Composite likelihood and two-stage estimation in family studies

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1 Biostatistics (2004), 5, 1,pp Printed in Great Britain Composite likelihood and two-stage estimation in family studies ELISABETH WREFORD ANDERSEN The Danish Epidemiology Science Centre, Statens Serum Institut, Artillerivej 5, 2300 Copenhagen S, Denmark and Department of Biostatistics, University of Copenhagen, Blegdamsvej 3, 2200 Copenhagen N, Denmark SUMMARY In this paper register based family studies provide the motivation for linking a two-stage estimation procedure in copula models for multivariate failure time data with a composite likelihood approach. The asymptotic properties of the estimators in both parametric and semi-parametric models are derived, combining the approaches of Parner (2001) and Andersen (2003). The method is mainly studied when the families consist of groups of exchangeable members (e.g. siblings) or members at different levels (e.g. parents and children). The advantages of the proposed method are especially clear in this last case where very flexible modelling is possible. The suggested method is also studied in simulations and found to be efficient compared to maximum likelihood. Finally, the suggested method is applied to a family study of deep venous thromboembolism where it is seen that the association between ages at onset is larger for siblings than for parents or for parents and siblings. Keywords: All possible pairs; Composite likelihood; Copula; Family studies; Optimal weights; Two-stage estimation. 1. INTRODUCTION In register-based family studies failure times on related individuals are observed and familial aggregation of a disease can be regarded as correlation of failure times within families. A family may be a group of exchangeable individuals such as siblings, but this is not always the case. The family may, for example, consist of parents and siblings. Both of these types of family studies have been the practical motivation for this work. There have been two main approaches when modelling correlated data, namely random effects models or marginal models, and this has also been the case for survival data (Lee et al., 1992; Wei et al., 1989; Oakes, 1989; Nielsen et al., 1992). This paper will, however, concentrate on copula models (Genest and MacKay, 1986), which offer a very flexible framework for combining the marginal approach with a model for the dependence within units. The joint survival function is modelled through the marginal survival functions and an association parameter. In family studies the main interest lies in the association between family members, but it is also important to be able to take possible confounders into account. Using the copula approach the association is estimated while covariates are included in the marginal models. A two-stage estimation procedure suggests itself to these models by first estimating the parameters in the marginal models and regarding them as fixed when estimating the association parameter. This Biostatistics 5(1) c Oxford University Press (2004); all rights reserved.

2 16 E. WREFORD ANDERSEN estimation procedure was suggested by Hougaard (1986) and has later been studied by Shih and Louis (1995) and Genest et al. (1995). Glidden (2000) concentrated on the case where the marginal model is a Cox model and the model for the association is based on the gamma model, whereas in Andersen (2003) a general choice of marginal model combined with a copula was studied. An extension of the copula models to hierarchical data was suggested by Bandeen-Roche and Liang (1996), but as noted by the authors this approach is not so well suited to families of parents and children because some choices of copula lead to unwanted constraints on the parameters. In this paper the copula approach is extended to include families consisting of members at different levels, e.g. parents and children, by combining the two-stage estimation procedure with the composite likelihood approach (Parner, 2001; Heagerty and Lele, 1998). Although the methods are motivated by family studies, they can also be used in other cases of correlated failure time data. The paper is organized as follows. Copula models are briefly described in Section 2. In Section 3 the composite likelihood approach for groups of siblings and families of parents and children is presented. The statistical properties of the estimators reached by two-stage estimation combined with composite likelihood are derived in Section 4. In Section 5 different choices of weights are discussed. Section 6 concerns the ascertainment of data. In Section 7 the properties of the suggested estimators are studied in simulations. An application to a family study of deep venous thromboembolism is described in Section 8. Section 9 contains a discussion. 2. COPULA MODELS Let (T 1,...,T K ) be the failure times from a family of exchangeable members and S 1,...,S K the marginal survival functions, possibly depending on covariates. The joint distribution of (T 1,...,T K ) is fully specified by the joint survival function S(t 1,...,t K ). When S(t 1,...,t K ) can be written in the form S(t 1,...,t K ) = C θ {S 1 (t 1 ),...,S K (t K )}, t 1,...,t K 0, (2.1) where C θ is a K dimensional survival function C θ :[0; 1] K [0; 1] with uniform margins and θ is a parameter or possibly a vector of parameters, then (T 1,...,T K ) is said to come from the C θ copula. Different choices of C θ give different joint distributions but the marginal models are unaltered. A special group of copulas is the Archimedean copula model family, where the copulas are of the form C θ (u 1,...,u K ) = φ θ {φθ 1 (u 1 ) + +φθ 1 (u K )} with 0 u i 1, i = 1,...,K,0 φ θ,φ θ (0) = 1, φ θ < 0, φ θ > 0. In this paper the main example of an Archimedean copula is Clayton s family. The survival times are (T 1,...,T K ) with marginal survival functions {S 1 (t 1 ),...,S K (t K )}=(u 1,...,u K ) and joint survival function C(u 1,...,u K ). Clayton s family is then given as C(u 1,...,u K ) ={u 1 θ 1 + +u 1 θ K (K 1)} 1 1 θ, θ > 1. Here φ(u) = (1+u) 1 1 θ is the Laplace transform of a gamma distribution with mean 1 and variance θ 1. The failure times T i and T h are positively associated when θ>1 and independent for θ THE COMPOSITE LIKELIHOOD APPROACH In the following it will be shown that the composite likelihood approach offers a very flexible way of analysing clustered failure time data. With the work done by Andersen (2003) it is possible to study groups

