Tests of independence for censored bivariate failure time data

Transcription

1 Tests of independence for censored bivariate failure time data Abstract Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of χ 2 -type tests for independence between pairs of failure times after adjusting for covariates. A bivariate accelerated failure time model is proposed for the joint distribution of bivariate failure times while leaving the dependence structures for related failure times completely unspecified. Theoretical properties of the proposed tests are derived and variance estimates of the test statistics are obtained using a resampling technique. Simulation studies show that the proposed tests are appropriate for practical use. Two examples including the study of infection in catheters for patients on dialysis and the diabetic retinopathy study are also given to illustrate the methodology. Keywords Accelerated failure time model Bivariate failure time data Independence test Resampling 1 Introduction Multivariate failure time data arise in many medical studies when each study subject can potentially experience multiple failures or when failures times may be clustered. In this article we focus on the bivariate case. It is of great interest to test whether the pair of failure times for the same subject or within the same cluster are independent or not (Clayton, 1978). A number of methods have been developed for testing independence between two variables. For example, when there is no censoring, the coefficient of concordance (Kendall, 1962) is widely used for independence tests in a bivariate distribution. Oakes (1982 a ) generalized Kendall s tau to censored bivariate data. Furthermore, the cross ratio has been extensively studied for testing independence in a 2 2 table (e.g. Mantel and Haenszel, 1959). And Clayton (1978) introduced the cross ratio concept into bivariate failure time data, which was further studied by 1

2 Oakes (1989) and by Hsu and Prentice (1996). Based on similar ideas, Ding and Wang (2004) also proposed a Mantel-Haenszel type of independence test for bivariate current status data. However, most independence tests proposed in literature for bivariate failure time data do not account for the effect of covariates. One exception is that Hsu and Prentice (1996) extended their proposed covariance test to accommodate covariates. In their paper, the marginal distributions of the pair of failure times were assumed to follow the proportional hazards model. In practice it is important to investigate the dependence relationship between the two related failure times conditional on some covariates. For example, in twins study, it is of great interest to know whether the familial genetic effects introduce additional association between the two failure times of twins after adjusting some common environmental factors. The proportional hazards model (Cox, 1972) has been widely used for analysis of right censored survival data. However, as noted by many authors, the proportional hazards model may not be appropriate for modeling survival times in some medical studies, and alternative models may be more suitable. One of the alternatives is the accelerated failure time (AFT) model, which relates the logarithm of the failure time linearly to the covariates (Kalbfleisch and Prentice, 1980; Cox and Oakes, 1984). The AFT model could be more attractive than the proportional hazards model for many applications due to its direct physical interpretation (Jin et al., 2003). In this article, we propose a bivariate AFT model for joint analysis of pairs of failure times. It naturally generalizes the conventional AFT model for univariate failure times to bivariate case and leaves the dependence structure between the related failure times completely unspecified. For such models, we derive a class of χ 2 -type independence tests for the bivariate AFT model and study both the theoretical and numerical properties of the proposed independence tests. The remainder of the article is organized as follows. In the next section, we specify the bivariate accelerated failure time model and derive the proposed χ 2 -type independence tests as well as their theoretical properties. Numerical studies including simulation results and two examples are given in Section 3 and 4, respectively. Some 2

