 17, 30-31, 59, 74-75, 198, 204, 263, 289

345

Z Zermelo, E., 100, 121 Zinnes, J. L., 78, 95, 97-98, 124, 134-137, 161, 176-177, 180, 182-184, 190, 197, 204-205, 212, 219, 222, 261, 290, 296, 313, 315

This Page Intentionally Left Blank

347

SUBJECT INDEX

A abundant data, 100 ACOVS, 139, 141 additive tree, 164 adjacency matrix, 272 admissible pseudo-distances, 16 AIC statistic, 134 algebraical relational system, 209 algorithm, alternating leastsquares, 57, 63 algorithm, branch-and-bound, 90 algorithm, EM, 108 algorithm, generalized Fisher scoring, 132 alternating least-squares algorithm, 57, 63 AMOMS, 150 analgesic pain reliever, over-thecounter, 306 analysis, covariance structure, 110 analysis, external, 4, 59, 294, 302, 317, 331 analysis, factor, 131, 150, 260 analysis, hierarchical cluster, 265 analysis, internal, 10, 59, 302, 317, 33 1 analysis, latent class, 101 analysis, latent structure, 99, 101 analysis, loglinear, 139 analysis, multivariate conjoint, 62,

302 analysis of covariance structures, 139, 141 analysis of moment structures, 150 analysis, parallelogram, 262 analysis, path, 150 approach, nonmetric, 9 approach to ties, primary, 25 approach, unconditional, 12 approximation, Patnaik, 184 assimilation effect, 4 asymptotically distribution free method, 153 attitude scaling, 3 attitude scaling, Thurstonian, 3 axiom, Luce’s choice, 104 axiomatic foundations of unfolding, 221 axiomatic measurement theory, 207

B backtracking, 244 Bernoulli trial, 81, 209 betweenness relation, 223 binary choice, 209 bivariate normal distribution, 197 bootstrap, 27 boundary, 34 boundary, orientation of a, 47

348

Subject Index

boundary triangle, 47 branch-and-bound algorithm, 90 branch-and-bound method, 244

C CANDELINC, 62, 302 category points, 6 central chi-square, 184 central F distribution, 184 characteristic monotonicity, 83, 272 chi-square, central, 184 chi-square distribution, noncentral, 183 choice axiom, Luce’s, 104 choice, binary, 209 choice model, individual, 100 choice model, stochastic multidimensional spatial, 162 choice theory, probabilistic, 207 choices, replicated, 178 class analysis, latent, 101 class, latent, 247 class model, latent, 99 cluster analysis, hierarchical, 265 coefficient of scalability, 268 comparative judgment, factorial model of, 320 comparison judgment, pair, 139 comparisons data, paired, 77-78, 123, 161, 291 comparisons, graded paired, 294 comparisons, method of paired, 317 comparisons, paired, 209 configuration, recovered, 200 conjoint analysis, multivariate, 62,

302 conjugate gradient method, 71, 305 consensus ranking, 237 consumer preference, latent, 298 contingency table, 34 contour, iso-utility, 293, 295 contrast effect, 4 COSAN, 148 covariance structure analysis, 110 covariance structures, analysis of, 139, 141 cross validation, 3, 27 cumulative scaling, multiple unidimensional, 272 curve, single-peaked, 6 curve, unimodal, 6

D data, abundant, 100 data, dichotomized, 259 data, dichotomous, 259 data, models of, 214 data, paired comparisons, 77-78, 123, 161, 291 data, pick anyln, 259 data, pick kln, 259 data, possible realization of empirical, 213 data, possible realization of the, 209 decision, majority, 237 decomposition rule, 52 degeneracy, 54, 260 degenerate solution, 10, 63 degeneration, 3, 6, 11 descent method, steepest, 71

Subject Index

design, incomplete, 292 deviance measure, 305 dichotomized data, 259 dichotomous data, 259 direction, stochastic preference, 317 discriminal dispersion, 3 19 discriminal process, 144, 318 dispersion, discriminal, 319 distance distribution, 15 distance model, 99 distance model, weighted, 263 distribution, bivariate normal, 197 distribution, central F , 184 distribution, distance, 15 dismbution, double-exponential, 103 dismbution, doubly noncentral F , 124, 183, 296 distribution free method, asymptotically, 153 distribution, multivariate normal, 125, 177, 180 distribution, noncentral chi-square, 183 distribution, normal, 184 distribution, smooth, 18 dominance matrix, 272 dominant J-scale, 227 double-exponential distribution, 103 doubly noncentral F distribution, 124, 183, 296