3 Composite likelihood for family studies 17 of exchangeable members, e.g. siblings, where the groups can have any size. However, in this paper the family members do not have to be exchangeable. This situation occurs when the families consist not only of siblings, but also family members on another level, e.g. parents and children or half-siblings who live in the same home. When the families are groups of siblings a composite likelihood approach is also a possibility, in which instead of one contribution from each group, each possible pair of siblings gives rise to a contribution. This means that software meant for analysing pairs can be used for groups of any size. This is, however, just a special case of the situation where the members are no longer exchangeable. 3.1 Groups of siblings The joint distribution for a family is given by the joint survival function which again is modelled through the marginal survival functions and a copula tying the marginal distributions together. In the two-stage estimation the parameters in the margins, β, are estimated in the first stage taking the clustering into account when estimating the variance of the parameters (Section 4). In the second stage the association parameter θ is estimated using the score equation from the joint likelihood, but with the estimates from stage one regarded as known. If the logarithm of the likelihood is log L(β, θ) = n l j (β, θ) then the score equation for θ is U θ ( ˆβ,θ) = θ l j( ˆβ,θ). Instead of using the joint likelihood in the second stage we suggest using score equations based on the bivariate distributions of all possible pairs of siblings. When the joint survival function is given by a copula as in (2.1) then the bivariate survival functions are given by the same copula. For example, the bivariate survival function for siblings 1 and 2 is S 12 (t 1, t 2 ) = C θ {S 1 (t 1 ), S 2 (t 2 )}. The composite likelihood proposed in the second stage of the estimation is based on these bivariate distributions. Let G j be the set of possible pairs for family j and L ih (β, θ) the likelihood for pair (i, h), then the composite likelihood is log L (θ, β) = (i,h) G j w ih log L ih (θ, β) = (i,h) G j w ih l ih (θ, β) (3.2) where w ih are positive weights. Weights are introduced to compensate for the composite likelihood, thereby putting more emphasis on the large families by comparison with the full likelihood. Parner (2001) showed that the estimates found by maximizing the composite likelihood L are asymptotically Normal under suitable assumptions. In this paper the composite likelihood (3.2) is used to find pseudo score equations for θ in the second stage of the estimation. Because there is just one association parameter, the model can be simplified so each pair in the family enters with the same weight in the composite likelihood (3.2) and the weights only depend on the family size. The choice of weights will be discussed in Section 5. The simplified version of the composite likelihood (3.2) is log L (θ, β) = w j (i,h) G j l ih (θ, β). (3.3) This can be fitted by software for bivariate data by listing separate bivariate observations for the k(k 1)/2 pairs coming from a sibling group of size k.

4 18 E. WREFORD ANDERSEN 3.2 Families of parents and children In the case where the families consist of members at different levels, e.g. parents and children, one possibility is to consider the hierarchical model suggested by Bandeen-Roche and Liang (1996). Assume, for instance, that the families consist of parents (1, 2) and children (3,...,K j ) with survival times given by T ={(T 1, T 2 ), (T 3,, T K j )} where separate survival functions for parents and children are Archimedian copulas S(t 1, t 2 ) = φ θ1 [φθ 1 1 {S 1 (t 1 )}+φθ 1 1 {S 2 (t 2 )}] S(t 3,...,t K j ) = φ θ2 [φθ 1 2 {S 3 (t 3 )}+ +φθ 1 2 {S K j (t K j )}]. To find the simultaneous distribution for the family, Bandeen-Roche and Liang (1996) suggested combining these two survival functions in a joint survival function using an Archimedian copula S(t 1,...,t K j ) = φ θ3 [φ 1 θ 3 {S(t 1, t 2 )}+φ 1 θ 3 {S(t 3,...,t K j )}]. (3.4) Bandeen-Roche and Liang (1996) gave conditions to ensure that (3.4) is a valid survival function. These conditions can in some cases lead to unwanted constraints on the association parameters. For instance, if 1 θ Clayton s family is chosen for each of the three Archimedian copulas, so φ θ1 (s) = (1 + s) 1 1, φ θ2 (s) = 1 θ (1 + s) θ and φ θ3 (s) = (1 + s) 1 3, then (3.4) is a survival function with the following constraints on the three parameters 1 <θ 3 <θ 1 and 1 <θ 3 <θ 2. (3.5) The parameters in Clayton s family can be interpreted in the following manner. One can define an association measure γ as γ(t i, t h ) = λ T i T h (t i T h = t h ) λ Ti T h (t i T h > t h ), where λ is the intensity of disease. Then γ measures the change in risk for person i if person h gets the disease at time t h compared to when person h is disease-free at time t h.itisonly in Clayton s family that γ is independent of time (i.e. constant) and moreover γ = θ. Inthe model with parents and children the three parameters can be thought of as θ 1 = γ(t 1, t 2 ), which is the association between parents, θ 2 = γ(t i, t h ) i, h 3, the association between two children, and θ 3 = γ(t i, t h ) i = 1, 2, h 3, the association between parents and children. The constraints (3.5) on the parameters imply that the association between parents is stronger than that between parents and children, which may not be plausible because parents and children are genetically more similar than parents. Instead of postulating a joint model such as (3.4), the suggestion here is to model the bivariate margins and combine the likelihood contributions in a composite likelihood. If H j is the set of possible pairs combining parents and children in family j, G j the set of possible pairs of children and l ih the logarithm of the likelihood for pair (i, h) then the logarithm of the composite likelihood can be written as log L (β, θ 1,θ 2,θ 3 ) = {w j 12 l 12(β, θ 1 ) + + (l,m) H j w j (i,h) G j w j ih l ih(β, θ 2 ) lm l lm(β, θ 3 )}. (3.6)