3 concluding remarks are provided in Section 5. Major technical derivations are contained in the Appendix. 2 The Proposed Methodology 2.1 The bivariate accelerated failure time model and parameters estimation Consider a study involving n independent subjects. Here we focus on data consisting of bivariate events, i.e. each study subject can potentially experience two types of failures. Extension to bivariate clustered failure time data will be discussed later. For i = 1,, n and k = 1, 2, let T ik be the time to the kth failure of the ith subject; let C ik be the corresponding censoring time, and Z ik be the p-dimensional vector of covariates. Throughout the paper, we assume that censoring is noninformative, i.e. (T i1, T i2 ) and (C i1, C i2 ) are independent conditional on (Z i1, Z i2 ). The joint distribution of the two types of failure times is formulated with the following bivariate accelerated failure time model, log T ik = β kz ik + ɛ ik, i = 1,, n; k = 1, 2, (1) where β k is a p-dimensional vector of regression parameters, and (ɛ i1, ɛ i2 ) (i = 1,, n) are independent random vectors with a common, but completely unspecified joint distribution. Thus, the two types of failure times marginally also follow the AFT model while the dependence structure between the pair of failure times is left completely unspecified. Define T ik = T ik C ik and δ ik = I(T ik C ik ). Here and in the sequel, a b = min(a, b), and I( ) is the indicator function. Then the observed data consists of ( T ik, δ ik, Z ik ) (k = 1, 2; i = 1,, n), which are n independent copies of ( T k, δ k, Z k ) (k = 1, 2). Furthermore, define W ik (β k ) = log T ik β k Z ik, C ik (β k ) = log C ik β k Z ik and W ik (β k ) = W ik (β k ) C ik (β k ) = log T ik β k Z ik. Let N ik (x, β k ) = δ ik I{ W ik (β k ) x} and Y ik (x, β k ) = I{ W ik (β k ) x} denote the counting and at-risk processes respectively, and let S (r) k (x, β k) = n 1 n Zr ik Y ik(x, β k ) (r = 0, 1). Then the weighted log-rank 3

4 estimating function for β k is given by U k, φk (β k ) = δ ik φ k { W ik (β k ), β k }[Z ik Z k { W ik (β k ), β k }], k = 1, 2, (2) where Z k (x, β k ) = S (1) k (x, β k)/s (0) k (x, β k) and φ k is a weight function satisfying Condition 5 of Ying (1993). Let ˆβ k, φk be a solution of U k, φk (β k ) = 0 and β 0k be the true value of β k (k = 1, 2). It has been established by a number of authors that n 1/2 ( ˆβ k, φk β 0k ) is asymptotically zero-mean normal (eg. Tsiatis, 1990; Ying, 1993). Furthermore, Lin and Wei (1992) showed that n 1/2 (ˆβ φ β 0 ) also converges in distribution to a zero-mean normal vector, where ˆβ φ = ( ˆβ 1, φ 1, ˆβ 2, φ 2 ) and β 0 = (β 01, β 02). In general, U k, φk (β k ) is neither continuous nor componentwise monotone in β k. Therefore, it is difficult to obtain the solution ˆβ k, φk, especially when β k is high dimensional. One simplification arises in the choice of φ k (x, β k ) = S (0) k (x, β k), which corresponds to the Gehan (1965) weight function. In this case, U k, φk (β k ) can be expressed as U k, G (β k ) = n 1 δ ik (Z ik Z jk )I{ W ik (β k ) W jk (β k )}, (3) j=1 which is monotone in each component of β k (Fygenson and Ritov, 1994). It is easy to show that the right-hand side of (3) is the gradient of the following convex function L k, G (β k ) = n 1 δ ik { W ik (β k ) W jk (β k )}, (4) j=1 where a = a I(a 0). And a minimizer ˆβ k, G of L k, G (β k ) can be easily obtained by the linear programming technique (Jin et al., 2003). To account for the potential dependence among multivariate failure times, Jin et al. (2006) recently proposed a new resampling approach for approximating the asymptotic variance-covariance matrix of ˆβ G ( ˆβ 1, G, ˆβ 2, G ). That is, define a new loss function L k, G(β k ) = n 1 δ ik { W ik (β k ) W jk (β k )} V i V j, (5) j=1 where V i (i = 1,, n) are are independent positive random variables with mean 1 and variance 1, and are independent of the observed data. Let 4 ˆβ k, G be a minimizer of

