Biostatistics (2002), 3, 1, pp. 57–75 Printed in Great Britain
Multivariate survival models induced by genetic frailties, with application to linkage analysis HONGZHE LI∗ Rowe Program in Human Genetics, School of Medicine, University of California, Davis, CA 95616, USA
[email protected] XIAOYUN ZHONG Department of Statistics, University of California, Davis, CA 95616, USA S UMMARY We derive a multivariate survival model for age of onset data of a sibship from an additive genetic gamma frailty model constructed basing on the inheritance vectors, and investigate the properties of this model. Based on this model, we propose a retrospective likelihood approach for genetic linkage analysis using sibship data. This test is an allele-sharing-based test, and does not require specification of genetic models or the penetrance functions. This new approach can incorporate both affected and unaffected sibs, environmental covariates and age of onset or age at censoring information and, therefore, provides a practical solution for mapping genes for complex diseases with variable age of onset. Small simulation study indicates that the proposed method performs better than the commonly used allele-sharing-based methods for linkage analysis, especially when the population disease rate is high. We applied this method to a type 1 diabetes sib pair data set and a small breast cancer data set. Both simulated and real data sets also indicate that the method is relatively robust to the misspecification to the baseline hazard function. Keywords: Age of onset; Conditional hazards ratio; Frailty model; Inheritance vector; Linkage analysis.
1. I NTRODUCTION Many complex human diseases have variable age of onset. Examples include breast cancer (Claus et al., 1990), prostate cancer (Carter et al., 1992), and some affective disorders such as bipolar mood disorder (Stine et al., 1995). Studies of these complex diseases show a significant age of onset correlation between family members. Furthermore, early age of onset of breast cancer, type 1 diabetes, and Alzheimer’s dementia has been associated with an increased risk in relatives (Claus et al., 1990; Caillat-Zucman et al., 1992; Meyer et al., 1998). So age of onset data for individuals within a family are correlated and censored for those unaffected individuals and, therefore, are multivariate survival data. Statistical methods for analysis of multivariate survival data have attracted much attention in recent years. One important class of models is the frailty models, which are extensions of the Cox model with random effects. Various frailty models can be defined by assuming different distribution for the random effects. These include the shared univariate gamma frailty model (Vaupel et al., 1979; Clayton and Cuzick, 1985; Oakes, 1989; Nielson et al., 1992) and the shared univariate positive stable frailty model (Hougaard, ∗ To whom correspondence should be addressed
c Oxford University Press (2002)
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1995). The univariate shared frailty models were further extended to include multivariate correlated frailties (Yashin and Iachine, 1995; Petersen, 1998a,b; Petersen et al., 1998). However, these models are not based on a theory on how genes or environments act together to risk of an individual developing diseases. To model the dependence of age of onset within a family, the Cox model with latent major gene effect was introduced (Gauderman and Thomas, 1994; Li et al., 1998; Li and Thompson, 1997; Siegmund and McKnight, 1998). These models can be regarded as the Cox model with discrete frailty. Korsgaard and Andersen (1998) and Li (1999) defined the additive genetic gamma frailty models when the frailties within a family are constructed using kinship coefficients or the inheritance vectors. Li (1999) also demonstrated that the additive gamma frailty model constructed by the inheritance vectors can be used for linkage mapping of complex diseases using nuclear family data. In this paper, we further extend the model in Li (1999) to include possible effects on the risk of developing disease due to loci not linked to the putative disease locus. We define an additive genetic frailty model for sibship data which include the effects due to both the targeted and other putative disease susceptibility genes. Based on this additive genetic frailty model, we derive a multivariate survival model for age of onset data for a sibship. The model depends on the inheritance vector at the disease locus for sibships and the identity-by-descent (IBD) sharing for sib pairs. We also define a retrospective likelihood for the data and propose a likelihood ratio test for linkage using sibship data. The rest of the paper is organized as follows. We first introduce an additive genetic gamma frailty model. We then derive the joint density and survival functions for age of onset data for a sibship and a sib pair. We additionally present the conditional hazard ratio function for a sib pair derived from the previous model. We then present the application of the proposed model to genetic linkage analysis for sibships by considering a retrospective likelihood ratio test. Finally, we present the results of application of the proposed methods to both simulated and real data examples and a brief discussion. 2. A N ADDITIVE GENETIC GAMMA FRAILTY MODEL FOR SIBSHIP DATA 2.1
Construction of genetic frailties for sibship