5 Composite likelihood for family studies 19 The composite likelihood in (3.6) is simplified in the same way as (3.2) so pairs of parents and children from the same family get the same weight, and likewise pairs of children from the same family get the same weight. This means that (3.6) can be written as log L (β, θ 1,θ 2,θ 3 ) = {w 1 j l 12 (β, θ 1 ) + w 2 j +w 3 j This is a composite likelihood of exactly the same type as (3.3). (i,h) G j l ih (β, θ 2 ) (l,m) H j l lm (β, θ 3 )}. (3.7) 4. TWO-STAGE ESTIMATION As in Andersen (2003), two-stage estimation will be used to find the estimates of the parameters (β, θ). The two-stage estimation suits the way the models are constructed using copulas to tie the marginal distributions together with an association parameter. In the first stage the parameters in the marginal model, β, are estimated taking the clustering into account when the variance of the estimate is calculated. In the second stage the estimates from the first stage are regarded as fixed in an estimating equation for the association parameter θ. Calculation of the variance of the estimated association parameter then takes into account the estimation uncertainty from the first stage. 4.1 Notation and some assumptions There are n families indexed j = 1,...,n with K j members in family j, indexed i = 1,...,K j. Let T ij be the failure time for person (i, j), C ij the censoring time and Z ij covariates. Define T j = {T ij, i = 1,...,K j } and similarly C j and Z j. Suppose (T j, C j ) Z j ( j = 1,...,n) are independent identically distributed random variables and T j is independent of C j conditional on Z j.weobserve X ij = min(t ij, C ij ) and δ ij = I (T ij C ij ). The composite likelihood used to find an estimate of the association parameter θ is based on the bivariate distributions, so a model is assumed for all the interesting combinations of pairs in the family. Let M j be the set of interesting pairs. The logarithm of the composite likelihood becomes log L (θ, β) = (i,h) M j w ih log L ih (θ, β) = (i,h) M j w ih l ih (θ, β) (4.8) where w ih are positive weights, L ih is the likelihood for pair (i, h) and l ih = log L ih. The composite likelihood (4.8) covers both of the cases in Sections 3.1 and The asymptotic distribution in the parametric case First assume that the margins are modelled parametrically depending on a finite number of parameters β, which may include effects of the covariates Z. In the first stage, β is estimated by solving U β (β) = K j i=1 δ ij β log f (x ij,β)+ (1 δ ij ) β log S ij(x ij,β)= U β. j (β) = 0.

6 20 E. WREFORD ANDERSEN This is also the score equation for β in the case of independence and is unrelated to the composite likelihood (4.8). In the second stage, the estimate ˆθ of the association parameter θ is found as the solution to the pseudo score equation for θ based on the composite likelihood (4.8) with the estimate from stage one ( ˆβ) plugged in, hence U θ ( ˆβ,θ) = θ log L = (i,h) M j θ w ih log L ih ( ˆβ,θ) = Let V β = varu β.1 (β 0), V θ = varu θ.1 (β 0,θ 0 ), V β,θ =cov{u β.1 (β 0), U θ U θ. j ( ˆβ,θ) = 0. (4.9).1 (β 0,θ 0 )} and R = Iβ 1 V β Iβ 1. PROPOSITION 4.1 Assume standard regularity conditions for the marginal models, the regularity conditions stated in the appendix and that n 1 β U β, n 1 β U θ and n 1 θ U θ converge to I β, I βθ and I θ at (β 0,θ 0 ) as n. Then n 1 2 ( ˆβ β 0, ˆθ θ 0 ) converges to a Normal distribution with mean (0, 0) and variance covariance ( where I 1 θ V = Iθ 1 V θ Iθ 1 R I βθ R + Iθ 1 V βθ Iβ 1 + Iθ 1 I βθ RI βθ I θ 1 I 1 θ RI βθ I 1 θ + Iβ 1 V V βθ Iβ 1 I βθ I θ 1 V βθ I 1 θ I 1 θ ), I βθ Iβ 1 V βθ I θ 1. The proof of Proposition 4.1 is a straightforward generalization of the proof of Theorem 1 in Shih and Louis (1995). The variance is estimated by ( Vβ V βθ ) V βθ n 1 V θ n = E{(U β.1, U θ.1 ) (U β.1, U θ.1 )} (U β. j, U θ. j ) (U β. j, U θ. j ). 4.3 The asymptotic distribution in the semi-parametric case In this section we derive the asymptotic distribution of the parameters in the semi-parametric model using the two-stage method and a composite likelihood for θ. The marginal intensity λ ij (t) for person i in family j follows a Cox model λ ij (t) = λ 0 (t) exp(β Z ij ), where the baseline intensity λ 0 (t) is an unknown function of t and Z ij is a vector of covariates for person (i, j). Itisalso possible to have a stratified model or a model without covariates, leaving the marginal model purely non-parametric. We denote the counting process as N ij (t) = I (X ij t,δ ij = 1), the indicator of risk Y ij (t) = I (X ij t), the maximum follow-up time τ, and the integrated baseline intensity 0 (t) = t 0 λ 0(s)ds. The composite log-likelihood is a sum over the possible pairs, log L = n (i,h) M j l{θ,β, 0 (X ij ), 0 (X hj )}.