E effect, assimilation, 4 effect, contrast, 4

349

effect, similarity, 143 election, presidential, 19 EM algorithm, 108 empirical data, possible realization of, 213 EQS, 148 equations, linear structural, 150 error-in-variable regression, 154 estimate, initial, 180 estimate, maximum likelihood, 86, 101, 106, 132, 177, 180, 247, 305 exponential, negative, 8 external analysis, 4, 59, 294, 302, 317, 331 F distribution, central, 184 F distribution, doubly noncentral, 124, 183, 296

F facet theory, 27 factor analysis, 131, 150, 260 factorial model, 144 factorial model of comparative judgment, 320 Fechnerian theory, 212 Fisher scoring algorithm, generalized, 132 Fisherian information matrix, 133 foundations of unfolding, axiomatic, 22 1 function, penalty, 166 function, real-valued utility, 208 function, single-peaked, 8 function, single-peaked preference, 237

350

Subject Index

G general unfolding model, 59 generalizability theory, 27 generalized Fisher scoring algorithm, 132 GENFOLD, 58 GENFOLD2,57, 302 graded paired comparisons, 294 gradient method, conjugate, 71, 305 GSTLJN, 166

H Hefner model, 180 hierarchical cluster analysis, 265 hierarchical tree, 163 higher order metric scale, 38 homomorphic mapping, 209

I ideal point model, probabilistic, 321 ideal point model, stochastic, 317 ideal point model, wandering, 123, 125, 149, 161-162, 296, 327 incomplete design, 292 index, uncomparability, 142 individual choice model, 100 inequality, ultrametric, 163 information matrix, Fisherian, 133 initial estimate, 180 internal analysis, 10, 59, 302, 317, 33 1 invariant, order, 3, 10, 17 inverse, Moore-Penrose, 133, 173 I-scale, 223, 243, 263

isochrest, 8 isotonic region, 33 isotonic regression, 9 iso-utility contour, 293, 295

J jackknife, 27 joint probabilistic midpoint unfolding theory, 84 J-scale, 237, 261 J-scale, dominant, 227 J-scale, qualitative, 45, 222, 237, 239, 263 J-scale, quantitative, 45, 222, 237, 239, 263 judgment, factorial model of comparative, 320 judgment, numerical, 178 judgment, pair comparison, 139 judgment, preference ratio, 177-178 judgment, same-different, 180 judgment school, social, 4 judgment, similarity, 180

K KYST, 10, 198

L ladder for technical occupations, prestige, 237 latent class, 247 latent class analysis, 101 latent class model, 99 latent consumer preference, 298 latent structure analysis, 99, 101 least-squares algorithm,

Subject Index

alternating, 57, 63 length tree, path, 164 likelihood estimate, maximum, 86, 101, 106, 132, 177, 180, 247, 305 likelihood seriation, maximum, 78 Liken-type rating scale, 260 linear programming, 238, 245 linear structural equations, 150 LISCOMP, 149 LISREL, 148 log linear model, 105 logistic regression, 27 loglinear analysis, 139 Luce’s choice axiom, 104

M majority decision, 237 mapping, homomorphic, 209 mapping, reverse, 58, 68 matrix, adjacency, 272 matrix, dominance, 272 matrix, Fisherian information, 133 maximum likelihood estimate, 86, 101, 106, 132, 177, 180, 247, 305 maximum likelihood seriation, 78 MDS model, probabilistic unfolding, 291 MDS model, probabilistic vector, 29 1 mean normalized stress, 9 measure, deviance, 305 measure of recovery, 198 measurement theory, axiomatic, 207 method, asymptotically distribution

35 1

free, 153 method, branch-and-bound, 244 method, conjugate gradient, 71, 305 method of paired comparisons, 317 method, quasi-Newton, 7 1 method, steepest descent, 71 method, unfolding, 222 metric scale, higher order, 38 midpoint, 34 midpoint monotonicity, 82 midpoint unfolding theory, joint probabilistic, 84 MINIRSA, 10, 45, 115, 228 moderate stochastic transitivity, 125, 143, 168, 296 moderate utility model, 123, 125, 170 moment structures, analysis of, 150 monotonic regression, 9 monotonicity, characteristic, 83, 272 monotonicity, midpoint, 82 monotonicity restriction, 9 Moore-Penrose inverse, 133, 173 MUDFOLD, 260 multidimensional spatial choice model, stochastic, 162 multidimensional unfolding model, 57 multidimensional unfolding model, probabilistic, 123-124 multiple scaling, 265 multiple unidimensional cumulntive scaling, 272 multiple unidimensional unfolding,