Consider a sibship with n sibs, 1, 2, . . . , n, and denote their parents as F for the father and M for the mother. Assuming that the father and mother are unrelated, there are only four unique alleles that are distinct by descent at a given locus. Consider the setting of Kruglyak et al. (1996): we have a series of markers on a chromosomal region which may harbor the disease-causing locus/loci. Suppose d is a point in this test chromosomal region. We are interested in testing whether there is a disease susceptible (DS) gene linked to locus d. Arbitrarily label the paternal, chromosomes containing the locus of interest by (1, 2), and label the maternal chromosomes by (3, 4). The inheritance vector (Kruglyak et al., 1996; Lander and Green, 1987) of a sibship at the d locus is the vector Vd = (v1 , v2 , . . . , v2 j−1 , v2 j , . . . , v2n−1 , v2n ), where v2 j−1 = 1 or 2, v2 j = 3 or 4 for j = 1, 2, . . . , n. The inheritance vector indicates which parts of the genome at locus d are transmitted to the n children from the father and the mother. We first define the additive genetic frailties due to gene linked to locus d for the father and mother as Z d F = Ud1 + Ud2 , Z d M = Ud3 + Ud4 , where Ud1 and Ud2 are used to represent the genetic frailties due to part of the genome on the two chromosomes of the father at locus d; Ud3 and Ud4 are analogous, though for the mother. For a given
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inheritance vector vd at the d locus for a sibship, we define the frailty for the jth sib as Z d j = Udv2 j−1 + Udv2 j for j = 1, 2, . . . , n. This definition is based on the fact that it is the parts of the genome of the parents that are transmitted to the sibs, and the inheritance vectors indicate which parts are transmitted. We further assume that the Ud1 , Ud2 , Ud3 and Ud4 are independently and identically distributed across different families as (νd /2, η), where the parameter η is the inverse scale parameter, and νd is the shape parameter. Then Z d j is distributed as (νd , η), for j = 1, 2, . . . , n. Taking into account possible genetic contributions to the disease not due to the single disease locus linked to d (for example, due to loci unliked to locus d, or contributions to shared familial effects), we add another random frailty term, U p , to the genetic frailty, and define the genetic frailty for the jth sib as Z j = Zd j + U p = Udv2 j−1 + Udv2 j + U p . We assume that U p is distributed as (ν p , η) over different sibships. Then Z j follows a (νd + ν p , η) distribution. It is easy to verify that both the conditional (on Vd ) and the marginal the means of the frailties are νd + ν p E(Z 1 ) = E(Z 2 ) = · · · = E(Z n ) = , η and both the conditional and the marginal variance of the frailties are Var(Z 1 ) = Var(Z 2 ) = · · · = Var(Z n ) =
νd + ν p . η2
So the parameter νd can be interpreted as the proportion of the variance of the genetic frailty which can be explained by gene linked to the locus d. The frailties for a sibship can be written into a matrix form as Z = HU,
(1)
where Z = {Z 1 , Z 2 , . . . , Z n } , a11 a12 a13 a14 1 .. H = , . an1 an2 an3 an4 1 U = Ud1 , Ud2 , Ud3 , Ud4 , U p , where a j1 = I (v2 j−1 = 1), a j2 = I (v2 j−1 = 2), a j3 = I (v2 j = 3), a j4 = I (v2 j = 4) for j = 1, 2, . . . , n, where I (.) is the indicator function. 2.2
The additive genetic gamma frailty model
Consider a sibship with n sibs. Let T j be the random variable of age at disease onset for the jth sib. Let (t j , δ j ) be the observed data where t j is the observed age at onset if δ j = 1, and age at censoring if δ j = 0.
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We assume that the hazard function of developing disease for the jth individual at age t j is modeled by the proportional hazards model with random effect Z j ,
λ j (t j |Z j ) = λ0 (t) exp(X j β)Z j , for j = 1, 2, . . . , n,
(2)
where λ0 (t) is the unspecified baseline hazard function, X j is a vector of observed covariates for the jth sib, and β is a vector of regression parameters associated with the covariates. Z j is the unobserved genetic frailty constructed by equation (1) in the previous section. Since Z 1 , Z 2 , . . . , Z n are dependent due to gene segregation and shared frailty, T1 , T2 , . . . , Tn are therefore dependent. Finally, to make the baseline hazard λ0 (t) identifiable, we let νd + ν p = η, which sets E(Z j ) = 1, j = 1, 2, . . . , n, and prevents arbitrary scaling in model (2). Under this restriction, there are two free parameters, νd and ν p , and Z d j ∼ (νd , νd + ν p ) and Z p ∼ (ν p , νd + ν p ). We may also consider reparametrization in terms of the two frailty variances, σd = Var(Ud j ) = νd /η2 and σ p = Var(U p ) = ν p /η2 . Let σdp = σd + σ p −2 −1 −2 −1 denote the variance of Z j . We then have Z d j ∼ (σd σdp , σdp ) and Z p ∼ (σ p σdp , σdp ). 3. M ULTIVARIATE SURVIVAL MODELS FOR SIBSHP AGE OF ONSET DATA 3.1
Multivariate survival models induced by genetic frailties
Assuming conditional independence and basing on model (2), we can see that conditioning on the frailty vector Z , the joint survival function for a sibship can be written as S(t1 , t2 , . . . , tn |Z 1 , Z 2 , . . . , Z n ) = exp[−1 (t1 )Z 1 − 2 (t2 )Z 2 − · · · − n (tn )Z n ], where
j (t j ) = 0 (t j ) exp(X j β), j = 1, 2, . . . , n. Then it is easy to verify that the marginal joint survival function by integrating out Z 1 , Z 2 , . . . , Z n is given by S(t1 , . . . , tn ) = Pr(t1 , δ1 = 0, . . . , tn , δn = 0) 4 ην p ηνd /2 ν p . = × n i=1 n j (t j )a ji + ηνd /2 j=1 j (t j ) + η j=1
In practice, observations are often censored and therefore we need not only the joint survival function but also combined densities and survivor functions. For a sibship with a affected sibs (indexed by j = 1, . . . , a) and n − a unaffected, the joint survival and density function is Pr(t1 , δ1 = 1, . . . , ta , δa = 1, ta+1 , δa+1 = 0, . . . , tn , δn = 0) = (−1)a
∂ a S(t1 , . . . , tn ) . ∂t1 , . . . , ∂ta
For sibship with all sibs affected, the joint density function is Pr(t1 , δ1 = 1, . . . , tn , δn = 1) = (−1)n
∂ n S(t1 , . . . , tn ) . ∂t1 , . . . , ∂tn
The closed forms of these expressions and the detailed derivations are given in the Appendix (equation A1).