7 Composite likelihood for family studies 21 In the first stage of the estimation the marginal models are fitted taking the clustering into account using the method of Lee et al. (1992). The resulting estimates are ˆβ and ˆ 0 (t). The estimate for β is found by solving the marginal score equation U β (β) = K j i=1 τ 0 { } Z ij S(1) (β, u) S (0) dn ij (u) = (β, u) U β. j = 0, (4.10) where S (0) (β, u) = n 1 n K j i=1 Y ij(u) exp(β Z ij ) and S (1) (β, u) = n 1 n K j i=1 Y ij(u)z ij exp(β Z ij ), while the estimator for 0 (t) is an Aalen Breslow type estimator t dn... (u) ˆ 0 (t, ˆβ) = 0 ns (0) ( ˆβ,u). Spiekerman and Lin (1998) have shown that under suitable regularity conditions, ˆβ is asymptotically Normal around the true value and n 1 2 { ˆ 0 (t, ˆβ) 0 (t)} converges to a zero-mean Gaussian random field. At the second stage of estimation the estimates from the first stage are plugged into the pseudo score function for θ, which is based on the composite log-likelihood (4.8). This creates the pseudo score function U θ for the parameter θ: U θ (θ, ˆβ, ˆ 0 ) = (i,h) M j θ w ihl{θ, ˆβ, ˆ 0 ( ˆβ,t i ), ˆ 0 ( ˆβ,t h )}. (4.11) The estimate ˆθ is found by solving the equation obtained by setting (4.11) equal to zero. Under the regularity conditions stated in the appendix the estimator of the association parameter has the following asymptotic distribution. PROPOSITION 4.2 n 1 2 ( ˆθ θ 0 ) converges to a Normal distribution with mean zero and variance Iθ 1 V (W )Iθ 1. The precise definition of V (W ) is found in the appendix. The proof of Proposition 4.2 follows closely that of Proposition 3.2 in Andersen (2003). The variance from Proposition 4.2 is estimated by inserting the estimates in the formulae in the appendix. 5. CHOICE OF WEIGHTS In the composite likelihood approach (3.2) and (3.6) the likelihood contributions are weighted together using positive weights, different choices of weights leading to different estimators. The maximum likelihood estimate using the full likelihood is the most efficient method, but is not always available. Andersen (2003) showed that the two-stage method has good efficiency and one would expect that the two-stage method using a composite likelihood in the second stage would be less efficient. The question is now whether it is possible to choose optimal weights so the loss of efficiency becomes as small as possible. This will be investigated informally for the two situations from Sections 3.1 and One association parameter First, we concentrate on the simplest case with a parametric model and just one association parameter as in Section 3.1. Even in this case the resulting variance of ˆθ is quite complicated (Propositions 4.1 and

8 22 E. WREFORD ANDERSEN 4.2) and the problem is simplified by assuming that the parameters from the first stage are known. This leaves the variance as V = Iθ 1 V θ Iθ 1. Lindsay (1988) has found an expression for optimal weights in the one-dimensional case, and since the problem is here reduced to only one dimension the suggested weights are calculated. From (3.3) one sees that the pseudo score equation for θ is U θ ( ˆβ,θ) = θ log L = w j (i,h) G j θ l ih( ˆβ,θ) = w j (i,h) G j S (ih) j ( ˆβ,θ), (5.12) where S (ih) j is the score contribution from pair (i, h) in family j. Let U be the score function for θ based on the full likelihood with the marginal parameters β assumed known, i.e. U = θ log L( ˆβ,θ). Lindsay (1988) shows that the optimal weights are w opt =[vars] 1 E(US), (5.13) where S is the vector of score contributions. In this set-up E(US) = E(S 2 ), where S 2 denotes the vector whose elements are the squared elements of S. The variance used for the weights (5.13) is a block matrix since the families are assumed to be independent and the size of each block depends on the size of the family. It is modelled using three parameters: σ 2 for the variance in the diagonal, ω for the covariance between score contributions from pairs with one person in common and ρ for the covariance between score contributions from pairs with nobody in common. If, for instance, the families have 2, 3, 4 or 5 members then the weights (w 2,w 3,w 4,w 5 ) are found by solving equation (5.13): w 2 = 1, w 3 = σ 2 /(σ 2 + 2ω), w 4 = σ 2 /(σ 2 + 4ω + ρ) and w 5 = σ 2 /(σ 2 + 6ω + 3ρ). The parameters σ 2,ω,and ρ are estimated by ˆσ 2 = 1 n 1 ˆω = 1 n 2 ˆρ = 1 n 3 S(ih) 2 j (ih) G j {((ih),(lm)) G 2 j i=l,h =m or i =l,h=m} S (ih) j S (lm) j {((ih),(lm)) G 2 j i =l,h =m} S (ih) j S (lm) j, where n 1 is the total number of pairs, n 2 the number of elements in the set {((ih), (lm)) G 2 j i = l, h = m or i = l, h = m}, j = 1,...,n and n 3 the number of elements in the set {((ih), (lm)) G 2 j i = l, h = m}, j = 1,...,n. 5.2 More than one association parameter In Section 3.2 families of parents and children were considered. The association parameter θ = (θ 1,θ 2,θ 3 ) (for parents, children, and parent child pairs) is now three-dimensional and the variance covariance matrix