5 L k, G (β k). Then the asymptotic distribution of n 1/2 (ˆβ G β 0 ) can be approximated by the conditional distribution of n 1/2 (ˆβ G ˆβ G ) given the observed data, where ˆβ G = ( ˆβ 1, G, ˆβ 2, G ). In addition, given β k, the cumulative hazard function Λ k of error terms ɛ k (k = 1, 2), i.e. P (ɛ k > x) = exp{ Λ k (x)}, can be consistently estimated by the Aalen-Breslowtype estimator ˆΛ k (x, β k ) = x dn ik (u) ns (0) k (u, β k). (6) Therefore, a consistent estimator ˆΛ k of Λ k can be obtained by plugging a consistent estimator of β 0k into (6), for example the Gehan-weight estimator ˆβ k, G, i.e. ˆΛk (x) = ˆΛ k (x, ˆβ k, G ). 2.2 The proposed χ 2 -type independence test Our interest is to test the null hypothesis H 0 : T 1 and T 2 are independent given covariates Z 1 and Z 2, or equivalently H 0 : ɛ 1 and ɛ 2 are independent. Under the null hypothesis H 0, we have P {W i1 (β 01 ) W i2 (β 02 ) > x Z i1, Z i2 } = exp[ {Λ 1 (x) + Λ 2 (x)}]. (7) Define M i (x, β 1, β 2, Λ 1, Λ 2 ) = N i (x, β 1, β 2 ) x I{ W i (β 1, β 2 ) u}d{λ 1 (u) + Λ 2 (u)}, where W i (β 1, β 2 ) = W i1 (β 1 ) W i2 (β 2 ), N i (x, β 1, β 2 ) = δ i (β 1, β 2 )I{ W i (β 1, β 2 ) x}, and δ i (β 1, β 2 ) = δ i1 if W i1 (β 1 ) W i2 (β 2 ) and δ i2 otherwise. Since (T i1, T i2 ) and (C i1, C i2 ) are independent given covariates (Z i1, Z i2 ), it is easy to show that M i (x, β 01, β 02, Λ 01, Λ 02 ) is a zero-mean process under H 0, where Λ 01 and Λ 02 are the true values of Λ 1 and Λ 2, respectively. Thus, we propose to use the following test statistic for H 0, that is H n = n 1/2 h(x)dm i {x, ˆβ 1, G, ˆβ 2, G, ˆΛ 1 (, ˆβ 1, G ), ˆΛ 2 (, ˆβ 2, G )}, (8) where h(x) is a positive deterministic weight function. For example, the simplest choice of h is h(x) 1. Here we use the Gehan-weight estimators ˆβ 1, G and ˆβ 2, G of β 1 and β 2 respectively in H n for simplicity. Other weighted log-rank estimators can also be accommodated in H n using the technique of Jin et al. (2003). It is easy to see that 5

6 when replacing the Gehan-weight estimators by the true values β 01, β 02, Λ 01 and Λ 02 in (8), each summand of H n has mean zero under H 0. Thus, by the central limit theorem, H n converges in distribution to a normal random variable with mean zero under H 0. Actually, H n defined in (8) also converges in distribution to a zero-mean normal random variable under H 0. We will establish the theoretical properties of H n in the following theorem. Theorem 1 Under some regularity conditions, as n goes to, H n converges in distribution under H 0 to a normal random variable with mean 0 and variance σ 2. Moreover, H n can be expressed asymptotically as a sum of independent random variables, i.e. H n = n 1/2 n ψ i + o p (1), where ψ i = h(x)dm i (x, β 01, β 02, Λ 01, Λ 02 ) B k, GA 1 k, G U ik, G h(x) s(0) (x, β 01, β 02 ) s (0) k (x, β dm ik (x, β 0k, Λ 0k ), 0k) and M ik (x, β, Λ) = N ik (x, β) x Y ik(u, β)dλ(u) (k = 1, 2). The quantities B k, G, A k, G, U ik, G, s (0) (x, β 1, β 2 ) and s (0) k (x, β k) are defined in the Appendix. The proof of Theorem 1 is given in the Appendix. It is easy to show that under H 0, H n is asymptotically a sum of zero-mean independent random variables and σ 2 = V ar(ψ 1 ). To use Theorem 1 for testing H 0, we need to find a consistent estimator for σ 2. Like the variances of the weighted log-rank estimators, σ 2 also involves the derivatives of the density functions of error terms ɛ k (k = 1, 2). Directly estimating σ 2 from data is very complicated and needs nonparametric smoothing techniques, which usually require large sample size to ensure good results. In addition, since the test statistic H n involves the estimates of finite and infinite dimensional parameters, the classical bootstrap method for estimating asymptotic variance may not be applicable here. Therefore, we propose a resampling method for computing the estimator ˆσ 2 of σ 2. To be specific, define Hn = n 1/2 h(x)dm i {x, ˆβ 1, G, ˆβ 2, G, ˆΛ 1 (, ˆβ 1, G), ˆΛ 2 (, ˆβ 2, G)}. (9) 6