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Table 1. Joint survival and density functions for sib pair (1, 2) who share 0, 1, 2 alleles IBD at locus d. Define ∗1 = 1 (t1 ) + η,∗2 = 2 (t2 ) + η, 12 = 1 (t1 ) + 2 (t2 ) + η. Note that η = νd + ν p Pr(t1 , δ1 Pr(t1 , δ1 Pr(t1 , δ1 Pr(t1 , δ1
= 0, t2 , δ2 = 1, t2 , δ2 = 0, t2 , δ2 = 1, t2 , δ2
S(t1 , t2 ) C1 (t1 , t2 ) C2 (t1 , t2 ) C(t1 , t2 )
Joint density and survival functions = 0) = S(t1 , t2 ) = 0) = C1 (t1 , t2 )S(t1 , t2 )λ1 (t1 ) = 1) = C2 (t1 , t2 )S(t1 , t2 )λ2 (t2 ) = 1) = [C(t1 , t2 ) + C1 (t1 , t2 )C2 (t1 , t2 )]S(t1 , t2 )λ1 (t2 )λ2 (t2 )
I B Dd = 0 η2νd ην p νp ∗ν ∗ν × 1 d 2 d 12 νp νd + ∗1 12 ν νd + p ∗2 12 νp 2 12
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ην p νp ∗ν /2 ∗ν /2 ν /2 × 12 1 d 2 d 12d νp νd /2 νd /2 + + ∗1 12 12 ν νd /2 + νd /2 + p ∗2 12 12 ν νd /2 + 2p 2 12 12 η3νd /2
I B Dd = 2
ηνd ν 12d νd 12 νd 12 νd 212
νp × η νp
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+ p 12 ν + p 12
+
νp 212
Bivariate case: sib pair data
For a sibship with two sibs (sib pair), we can derive the bivariate survival function and the joint density and survival function for a sib pair who shares 0, 1 and 2 alleles IBD at the putative disease locus linked to d. These joint functions are presented in Table 1. Notice that when νd = 0, the joint survival function does not depend on the number of alleles IBD at the d locus, indicating there is no linkage between disease and the d locus.
3.3
The conditional hazard ratio function
The recurrence risk ratio λ S , an important parameter often used in linkage analysis, is defined as the conditional probability of developing disease for the first sib given that the second sib was affected and the unconditional probability of developing the disease in the population (Risch, 1990a,b). This parameter plays an important role in determining the power of the affected sib pair (ASP) method (Risch, 1990a). It is, however, important to note that for diseases with variable age at onset and age-dependent penetrance, the parameter λ S ignores both current age of the first sib and the age at onset of the second sib. An alternative measurement of familial aggregation is the conditional hazards ratio, which is defined as follows. Consider sib pair (1, 2). Let (T1 , T2 ) be the random variables of age at onset of the two sibs respectively, and define φ(t1 , t2 ) =
λ(t1 |T2 = t2 ) λ(t1 |T2 > t2 )
to be the conditional hazard ratio, where λ(t1 |T2 = t2 ) is the instantaneous probability of the first sib having the disease at age t1 given that the second sib is affected with disease at age t2 , and λ(t1 |T2 > t2 ) is defined similarly given that the second sib is disease-free at age t2 (Oakes, 1989; Clayton, 1978).