352

Subject Index

259 multiple-judgment sampling, 139-140 multivariate conjoint analysis, 62, 302 multivariate normal distribution, 125, 177, 180

N negative exponential, 8 Newton-Raphson procedure, 101 noncentral chi-square distribution, 183 noncentral F distribution, doubly, 124, 183, 296 noncentrality parameter, 183 nonmetric approach, 9 nonscalability, 264 normal distribution, 184 normal distribution, bivariate, 197 normal distribution, multivariate, 125, 177, 180 normalization, 9 normalized stress, mean, 9 numerical judgment, 178 numerical relational system, 209

0 occupations, prestige ladder for technical, 237 of covariance structures, analysis, 139, 141 order invariant, 3, 10, 17 order metric scale, higher, 38 order, weak, 225 orientation of a boundary, 47 orientations, value, 111

over-the-counter analgesic pain reliever, 306

P pain reliever, 65 pain reliever, over-the-counter analgesic, 306 pair comparison judgment, 139 paired comparisons, 209 paired comparisons data, 77-78, 123, 161, 291 paired comparisons, graded, 294 paired comparisons, method of, 317 parallelogram analysis, 262 parameter, noncentrality, 183 party preferences, political, 221 path analysis, 150 path length tree, 164 Patnaik approximation, 184 penalty function, 166 Pendergrass-Bradley model, 102 pick anyln data, 259 pick kln data, 259 point model, probabilistic ideal, 321 point model, stochastic ideal, 317 point model, wandering ideal, 123, 125, 149, 161-162, 296, 327 point, statement, 6 points, category, 6 political party preferences, 221 possible realization of a theory, 208 possible realization of empirical data, 213 possible realization of the data,

Subject Index

209 preference direction, stochastic, 317 preference function, single-peaked, 237 preference, latent consumer, 298 preference ratio judgment, 177-178 preference strength, 7 preferences, political party, 22 1 PREFMAP, 59 PREFMAP2, 59 presidential election, 19 prestige ladder for technical occupations, 237 prestige, social, 237 primary approach to ties, 25 probabilistic choice theory, 207 probabilistic ideal point model, 32 1 probabilistic midpoint unfolding theory, joint, 84 probabilistic multidimensional unfolding model, 123-124 probabilistic unfolding, 80, 207 probabilistic unfolding MDS model, 291 probabilistic unidimensional unfolding, 77 probabilistic vector MDS model, 29 1 probabilistic vector model, 321 procedure, Newton-Raphson, 101 procedure, unfolding, 33 process, discriminal, 144, 318 programming, linear, 238, 245 property-fitting, 303 pseudo-distances, 9

353

pseudo-distances, admissible, 16

Q Q-methodology, 27 qualitative J-scale, 45, 222, 237, 239, 263 qualitative solution, 34 quantitative J-scale, 45, 222, 237, 239, 263 quantitative solution, 34 quasi-independent model, 267 quasi-Newton method, 7 1

random sample, 180 random utility model, 102, 162 ranking, 99 ranking, consensus, 237 ranking model, strict utility, 102 rating scale, Likert-type, 260 ratio judgment, preference, 177-178 raw stress, 9 realization of a theory, possible, 208 realization of empirical data, possible, 213 realization of the data, possible, 209 real-valued utility function, 208 recognition response, 180 recovered configuration, 200 recovery, measure of, 198 region, isotonic, 33 regression, error-in-variable, 154 regression, isotonic, 9 regression, logistic, 7-7

Subject Index

354

regression, monotonic, 9 relation, betweenness, 223 relational system, algebraical, 209 relational system, numerical, 209 reliever, over-the-counter analgesic pain, 306 reliever, pain, 65 replicated choices, 178 reproducibility, 5 resampling, 3, 27 response model, unimodal, 3 response, recognition, 180 response strength, 7 restriction, monotonicity, 9 restriction, smoothness, 3 reverse mapping, 58, 68 rule, decomposition, 52