9 Composite likelihood for family studies 23 could be defined as optimal if it is smaller than other variance covariance matrices when ordering the matrices by positive definite differences. However, in Lindsay (1988) it is mentioned that an optimal choice of weights is not usually globally attainable. Since the three likelihood contributions depend on separate parameters one possible approach is to treat the choice of weights as three separate problems. The pseudo score equations for θ are found from (3.7): U θ1 ( ˆβ,θ 1 ) = log L = w 1 l 12 ( ˆβ,θ 1 ) (5.14) θ 1 θ 1 U θ2 ( ˆβ,θ 2 ) = log L = θ 2 U θ3 ( ˆβ,θ 3 ) = log L = θ 3 w 2 j (i,h) G j w 3 j (l,m) H j θ 2 l ih ( ˆβ,θ 2 ) (5.15) θ 3 l lm ( ˆβ,θ 3 ). (5.16) Here w 1 is independent of family number, since θ 1 is the association parameter for parents and estimation is always based on the pair of parents, so the natural choice is w 1 = 1. Let S 2 be the vector of score contributions S (2) (ih) j for θ 2 and S 3 the vector of score contributions S (3) (lm) j for θ 3. Then one could choose weights as in (5.13) w 2 =[vars 2 ] 1 E(US 2 ) (5.17) w 3 =[vars 3 ] 1 E(US 3 ). (5.18) Again the variances used to calculate the weights are block matrices since the families are assumed to be independent and the size of each block depends on the size of the family. They are modelled as in Section 5.1 with separate parameters in the two variances. If, for example, the families have 1, 2, or 3 children then the weights for θ 2, (w (2) 2,w(3) 2 ), are now found by solving equation (5.17). Similarly, the weights for θ 3, (w (1) 3,w(2) 3,w(3) 3 ), are found by solving equation (5.18). This leads to w (2) 2 = 1, w (3) 2 = σ2 2/(σ ω 2), w (1) 3 = σ3 2/(σ ω 3), w (2) 3 = σ3 2/(σ ω 3 + ρ 3 ) and w (3) 3 = σ 2 3 /(σ ω 3 + 2ρ 3 ). The parameters σ 2 2,ω 2,σ 2 3,ω 3 and ρ 3 are estimated in the following way: ˆσ 2 2 = 1 n 1 ˆω 2 = 1 n 2 ˆσ 2 3 = 1 n 3 ˆω 3 = 1 n 4 ˆρ 3 = 1 n 5 (ih) G j S (2) (ih) j S(2) (ih) j {((ih),(lm)) G 2 j i=l,h =m or i =l,h=m} S (2) (ih) j S(2) (lm) j (ih) H j S (3) (ih) j S(3) (ih) j {((ih),(lm)) H 2 j i=l,h =m or i =l,h=m} S (3) (ih) j S(3) (lm) j S (2) (ih) j S(2) (lm) j, {((ih),(lm)) H 2 j i =l,h =m}

10 24 E. WREFORD ANDERSEN where n 1 is the total number of pairs in G j, j = 1,...,n, n 2 the number of elements in the set {((ih), (lm)) G 2 j i = l, h = m or i = l, h = m}, j = 1,...,n, n 3 is the total number of pairs in H j, j = 1,...,n, n 4 the number of elements in the set {((ih), (lm)) H 2 j i = l, h = m or i = l, h = m}, j = 1,...,n and n 5 the number of elements in the set {((ih), (lm)) H 2 j i = l, h = m}, j = 1,...,n. The weights calculated in this way are used in the second stage of estimation when θ = (θ 1,θ 2,θ 3 ) is estimated setting (5.14) (5.16) equal to zero simultaneously. 6. A SAMPLED DATASET Until now it has been assumed that the dataset is a random sample of families. For rare diseases this design may be inefficient, and different sampling schemes may be considered. One possible strategy could be to sample all families with at least one case and a random sample of families without a case. Adapting the results from Binder (1992) this sampling scheme has been considered in Andersen (2003) who suggests weighting the estimating equations by the inverse sampling probabilities. Let π j be the sampling probability for family j, ξ j = 1iffamily j is chosen and 0 otherwise, and n the total number of families in the population. Taking the parametric case as an example, the estimating equation for β in the first stage becomes Ũ β = k i=1 ξ j π j U ij β. The estimating equation for θ, (4.9) or (4.11), is weighted with the inverse sampling probabilities in the same way, which means that in the second stage of estimation there are two sets of weights, one to take the sampling into account and one for the pairwise comparisons, leading to Ũ θ = ξ j π j U j θ. The estimates derived from the weighted analysis are still asymptotically Normal with a distribution derived in exactly the same way as in Andersen (2003). It is important to have a good approximation to the true sampling probability π j for a family j as simulations have shown that misspecified weights give biased results. For the suggested sampling scheme the probability is known to be 1, when there is at least one case in the family. The sampling probability π j is { 1 if there is at least one case in family j π j = m k N k if family j is of size k and no case in family j, where m k is the number of families of size k in the sample and N k is the number of families of size k in the population. In practice it can be a problem to determine N k,but an approximation can be found when the families are constructed from a random sample of individuals. Let X be the number of families constructed on the basis of the random sample of persons drawn from a population of N individuals. Then the number of families is X = k 1 m k and the number of individuals is N = k 1 kn k.anadhoc approximation to N k is Ñ k = (m k N)/(kX), which preserves the correct number of individuals, N = k 1 k Ñ k.