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H. L I AND X. Z HONG It is easy to verify that Pr(t1 , δ1 = 0, t2 , δ2 = 0) Pr(t1 , δ1 = 1, t2 , δ2 = 1) Pr(t1 , δ1 = 0, t2 , δ2 = 1) Pr(t1 , δ1 = 1, t2 , δ2 = 0) C1 (t1 , t2 )C2 (t1 , t2 ) + C(t1 , t2 ) = C1 (t1 , t2 )C2 (t1 , t2 ) C(t1 , t2 ) =1 + , C1 (t1 , t2 )C2 (t1 , t2 )
φ(t1 , t2 ) =
(3)
where C1 (t1 , t2 ), C2 (t1 , t2 ) and C(t1 , t2 ) are defined in Table 1 for sib pairs who share 0, and 2 alleles IBD at the locus d. It is easy to see that this ratio is, as expected, increasing with respect to the number of alleles shared IBD, and when νd = 0, φ(t1 , t2 ) is a constant (1 + 1/η), not depending on the numbers of alleles IBD at the d locus. When ν p = 0, the conditional hazard ratios are reduced to those derived in Li (1999). 4. A LIKELIHOOD APPROACH FOR GENETIC LINKAGE ANALYSIS We propose a likelihood approach for genetic linkage analysis based on the multivariate survival model derived in the previous sections. 4.1
A retrospective likelihood ratio test
The multivariate survival model proposed above can be used to construct a likelihood ratio based test for linkage analysis. As can be seen from Table 1, when νd = 0, the conditional hazard ratio function (3) between a sib pair and the joint density and survival function for a sibship does not depend on the number of alleles IBD at the locus d or the inheritance vector at the locus d; therefore, test of linkage between locus d and the disease can be formulated as testing H0 : νd = 0. Under H0 , Ud j = 0 with probability of 1; therefore, the sibship has the shared frailty U p . Consider sibship i with n i sibs. Let (ti , δi ) = (ti1 , δi1 , ti2 , δi2 , . . . , tini , δin i ) be the observed age of onset/age at censoring data. Let Mi be the marker data for the ith sibship. Following the derivation of Whittemore (1996), the data (Mi , ti , δi ) can be formulated as a retrospective likelihood of marker information Mi conditional on the phenotypes (ti , δi ). An advantage of using retrospective likelihood is that the test statistic derived from the likelihood is free of ascertainment bias, if families are ascertained through their phenotypes (Whittemore, 1996). The retrospective likelihood for the ith sibship can be written as L i (νd , ν p , 0 (t), β) = Pr(Mi |ti , δi ) v Pr(ti , δi |Vd = vd ) Pr(Vd = vd |Mi ) = d × Pr(Mi ), vd Pr(ti , δi |Vd = vd ) Pr(Vd = vd ) where Pr(ti , δi |Vd = vd ) is given in the appendix (equation A1), Pr(Vd = vd ) is the prior probability of the inheritance vector Vd , and Pr(Vd = vd |Mi ) can be calculated using multipoint methods (for example, those of Lander and Green (1987) and Kruglyak et al. (1996)). For sib pair data, the retrospective likelihood can be written as L i (νd , ν p , 0 (t), β) = Pr(Mi |ti , δi ) 2 k=0 Pr(ti1 , δi1 , ti2 , δi2 |I B Dd = k) Pr(I B Dd = k|Mi ) = Pr(Mi ), 2 k=0 Pr(ti1 , δi1 , ti2 , δi2 |I B Dd = k) Pr(I B Dd = k)
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where Pr(ti1 , δi1 , ti2 , δi2 |I B Dd = k) is given in Table 1, Pr(I B Dd = k) is the prior probability of sib pair sharing k alleles IBD. It is easy to see from the equation (A1) in the Appendix and Table 1 that the hazard functions are canceled out from both denominator and numerator; therefore, this likelihood function depends only on the cumulative hazards function. This indicates that only the cumulative hazards function need to be estimated or provided. When νd = 0, L i (0, ν p , 0 (t)) = Pr(Mi ), so the likelihood ratio statistic for the ith sibship is given by v Pr(ti , δi |Vd = vd ) Pr(Vd = vd |Mi ) L Ri (νd , ν p , 0 (t), β) = d . vd Pr(ti , δi |Vd = vd ) Pr(Vd = vd ) Assuming we have K sibships, we can then define a Lod score at locus d as Lodd = max
K
log10 L Ri (νd , ν p , 0 (t), β),
(4)
i=1
where the maximization is taken over the parameters {νd , ν p , 0 (t), β}. This paper focuses on the case that X j = 1 in model (2) because estimation of β is difficult in this frailty context and requires further investigation. Based on the general theory of likelihood ratio tests when the null value is on the boundary (Self and Liang, 1987), under H0 , 2 log(10)Lodd follows a mixture with equal probability of point mass 0 and χ12 distribution. 4.2 Estimation of the baseline hazard Many family data used for linkage analysis are collected because there are multiple cases in the family. Affected sib pair or sibships are two examples. For these data, it is often difficult to estimate the baseline hazard λ0 (t) from the data available. Fortunately, for some diseases such as breast cancer, we often know the population hazard λ p (t) or cumulative hazard p (t). By the innovation theorem (Bremaud, 1981), the population hazard is related to the λ0 (t) in our model by λ p (t) = E(Z |T > t)λ0 (t). This implies that the population hazard is the average hazard of the surviving individuals. One can verify the following relationships: λ p (t) = p (t) = and
1
λ0 (t), 1 + η−1 0 (t) η log[1 + η−1 0 (t)],
p (t) 0 (t) = η exp −1 . η
In this case, we can use the population cumulative hazard function in the Lod score statistic defined in equation (4), and maximize the likelihood ratio over parameters νd , ν p , and β if covariates are considered. This maximization is performed using the quasi-Newton method (Dennis and Mei, 1979). 5. A PPLICATION TO SIMULATED AND REAL DATA SETS This section presents results of application of the proposed methods to simulated and real data examples.