S same-different judgment, 180 sample, random, 180 sampling, multiple-judgment, 139-140 sampling scheme, 83 scalability, 2i scalability, coefficient of, 268 scalability, simple, 143 scalar products model, 292 scale, higher order metric, 38 scale, Likert-type rating, 260 scaling, attitude, 3 scaling, multiple, 265 scaling, multiple unidimensional cumulative, 272 scaling, Thurstonian, 95, 317 scaling, Thurstonian attitude, 3 scheme, sampling, 83

school, social judgment, 4 scoring algorithm, generalized Fisher, 132 seriation, 77 seriation, maximum likelihood, 78 shifted single-peakedness, 8 similarity effect, 143 similarity judgment, 180 SIMOPT, 245 simple scalability, 143 simple unfolding model, 59 single-dipped, 20 single-peaked curve, 6 single-peaked function, 8 single-peaked preference function, 237 single-peakedness, 223 single-peakedness, shifted, 8 smooth distribution, 18 smooth succession, 6, 18 smooth transformation, 18 smoothness restriction, 3 social judgment school, 4 social prestige, 237 solution, degenerate, 10, 63 solution, qualitative, 34 solution, quantitative, 34 spatial choice model, stochastic multidimensional, 162 SSTUN, 166 stability, 27 starting value, 188 statement point, 6 statistic, AIC, 134 steepest descent method, 71 stochastic ideal point model, 317 stochastic multidimensional spatial

Subject Index

choice model, 162 stochastic preference direction, 3 17 stochastic transitivity, moderate, 125, 143, 168, 296 stochastic transitivity, strong, 124, 143, 168, 296 stochastic transitivity, weak, 210 stochastic tree unfolding, 161 stochastic tree unfolding model, 162 stochastic unfolding model, 261 strength, preference, 7 strength, response, 7 stress, mean normalized, 9 stress, raw, 9 strict utility ranking model, 102 strict utility theory, 212 strong stochastic transitivity, 124, 143, 168, 296 structural equations, linear, 150 structure analysis, covariance, 110 structure analysis, latent, 99, 101 structure, unfolding, 221, 224 structures, analysis of covariance, 139, 141 structures, analysis of moment, 150 substitutability, 2 10 succession, smooth, 6, 18 system, algebraical relational, 209 system, numerical relational, 209

T table, contingency, 34 technical occupations, prestige ladder for, 237 theory, axiomatic measurement,

355

207 theory, facet, 27 theory, Fechnerian, 212 theory, generalizability, 27 theory, joint probabilistic midpoint unfolding, 84 theory, possible realization of a, 208 theory, probabilistic choice, 207 theory, strict utility, 212 theory, unfolding, 77 theory, utility, 207 theory, weak utility, 212 Thurstonian attitude scaling, 3 Thurstonian scaling, 95, 317 ties, primary approach to, 25 TORSCA, 198 transformation, smooth, 18 transitivity, moderate stochastic, 125, 143, 168, 296 transitivity, strong stochastic, 124, 143, 168, 296 transitivity, weak stochastic, 210 tree, additive, 164 tree, hierarchical, 163 tree, path length, 164 tree, ultrametric, 163 tree unfolding model, stochastic, 162 tree unfolding, stochastic, 161 trial, Bernoulli, 81, 209 triangle, boundary, 47

U ultrametric inequality, 163 ultrametric tree, 163 uncomparability index, 142

Subject Index

356

unconditional approach, 12 UNFOLD,237,243 unfolding, axiomatic foundations of, 221 unfolding MDS model, probabilistic, 291 unfolding method, 222 unfolding model, 58, 123, 182, 222, 237, 259, 294 unfolding model, general, 59 unfolding model, multidimensional, 57 unfolding model, probabilistic multidimensional, 123-124 unfolding model, simple, 59 unfolding model, stochastic, 261 unfolding model, stochastic tree, 162 unfolding model, weighted, 59 unfolding, multiple unidimensional, 259 unfolding, probabilistic, 80, 207 unfolding, probabilistic unidimensional, 77 unfolding procedure, 33 unfolding, stochastic tree, 161 unfolding structure, 22 1, 224 unfolding theory, 77 unfolding theory, joint probabilistic midpoint, 84 unfolding, unidimensional, 238 unidimensional cumulative scaling, multiple, 272 unidimensional unfolding, 238 unidimensional unfolding, multiple, 259 unidimensional unfolding,

probabilistic, 77 unimodal curve, 6 unimodal response model, 3 utility function, real-valued, 208 utility model, moderate, 123, 125, 170 utility model, random, 102, 162 utility ranking model, strict, 102 utility theory, 207 utility theory, strict, 212 utility theory, weak, 212

v validation, cross, 3, 27 value orientations, 111 value, starting, 188 variance-component models, 150 vector MDS model, probabilistic, 29 1 vector model, 58, 99, 263, 292 vector model, probabilistic, 321 vector model, wandering, 125, 144, 161-162, 294, 324

W wandering ideal point model, 123, 125, 149, 161-162, 296, 327 wandering vector model, 125, 144, 161-162, 294, 324 weak order, 225 weak stochastic transitivity, 210 weak utility theory, 212 weighted distance model, 263 weighted unfolding model, 59