11 Composite likelihood for family studies SIMULATION STUDIES Some simulation studies were conducted to assess the statistical properties of the proposed method specifically for the sib groups and families of parents and children. A set of simulations was carried out comparing full maximum likelihood, the two-stage estimation using the full family as suggested in Andersen (2003) and the composite likelihood approach suggested here with different choices of weights for different distributions of sib group size. As expected, the maximum likelihood method is the most efficient, followed by the two-stage method where the full sib group is used in the second stage. For the two-stage method with the composite likelihood in the second stage different sets of weights were chosen with the optimal weights doing slightly better than the others. The loss of information is largest in the case with a big proportion of large families (25% of the families have five members), which does not seem surprising, but with the optimal weights the efficiency is still 94% compared to maximum likelihood. In most practical applications in Denmark the average size of sib groups will be closer to two than five. Simulation studies were also performed for datasets consisting of parents and children and again the method using a composite likelihood in the second stage of estimation had an efficiency of more than 90% compared to maximum likelihood, and showed little bias. A detailed description of the simulation results can be found at oupjournals.org. 8. A FAMILY STUDY OF DEEP VENOUS THROMBOEMBOLISM Several studies have shown that there are genetic and acquired risk factors for developing deep venous thrombosis and pulmonary embolism (in the following, thromboembolism) (Rosendaal, 1999; Seligsohn and Lubetsky, 2001). The analysis presented in this section is an exploratory analysis of the different amounts of familial aggregation in pairs of parents, parents and children, and sibling pairs. The study makes use of the nation-wide Danish registers based on the Civil Registration System and the Danish National Registry of Patients. Everybody in Denmark is registered in the Civil Registration System with a personal identification number, which is used in all registers. The Civil Registration System also includes a link to parents, making it possible to identify families. The Danish National Registry of Patients started in 1977 and includes information on all admissions to Danish hospitals. The study base was constructed by taking all patients from the Danish National Registry of Patients who were born in the period and with a given set of diagnoses (5329 patients). A random sample of persons, born in the same period and alive on 1 January 1977, was drawn from the population using the Civil Registration System. The parents and siblings of these sampled persons were identified using the link in the Civil Registration System. Together with the original sample of persons, these now constitute the study base. The events were identified in the Danish National Registry of Patients in the period 1 January 1977 until 31 December All in all there were families where the children had both parents in common. Within the study period, 3339 first-time events of thromboembolism were identified. When studying familial aggregation the families with several events contain most information. In our data, 2871 families had one person with an event, 208 had two, 16 had three and one family has four persons with a diagnosis of thromboembolism. Clayton s distributional family was chosen for each of the three types of pairs. In the case where the

12 26 E. WREFORD ANDERSEN 5329 from DNRP born random sample (CRS) born , alive 1/ Parents and sibs in CRS Fig. 1. Construction of the study base using the Danish National Registry of Patients (DNRP) and the Civil Registration System (CRS). family consists of two parents (T 1, T 2 ) and k 2 children (T 3,...,T k ) this means that S(t 1, t 2 ) ={S(t 1 ) (1 θ 1) + S(t 2 ) (1 θ 1) 1} 1 1 θ 1 S(t i, t j ) ={S(t i ) (1 θ2) + S(t j ) (1 θ2) 1 θ 1} 1 2, i, j = 3,...,k (8.19) S(t i, t j ) ={S(t i ) (1 θ3) + S(t j ) (1 θ3) 1 θ 1} 1 3, i = 1, 2 j = 3,...,k. Here θ 1 is the association between parents, θ 2 the association between two siblings and θ 3 the association between a parent and a child. There is delayed entry because the Danish National Registry of Patients started in This is taken into account by using the conditional survival function. If v i,v j denote the ages at entry, then the conditional survival function is P(T i > t i, T j > t j T i >v i, T j >v j ) = S(t i, t j ) S(v i,v j ). (8.20) Time-dependent covariates are now difficult to handle correctly since the conditional survival function (8.20) still depends on the time from birth until the person entered the study. In this example calendar period is part of the model and it is assumed that the risk of thromboembolism was the same before the register started as in the first period from The data are sampled as described in Section 6, and this is taken into account in the analyses. For the first stage of the analysis population rates have been used and assumed known. This means that there is no extra variation to take into account in the second stage leading to possible underestimation of the variance. However, since the dataset is so large the estimates from the marginal analysis will be close to the population rates. In the second stage the model (8.19) was fitted to the data taking the sampling and delayed entry into account. Two different sets of weights were chosen to account for the pairs: all weights set to 1 or the optimal weights from Section 5.2. The results are seen in Table 1. Table 1 shows that the association is largest for siblings with ˆθ 2 = 10.0, which is significantly larger than 1. The association for parents is smallest with an estimate of ˆθ 1 = 2.5 and a confidence interval including 1 when the optimal weights are chosen. This means that the constraints from (3.5) do not hold and it would not be possible to fit a joint model of the type (3.4) using Clayton s family for each copula. Since the association for parents is smaller than the association for the other types of pairs, this could