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Simulated data examples
For the following simulations, we generated age of onset data from the true models where we assumed a Weibull baseline hazard, λ0 (t) = (t/b)τ . The age of onset for each individual was generated from the true model (2), and the current age of each individual is generated from a uniform distribution U(60, 80). An individual with a simulated age of onset earlier than his/her simulated current age is defined as affected, and an individual with a simulated age of onset later than his/her simulated current age is defined as censored at the current age. We considered two sampling schemes. The first sampling scheme includes only affected sib pairs and, for each replication, the first 100 affected sib pairs (ASPs) are ascertained. The second sampling scheme includes both affected and discordant sib pairs (DSPs), and for each replication, the first 100 affected or discordant sib pairs are ascertained. For both schemes, the simulation includes 500 replications of 100 sib pairs. For each replication, fully informative and tightly linked markers are simulated. For the first simulation, we chose b = 80, τ = 5, and νd = 0.3, ν p = 0. The corresponding baseline hazard curve λ0 (t) and the baseline survival curve S0 (t) = exp[−0 (t)], together with the corresponding population hazard (λ p (t)) and population disease-free survival (S p (t) = exp[− p (t)]) curves are shown in Figures 1(a) and (b). We used the true population cumulative hazard rates in our analysis. Figure 1(c) plots the proposed Lod scores versus the mean IBD Z -values (Blackwelder and Elston, 1985) for the 500 replications of 100 ASPs and Figure 1(d) plots the proposed Lod scores versus the t-statistics for the 500 replications 100 mixture of ASPs and DSPs, where the number of DSPs range from 60 to 85, and the t-statistic is for the Haseman–Elston (1972) test for qualitative trait. In general, the larger the Z -values or the t-statistics, the larger the Lod scores. However, the proposed method is more powerful than the mean IBD test for ASPs and more powerful than the Haseman–Elston test for the mixture of ASPs and DSPs (see Table 2). For the second simulation, we chose the parameters b = 60, τ = 5 in the baseline hazard function, and νd = 0.8, ν p = 0 in the frailty distribution. Figures 2(a) and (b) show the baseline hazard and disease-free survival curves. For this simulation, the disease incidence and also the prevalence is high. Figure 2(c) plots the proposed Lod scores versus the mean IBD Z -values for 500 replications of 100 ASPs, and Figure 2(d) plots the Lod score against the values of the t-statistic for 100 mixture of ASPs and DSPs (the number of DSPs ranges from 29 to 57). In general, the larger the Z -values or the t-statistics, the larger the Lod scores. It is interesting to note that for diseases with high prevalence as simulated in this example, collecting DSPs could have better power in detecting linkage than only collecting ASPs. In both sampling schemes, the proposed method is more powerful that the mean IBD test or the Haseman–Elston test (see Table 2). To examine the robustness of the power of the proposed test to the misspecification of the baseline hazard function, we performed a simulation study. We considered the same models and sampling schemes as previously, but assessed the power of the proposed test by assuming different baseline hazard functions. We assume λ0 (t) = (t/b)τ , with b = 10, 20, . . . , 90, and τ = 2, 3, . . . , 7, for a total of 42 different baseline hazard functions. The corresponding survival curves are plotted in Figure 3. Figure 4 shows the power of the proposed test for different models, sampling schemes and different baseline hazard functions assumed in the analyses. Overall, as expected, specification of the baseline hazard functions that are close to the true function give the best power. For true models with b = 80, the power of detecting linkage is essentially the same for a wide range of assumed baseline hazard functions with b = 70, 80, 90. However, when the baseline function is greatly misspecified, the power can be lower. For the true models with b = 60 when the population disease incidence rate is very high, the power of the proposed test is still higher than the mean IBD test for ASPs or the Haseman–Elston test for mixture of ASPs and DSPs for many of the misspecified baseline hazard functions used in the analysis (see Figures 4(c) and (d)). However, the power of the proposed test is lower when the baseline hazard function is greatly misspecified.
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Multivariate survival models for linkage analysis
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Fig. 1. Simulation 1. (a) Hazard functions. (b) Survival functions. (c) Lod scores versus the mean IBD Z -values for 100 ASPs. (d) Lod scores versus the t-statistics for 100 ASPs and DSPs.