7 Then we have Theorem 2 Under some regularity conditions, as n goes to, the conditional distribution of Q n of H n under H 0, where given the observed data converges almost surely to the limiting distribution Q n = H n H n + n 1/2 [ h(x)dm i {x, ˆβ 1, G, ˆβ 2, G, ˆΛ 1 (, ˆβ 1, G ), ˆΛ 2 (, ˆβ 2, G )} h(x) S(0) (x, ˆβ 1, G, ˆβ 2, G ) S (0) k (x, ˆβ dm ik {x, ˆβ k, G, ˆΛ k (, ˆβ ] k, G )} (V i 1), k, G ) with S (0) (x, β 1, β 2 ) = n 1 n I{ W i (β 1, β 2 ) x} and V i (i = 1,, n) are the same as those used in (5) for computing ˆβ k, G. A similar resampling technique was also used by Lin et al. (1994) for constructing confidence bands of survival functions under the proportional hazards model. major difference between the proposed resampling method and Lin et al. (1994) s is that the same set of perturbation random variables {V i } s were also used for computing the resampling estimators One ˆβ k, G, k = 1, 2. Consequently, the proper resampling scheme needs to account for the sampling variations due to both the rank estimation of β k and the estimation of the infinite dimensional parameters Λ k (x), k = 1, 2. The proof of Theorem 2 is given in the Appendix. To obtain the estimator of σ 2, we repeatedly generate the random variables (V 1,, V n ), say M times, and calculate Q n,j (j = 1,, M) for each generated set. Then ˆσ 2 can be obtained using the sample variance of Q n,j (j = 1,, M), and the null hypothesis H 0 is rejected at level α when H 2 n/ˆσ 2 > χ 2 1(α), where χ 2 1(α) is the upper αth quantile of the χ 2 1 distribution. 2.3 Extension to bivariate clustered failure time data Bivariate clustered failure time data is another common type of bivariate failure time data. For example, in twins study the pair of failure times of twins are usually correlated. The model (1) proposed in Section 2.1. for bivariate events data can be easily tailored to accommodate bivariate clustered failure time data. To be specific, 7

8 we set β 1 = β 2 = β and Λ 1 = Λ 2 = Λ in (1). Then under the working independence assumption, the weighted log-rank estimating function for β can be written as U φ (β) = δ ik φ{ W ik (β), β}[z ik Z{ W ik (β), β}], (10) where Z(x, β) = S (1) (x, β)/s (0) (x, β) with S (r) (x, β) = n 1 n 2 Zr ik Y ik(x, β) (r = 0, 1) and φ is a weight function. The Gehan-weight estimating function corresponds to the choice of φ(x, β) = S (0) (x, β), i.e. U G (β) = n 1 δ ik (Z ik Z jl )I{ W ik (β) W jl (β)}, (11) j=1 l=1 which is the gradient of L G (β) n 1 δ ik { W ik (β) W jl (β)}. (12) j=1 l=1 Let ˆβ G be a minimizer of L G (β). Jin et al. (2006) showed that n 1/2 ( ˆβ G β 0 ) converges in distribution to a zero-mean normal vector, where β 0 is the true value of β. And the asymptotic variance-covariance matrix of ˆβ G can be estimated by the resampling method. To be specific, define L G (β) n 1 j=1 l=1 where V i (i = 1,, n) are defined the same as before. Let δ ik { W ik (β) W jl (β)} V i V j, (13) ˆβ G be a minimizer of L G (β). Then the limiting distribution of n 1/2 ( ˆβ G β 0 ) can be approximated by the conditional distribution of n 1/2 ( ˆβ G ˆβ G ) given the observed data. Furthermore, Λ(x) can be consistently estimated by Λ(x) = ˆΛ(x, ˆβ G ), where ˆΛ(x, β) = x dn ik (u) ns (0) (u, β). (14) Now we are ready to derive independence tests for bivariate clustered failure time data. Define M i (x, β, Λ) = Ñi(x, β) 2 x I{ W i (β) u}dλ(u), where W i (β) = W i1 (β) W i2 (β), Ñ i (x, β) = δ i (β)i{ W i (β) x}, and δ i (β) = δ i1 if Wi1 (β) W i2 (β) 8