13 Composite likelihood for family studies 27 Table 1. Estimates of the association between parents (θ 1 ), children (θ 2 ) and parent/child (θ 3 )inthe application to deep venous thromboembolism Pairwise weights Parameter Estimate Std error θ(95% Conf. int.) Kendall s τ (1,1,1) log θ (1.07; 5.84) log θ (6.31; 15.89) log θ (3.51; 5.03) optimal log θ (0.98; 6.35) log θ (6.51; 15.37) log θ (3.39; 4.99) indicate a genetic factor in the familial aggregation. A simple genetic model would imply that θ 2 = θ 3,as a parent and a child share 50% of their genes on average as do two siblings. Testing this hypothesis, using awald type test in the setting with optimal weights, results in a test statistic of and a test probability of , hence the hypothesis is rejected. The optimal choice of weights improves the standard error for the association among siblings and the relative improvement is larger than the loss of precision for the parent child pairs, suggesting that the optimal weights are a good choice. All calculations were carried out using SAS version DISCUSSION In this paper a two-stage procedure combined with a composite likelihood in the second stage has been studied, with particular reference to the two cases of sib groups and families consisting of parents and children. The sib groups can already be studied using a two-stage method as in Andersen (2003), but with the method presented here it is only necessary to study all possible pairs. This means that software designed for pairs can be used. The loss of efficiency compared to the methods using the full likelihood depends on the amount of information outside the pairs and the choice of weights. Simulations indicate that choosing the optimal weights from Lindsay (1988) is sensible. When 25% of sibling groups have five members the efficiency is still above 90%. For the families of parents and children the composite likelihood approach gives a flexible framework in which to model this type of data. Other approaches have been suggested. The hierarchical models in Bandeen-Roche and Liang (1996) are also based on copulas but they can give unwanted constraints on the parameters. Additive and multiplicative frailty models have also been suggested (e.g. Petersen, 1998, Yashin, 1995). In these models the bivariate margins are not generally shared frailty models. Li and Zhong (2002) suggested an additive genetic gamma frailty model, which can be used in family studies where genetic information is available. The composite likelihood approach has also been studied by Parner (2001), but in this paper the composite likelihood approach is linked to the two-stage estimation. The weights suggested in Section 5 are not optimal in a mathematically precise sense, but they are optimal in some simple situations and also seem to perform well in more complicated cases. The situation where the families consist of members at the same level is very close to the one-dimensional case studied by Lindsay (1988) where the weights are truly optimal. They are not difficult to calculate, and simulations have shown that they perform well. In the case of family members at different levels the weights are more complicated to calculate and the advantage is not as clear. In such cases, one might consider a simpler choice of weights.

14 28 E. WREFORD ANDERSEN In summary, the composite likelihood approach combined with the two-stage estimation promises to be a useful tool in the analysis of sibling groups. It also presents new possibilities when studying families with members at different levels. ACKNOWLEDGEMENTS Work for this paper was started while the author was visiting the MRC Biostatistics Unit, Cambridge, UK. The author would like to thank Per Kragh Andersen and David Clayton for their valuable suggestions and comments and Henrik Toft Sørensen and Jørn Olsen for making the data on deep venous thromboembolism available. The activities of the Danish epidemiology Science Centre are supported by a grant from the Danish National Research Foundation. APPENDIX We first give some notation. Let M be the number of possible pairs, X 1 = (X 11,...,X 1M ),...,X n = (X n1,...,x nm ) n independent identically distributed replications of M pairs Y = (Y 1,...,Y M ), L ik (β, θ k ) the likelihood function for X ik, U ik (β, θ k ) the score function, U ik (β, θ k ) = log L ik. θ k Define the Fisher information matrix for the kth pair and the observed information matrix 2 i k (θ k ) = E 0 { θ k θ k log L ik (β, θ k )} j k (θ k ) = 1 n 2 θ k θ k L ik (β, θ). i=1 The following assumptions (A.1) concerning the bivariate models are adapted from Parner (2001) and assumed for Proposition 4.1. ASSUMPTION A.1 1. The functions θ k θ k L ik (θ k ), θ k θ k log L ik (θ k ) are locally, uniformly in θ k, dominated by integrable functions. 2. If is unbounded then for any sequence {θ kn } n in such that θ kn, log L ik (θ kn ), P a.s. i=1 3. For any sequence {θ kn } n in such that θ kn θ k then 1 log L ik (θ kn ) E 0 {log L 1k (θ k )}, P a.s. n i=1