5.2
Type 1 diabetes data example
Type 1 or insulin-dependent diabetes mellitus (IDDM) is a complex disorder in which both genetic and environmental factors contribute to the development of the disease (Davies et al., 1996). Genetic linkage studies of type 1 diabetes have produced a number of conflicting results, suggesting a high degree of locus heterogeneity in this disease. For example, (Paterson and Petronis, 2000) observed linkage to the region on chromosome 4q16.3 in sib pairs both diagnosed over the age of 10 years, but there was no evidence for linkage in sib pairs diagnosed before age 10 years. This indicates the importance of incorporating age of onset data into linkage analysis for IDDM. Davies et al. (1996) performed a multipoint linkage mapping of chromosome 6q in IDDM, where a total of 429 sib pairs (including 299 UK families and 130 USA families) were typed for 39 markers in a 90cM region of chromosome 6q (average spacing 2.5 cM). Out
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H. L I AND X. Z HONG Table 2. Power comparison based on 500 simulations of 100 sib pairs each Test
Significance level α = 0.05 α = 0.001 Simulation 1, b = 80, τ = 5, η = 0.3 Lod score+ (ASPs) 0.91 0.43 Mean IBD∗ (ASPs) 0.88 0.38 Lod score (ASPs + DSPs) 0.81 0.25 HE test∗∗ (ASPs + DSPs) 0.57 0.11 Simulation 2, b = 60, τ = 5, η = 0.8 Lod score (ASPs) 0.39 Mean IBD (ASPs) 0.27 Lod score (ASPs + DSPs) 0.64 HE test (ASPs + DSPs) 0.28
0.03 0.02 0.11 0.02
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of the 299 UK sib pairs, 245 ASPs have age at disease onset data available. Figure 5 plots the ages of onset of these 245 sib pairs. Clearly, there is a wide range of IDDM onset ages. We applied our proposed method to these 245 ASPs with known age of onset to examine how sensitive our method is to the specification of the baseline hazard function for a real data set. We first computed the probability of IBD sharing between ASPs for each marker using the multipoint linkage program MAPMAKER/SIBS (Kruglyak and Lander, 1995). To examine the effects of specifying the baseline hazard function, we assume the same Weibull baseline hazard function as for the simulated data, λ0 (t) = (t/b)τ , with b ranges from 30 to 90, and τ ranges from 2 to 7. See Figure 3 for the plots of the 42 disease-free survival curves assumed in our analysis. For each marker along the 6q region, Figure 6(a) presents the minimum and maximum Lod scores over all the 42 different baseline hazard functions. As a comparison, Figure 6(b) shows the mean IBD Z -statistic using the same data without accounting for age of onset. In general, results using the proposed method agree with those obtained by the mean IBD test. It is important to note that the ranges of the Lod scores over different specifications of the baseline hazard function are small, indicating the relative robustness of our proposed method to the misspecification of the baseline hazard function. For example, for marker D6S264 in the proterminal region, 6q27, which showed linkage to IDDM, Figure 6(a) shows the Lod scores for various b values from 30 to 90 and various τ values from 2 to 7. The resulted Lod scores range from 1.81 to 2.04 ( p = 0.0019 to 0.0011), all indicating linkage between this marker and IDDM. As a comparison, the mean IBD Z -value is 2.26 ( p = 0.012), also indicating linkage. For marker D6S250 which was not linked to IDDM from the results of Davies et al. (1996), Figure 6(b) shows the Lod scores for the same sets of b and τ values. The resulted Lod scores range from 0.117 to 0.15 ( p = 0.23 to 0.20), all indicating no linkage between marker D6S250 and IDDM. As a comparison, the mean IBD Z -value is 0.56 ( p = 0.29). This example indicates that mild misspecification of the baseline hazard λ0 (t) in analysis does not induce false linkage result or reduce power of linkage test greatly. 5.3
Breast cancer data example
We applied the proposed methods to a small breast cancer data set. From Hall et al. (1990), we identified 17 affected sib pairs with fully informative IBD sharing at locus D17S74 and with known ages of onset.