9 and δ i2 otherwise. In addition, let H n = n 1/2 n h(x)d M i {x, ˆβ G, ˆΛ(, ˆβ G )}, H n = n 1/2 n h(x)d M i {x, ˆβ G, ˆΛ(, ˆβ G )} and Q n = H n H n + n 1/2 [ h(x)d M i {x, ˆβ G, ˆΛ(, ˆβ G )} h(x) 2 S (0) (x, ˆβ G ) S (0) (x, ˆβ G ) dm ik{x, ˆβ G, ˆΛ(, ˆβ ] G )} (V i 1), where S (0) (x, β) = n 1 n I{ W i (β) x}. Then Theorem 3 Under some regularity conditions, as n goes to, Hn converges in distribution under H 0 to a normal random variable with mean 0 and variance σ 2. And the conditional distribution of Q n the limiting distribution of Hn under H 0. given the observed data converges almost surely to To estimate σ 2, we repeatedly generate the independent random variables V i (i = Q 1,, n), say M times, and calculate n,j for each generated set. Now σ 2 can be approximated by the sample variance ˆ σ 2 of Q n,j (j = 1,, M) and we reject H 0 at level α when H 2 n/ˆ σ 2 > χ 2 1(α). 3 Simulation Study The performance of the proposed independence test for bivariate failure time data was assessed in a series of simulations studies. The bivariate accelerated failure time model specified in (1) is used to generate pairs of failure times. Under each setting, two independent covariates were generated with the first covariate following a Bernoulli distribution with success probability 0.5 and the second one following a uniform distribution on ( 1, 1). For bivariate events data, the pair of failure times shared the same set of covariates, i.e. Z i1 = Z i2 = (Z i1,1, Z i1,2 ). But for bivariate clustered failure time data, Z i1 and Z i2 are independent draws. The first set of simulation studies were conducted for bivariate normal errors, i.e. the joint distribution of two error terms (ɛ i1, ɛ i2 ) is from a bivariate normal distribution 9

10 with mean zero and variance-covariance matrix σ2 1 θσ 1 σ 2 θσ 1 σ 2 σ2 2. Here we set σ 1 = σ 2 = 0.5 and chose θ = 0.5, 0, 0.5. Note that θ = 0 corresponds to the case that the pair of failure times T i1 and T i2 are independent given covariates Z i1 and Z i2. In the second set of simulations, (ɛ i1, ɛ i2 ) were generated from a Clayton-Oakes model (Clayton, 1978; Oakes, 1982 b, 1986), i.e. the joint survival function of (ɛ i1, ɛ i2 ) is P (ɛ i1 > x 1, ɛ i2 > x 2 ; θ) = [exp{θλ 1 (x 1 )} + exp{θλ 2 (x 2 )} 1] θ 1. where the dependence parameter θ 0 and θ = 0 corresponds to the independence between the two error terms. As in the second setting, we set Λ 1 (x) = Λ 2 (x). And chose the extreme value and logistic distributions for the marginal distributions of the two error terms. For bivariate events data, the two regression parameters β 1 = (β 11, β 12 ) = (1, 1), β 2 = (β 21, β 22 ) = (0.5, 0.5) ; while for bivariate clustered failure time data, the regression parameters β = (1, 1). Under each setting, the censoring times C ik were generated from a uniform distribution on (0, c k ), where c k (k = 1, 2) were chosen such that the expected proportion of censoring for each type of failures is 25% and 40%. (Insert Tables 1-5 here) Tables 1-3 summarize the results from the first and second sets of simulation studies for bivariate events data with sample size n = 100 or 200; while Tables 4 and 5 summarize the results for bivariate clustered failure time data with n = 100. Each entry in the table was based on 500 simulated data sets. For computational convenience, when constructing the χ 2 -type independence test statistics proposed in Section 2.3. and 2.4., we chose the weights h(x) 1. For estimating the variances of the proposed test statistics, we generated M = 500 realizations of n independently distributed unit exponential random variables V i, i = 1,, n, under each settings. From the simulation results, we see that the proposed independence tests give the appropriate type one 10