15 Composite likelihood for family studies For any sequence {θ kn } n in where θ kn θ k then j kn (θ kn ) i k (θ k ). 5. The parameter θ k can be identified from the distribution of Y k. 6. The Fisher information matrix i k (θ k ) is positive definite. 7. The expectations E 0 {U jl (θ 0 j )U kh (θ 0k )} 2 < for j, k = 1,...,K, l = 1,...,dim(θ j ) and h = 1,...,dim(θ k ). DEFINITION A.1 Let the quantities in Proposition 4.2 be W θ (θ, u,v 1,...,v K j ) = w ih θ l(θ, u,v i,v h ), (i,h) M j V θ (θ, u,v 1,...,v K j ) = w 2 ih θ (i,h) M 2 l(θ, u,v i,v h ), j V i (θ, u,v 1,...,v K j ) = i=l ori=h,(l,h) M j w lh I θ = E{ V θ (θ 0,β 0, 0 )} M ij (t) = N ij (t) exp(β Z ij ) The following assumptions are used in the proof of Proposition 4.2. t 0 2 θv i l(θ, u,v l,v h ) Y ij (s)λ 0 (s)ds. ASSUMPTION A.2 Assume the regularity conditions from Spiekerman and Lin (1998), Assumption A.1 and that θ l(θ, u,v i,v h ) and 2 l(θ, u,v θ 2 i,v h ) are continuous and bounded functions of u, v i and v h. The variance from Proposition 4.2 is consistently estimated by inserting the estimates in the formulae in the following way: Ŵ j = W θ { ˆθ, ˆβ, ˆ 0 ( ˆβ, X 1 j ),..., ˆ 0 ( ˆβ, X K j j)} ˆ j = Î θβ Îβ 1 Û β Î θ = n 1 Î θβ = n 1 Î i (t i ) = n 1 n n n. j + K j i=1 t d ˆM. j (u) ˆ j (t) = 0 S (0) ( ˆβ) τ 0 Î i (t i )d ˆ j (t i ) θ W θ { ˆθ, ˆβ, ˆ 0 ( ˆβ, X 1 j ),..., ˆ 0 ( ˆβ, X K j j)} β W θ { ˆθ, ˆβ, ˆ 0 ( ˆβ, X 1 j ),..., ˆ 0 ( ˆβ, X K j j)} Y ij (t i )V i { ˆθ, ˆβ, ˆ 0 ( ˆβ, X 1 j ),..., ˆ 0 ( ˆβ, X K j j)} { t 0 } E( ˆβ,u)d ˆ 0 (u, ˆβ) Îβ 1 Û β. j.

16 30 E. WREFORD ANDERSEN REFERENCES ANDERSEN, E. W.(2003). Two-stage estimation in copula models used in family studies. Lifetime Data Analysis (accepted). BANDEEN-ROCHE, K. J. AND LIANG, K.-Y. (1996). Modelling failure-time associations in data with multiple levels of clustering. Biometrika 83, BINDER, D.A.(1992). Fitting Cox s proportional hazards models from survey data. Biometrika 79, GENEST, C., GHOUDI, K. AND RIVEST, L. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82, GENEST, C.AND MACKAY, J.(1986). The joy of copulas. The American Statistician 40, GLIDDEN, D.V.(2000). A two-stage estimator of the dependence parameter for the Clayton Oakes model. Lifetime Data Analysis 6, HEAGERTY, P. J. AND LELE, S. R.(1998). A composite likelihood approach to binary spatial data. Journal of the American Statistical Association 93, HOUGAARD, P.(1986). A class of multivariate failure time distributions. Biometrika 73, LEE, E. W., WEI, L. J. AND AMATO, D. A.(1992). Cox-type regression analysis for large numbers of small groups of correlated failure time observations. In Klein, J. and Goel, P. (eds), Survival Analysis: State of the Art, Dordrecht: Kluwer, pp LI, H. AND ZHONG, X.(2002). Multivariate survival models induced by genetic frailties, with application to linkage analysis. Biostatistics 3, LINDSAY, B.G.(1988). Composite likelihood methods. Contemporary Mathematics 80, NIELSEN, G. G., GILL, R. D., ANDERSEN, P. K. AND SØRENSEN, T. I. A.(1992). A counting process approach to maximum likelihood estimation in frailty models. Scandinavian Journal of Statistics 19, OAKES, D.(1989). Bivariate survival models induced by frailties. Journal of the American Statistical Association 84, PARNER, E.(2001). A composite likelihood approach to multivariate survival data. Scandinavian Journal of Statistics 28, PETERSEN, J.H.(1998). An additive frailty model for correlated life times. Biometrics 54, ROSENDAAL, F.(1999). Venous thrombosis: a multicausal disease. The Lancet 353, SELIGSOHN, U. AND LUBETSKY, A.(2001). Genetic susceptibility to venous thrombosis. New England Journal of Medicine 344, SHIH, J. H. AND LOUIS, T. A.(1995). Inferences on association parameter in copula models for bivariate survival data. Biometrics 51, SPIEKERMAN, C.F.AND LIN, D.Y.(1998). Marginal regression models for multivariate failure time data. Journal of the American Statistical Association 93, WEI, L. J., LIN, D. Y. AND WEISSFELD, L.(1989). Regression analysis of multivariate incomplete failure time data by modelling marginal distributions. Journal of the American Statistical Association 84, YASHIN, A., VAUPEL, J. AND IACHINE, I.(1995). Correlated individual frailty: an advantageous approach to survival analysis of bivariate data. Mathematical Population Studies 5, [Received June 10, 2002; first revision March 17, 2003; second revision April 14, 2003; accepted for publication May 7, 2003]