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We used SEER database to obtain p (t). Our method resulted a Lod score of 0.74 ( p = 0.032), and mean IBD Z -value is 1.71 ( p = 0.044), both indicating linkage between the D17S74 marker and breast cancer. 6. D ISCUSSION We have constructed the additive genetic frailties for a sibship using the inheritance vector information, and have defined an additive genetic gamma frailty model for age of onset data for a sibship. We have also derived the joint density and survival functions for age of onset data of a sibship and showed that the proposed models can be used for a likelihood-ratio-based genetic linkage analysis. This method can incorporate age of onset and environmental covariates data into linkage analysis. One of the main
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advantage of the proposed method is that we do not need any assumptions on genetic models such as mode of inheritance or the penetrance functions. Application to simulated and real data sets indicates that the proposed methods behave well and can potentially gain power in detecting genetic linkage. We also derived the conditional hazard ratio for the sib pairs who share different numbers of alleles IBD under this model and showed that this parameter is a function of ages of onset/censoring. Like the recurrence risk ratio parameter, we believe that the conditional hazard ratio plays an important role in determining the power of dissecting complex diseases with variable age of onset. Further research along this line is warranted. We presented the models in the context of sibship data; however, the proposed model can be easily extended to linkage analysis basing on nuclear to medium-sized families. For large families, an alternative approach is to consider all the relative pairs and use the idea of pseudolikelihood in the spirit of Li and Huang (1998). The asymptotic results in Li and Huang (1998) can be used for performing a pseudolikelihood ratio test for linkage. We are currently exploring this approach for arbitrary pedigrees. The model can also be extended to include frailties due to multiple disease loci (Li, 1999). The model can be easily adapted to incorporate environmental covariates and gene–environment interactions by including product terms of covariates and genetic frailty. Detailed investigation of the effects of covariates and/or gene–environment interaction on linkage analyses in the framework of the proposed model is one direction for future research. Although the proposed additive genetic gamma frailty model is quite flexible, being able to model a number of individuals with varying degrees of association due to genetic sharing, and also being able to handle technical restriction of censoring, the model does have some limitations. First, for mathematical and computational reasons, we used the gamma distributions with common scale parameter to model the genetic frailties, although methods for other distributions such as the positive stable distribution (Hougaard, 1995) and the log-normal distribution can be developed similarly. The problem of the influence
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Fig. 4. Comparison of the power of the proposed test for different specification of the baseline hazard function used in calculating the Lod score based on 500 simulations of 100 sib pairs each, for α = 0.05 and 0.001. Assume that λ0 (t) = (t/b)τ , where b takes values of 30, 40, . . . , 90, and τ takes values of 2, 3, . . . , 7. For a given value of b, different point corresponds to different τ value. (a) 100 ASPs, b = 80, τ = 5, η = 0.3. (b) 100 mixture of ASPs and DSPs, b = 80, τ = 5, η = 0.3. (c) 100 ASPs, b = 60, τ = 5, η = 0.8. (d) 100 mixture of ASPs and DSPs, b = 60, τ = 5, η = 0.8.
of the choice of frailty distribution on power and validity of linkage testing deserves further investigation. Second, the current model assumes that there are no missing age at onset data, and no missing covariate data when considering covariate effects. One remedy to the missing data problem is to ignore individuals with missing data in the likelihood calculation. This approach obviously will lose some statistical power. Another possibility is to impute the missing data using a Monte Carlo EM approach as in Li and Thompson (1997), where the missing data variables are dealt with by sampling them in the Gibbs step. Third, we
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assume that the genetic frailty is additive of three frailty components. However, it is by no means the only way of modeling sibship age-of-onset data. An alternative is to assume that these three frailties act multiplicatively on the hazard. With this structure, the gamma frailty distribution is less attractive because it is not invariant to multiplication. An appealing alternative would be the log-normal distribution because the logarithms of the frailies, interpreted as covariates in a Cox model, would be normal. Fourth, for affected sib pairs or affected sibship data, we assume that the population disease rates or hazard rates are known for the disease being investigated. In practice, this may not be known exactly. However, our simulated data and IDDM example indicate that unless the hazard rate is greatly misspecified, our method should give reasonable power of testing genetic linkage. Finally, it is interesting to relate our methods to other published methods. Commenges (1994, 1995) proposed the use of additive random effects to model genetic effects of alleles shared IBD on disease risk, and he proposed testing the null hypothesis of no linkage based on a score test for testing the variance of the random effects is 0, which is similar in spirit to testing νd = 0 in our proposed method. The method can be applied to age of onset data by using the residuals. However, his score test is derived from a prospective likelihood and is based on only pair-wise allele sharing. Our proposed method is also related to the variance components linkage analysis methods for quantitative traits (Goldgar, 1990; Schork, 1993; Amos, 1994; Blangero and Almasy, 1997). These methods test the null hypothesis of no linkage by testing the additive variance due to a specific quantitative trait locus equal to zero. The method proposed in this paper can be thought of a variance-component linkage method for age-at-onset data. The additive genetic gamma frailty model corresponds well to the variance components model for normally distributed data in the sense that association between age at onset is induced by letting individuals within a family share some but not all the variance components. However, instead of describing the age of onset or some transformation of age of onset directly in a variance components model as in Daw et al. (1999), frailty components are combined additively which act multiplicatively on the individual hazard rates. This both extends the ordinary survival analysis with its emphasis on hazard modeling, and incorporates a variance components model to account for the dependence of age at onset among individuals within a family.
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Fig. 6. IDDM example. (a) Maximum and minimum LOD scores for IDDM data for 39 markers in a 90cM region of chromosome 6q. (b) Mean IBD Z -statistics for IDDM data for 39 markers in a 90cM region of chromosome 6q. (c) Lod score for IDDM data using different baseline hazard for linked marker D6S264. (d) Lod score for IDDM data using different baseline hazard for unlinked marker D6S250. Note that the y-scales for (c) and (d) are different.
ACKNOWLEDGMENTS This research was supported by grants from the National Institutes of Health (ES09911, HL65411, AI49213, AR44804) and by a UC Davis Health System Research Award grant. I thank the associate editor and the referee for many insightful comments and suggestions, and Dr John Todd for providing age of onset data for the IDDM sib pairs.
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H. L I AND X. Z HONG A PPENDIX Joint density and survival functions for sibship with censored observations The joint survival and density function for a sibship with a affected and n − a unaffected sibs is Pr(t1 , δ1 = 1, . . . , ta , δa = 1, ta+1 , δa+1 = 0, . . . , tn , δn = 0) ∂ a S(t1 , . . . , tn ) = (−1)a ∂t1 , . . . , ∂ta n a k ∂ A j log S(t1 , . . . , tn ) = (−1)a S(t1 , . . . , tn ) × ∂t A j k=1 A ,...,A j=1 1
= (−1)a S(t1 , . . . , tn ) × ×
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where λ j (t j ) = λ0 (t j ) exp(X j β), j = 1, . . . , a,a = 1, . . . , n − 1. A1 ,...,Ak is taken over all possible k partitions {A1 , . . . , Ak } of {1, . . . , a}. n A j is the number of elements of A j . ‘∂t A j ’ means ‘∂ti1 , . . . , ∂tin A ’, where A j = {i 1 , . . . , i n A j }. Here the k partition {A1 , . . . , Ak } of {1, . . . , a} is defined j as (1) Ai is a nonempty subset of {1, . . . , a}, i.e. Ai ⊆ {1, . . . , a} and Ai = φ, ∀i = 1, . . . , k, (2) Ai A j = φ, ∀i = j, i, j = 1, . . . , k, k (3) i=1 Ai = {1, . . . , a}. Note that joint density function for sibship with all n sibs being affected is a special case of Equation (A1) with a = n. To derive the results of (A1), one only needs to show n a k ∂ a S(t1 , . . . , tn ) ∂ A j log S(t1 , . . . , tn ) = S(t1 , . . . , tn ) × ∂t1 , . . . , ∂ta ∂t A j k=1 A ,...,A j=1 1
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Equation (A3) is straightforward using the closed-form expression of S of equation (3). For any k, we
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have ∂ k log S(t1 , . . . , tn ) ∂k = ∂t1 , . . . , ∂tk ∂t1 , . . . , ∂tk 4 n n νd × (2νd + ν p ) log η − m (tm )ami + η − ν p log m (tm ) + η log 2 i=1 m=1 m=1 k k 4 νd j=1 λ j (t j )a ji j=1 λ j (t j ) k = (−1) (k − 1)! + ν p n k . 2 i=1 n m (tm )ami + η k m=1 m=1 m (tm ) + η Equation (A2) can be easily seen from the following simple cases of a = 1, 2, 3. Using this result, we have S(t1 , . . . , tn ) = exp(log S(t1 , . . . , tn )), ∂ S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) = exp(log S(t1 , . . . , tn )) , ∂t1 ∂t1 ∂ 2 S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) ∂ 2 log S(t1 , . . . , tn ) = S(t1 , . . . , tn ) + ∂t1 ∂t2 ∂t1 ∂t2 ∂t1 ∂t2 = S(t1 , . . . , tn ) ×
n 2 k ∂ A j log S(t1 , . . . , tn ) ; ∂t A j k=1 A ,...,A j=1 k
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∂ 3 S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) = exp(log S(t1 , . . . , tn )) ∂t1 ∂t2 ∂t3 ∂t1 ∂t2 ∂t3 ∂ 2 log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) ∂ 2 log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) + ∂t1 ∂t2 ∂t3 ∂t1 ∂t3 ∂t2 ∂ 2 log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) ∂ 3 log S(t1 , . . . , tn ) + + ∂t2 ∂t3 ∂t1 ∂t1 ∂t2 ∂t3 ∂ log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) = S(t1 , . . . , tn ) ∂t1 ∂t2 ∂t3 +
∂ 2 log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) ∂ 2 log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) + ∂t1 ∂t2 ∂t3 ∂t1 ∂t3 ∂t2 ∂ 2 log S(t1 , . . . , tn ) ∂ log S(t1 , . . . , tn ) ∂ 3 log S(t1 , . . . , tn ) + + ∂t2 ∂t3 ∂t1 ∂t1 ∂t2 ∂t3 n 3 k A ∂ j log S(t1 , . . . , tn ) = S(t1 , . . . , tn ) × . ∂t A j k=1 A ,...,A j=1 +
1
k
Note that these expressions are greatly simplified for the sib pair data (see Table 1). R EFERENCES A MOS , C. I (1994). Robust variance components approach for assessing genetic linkage in pedigrees. American Journal of Human Genetics 54, 535–543. B LACKWELDER , W. C. AND E LSTON , R. C. (1985). A comparison of sib-pair linkage tests for disease susceptibility loci. Genetic Epidemiology 2, 85–98.
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