11 errors (the α-level was chosen as 5%) in all the settings when the pair of failure times are really independent given covariates and also have reasonable powers to detect the association between the pair of failure times when there is. And the results improve when the sample size increases and the level of censoring decreases. 4 Two Examples To illustrate the independence test of Section 2, we also considered two examples. For bivariate events data, we studied the data of infection in catheters for patients on dialysis (McGilchrist and Aisbett, 1991). The dataset contained time (days) to infection in kidney catheters for 38 patients on dialysis from two observation periods. Three covariates, age at the time of insertion, gender (1 for male and 0 for female) and type of disease (0 = GN, 1 = AN, 2 = PKD, and 3 = other), were also included. The bivariate AFT model (1) was used for the pair of event times from the two observation periods. For bivariate clustered failure time data, we considered the diabetic retinopathy data (Huster et al., 1989). The dataset contains 197 patients, who were a 50% random sample of the patients with high-risk diabetic retinopathy as defined by the Diabetic Retinopathy Study (DRS). Each patient had one eye randomized to laser treatment and the other eye served as an untreated control. For each eye, the event of interest was the time from initiation of treatment to the severe visual loss (call it blindness ). Besides treatment indicator (1 = treatment; 0 = control), four other covariates: laser type (0 = xenon, 1 = argon), age at diagnosis of diabetics, type of diabetics (0= juvenile, 1 = adult) and risk group (6-12), are also included in the bivariate AFT model. The proposed independence tests were constructed for both datasets using two different weight functions: constant weight function, i.e. h(x) = 1, and data-dependent weight function, i.e. h(x) = S (0) (x, ˆβ 1G, ˆβ 2G ) for the bivariate events data and h(x) = S (0) (x, ˆβ G ) for the bivariate clustered failure time data. The corresponding test statistics are denoted as Hn 1 for the constant weight function and Hn 2 for the data-dependent weight function. Their results are summarized as follows: 11

12 Example Test Statistic SE p-value Catheter Infection (Hn) Catheter Infection (Hn) Diabetic Retinopathy Data (Hn) Diabetic Retinopathy Data (Hn) Here SE is the estimated standard error of the corresponding test statistic with resampling size M = 500. From the above results, for catheter infection data the pair of event times are independent given the covariates: age, gender and types of disease; while for diabetic retinopathy data, the two failure times of the pair of eyes are correlated even after adjusting for the treatment indicator, laser type, age at diagnosis of diabetics, type of diabetics and risk group. 5 Discussion We have proposed a class of χ 2 -type independence tests for bivariate events data with the adjustment of covariates. The pair of failure times are modelled through the bivariate accelerated failure time model and the dependence structure between the two related failure times is completely unspecified. The large sample properties of the proposed tests are established. And a resampling technique is used for obtaining the estimates of the asymptotic variances of the proposed test statistics. The proposed independence tests can also be extended to multivariate failure time data by using pair-wise comparison. In this article, the proposed independence test statistics was introduced in an ad hoc fashion. Only the independence along the diagonal line is used for testing the null hypothesis H 0. That is, we only use the fact that, under H 0, P {W i1 (β 01 ) W i2 (β 02 ) > x Z i1, Z i2 } = exp[ {Λ 1 (x) + Λ 2 (x)}]. More comprehensive tests may be constructed, for example, the covariance test proposed by Hsu and Prentice (1996) for the proportional hazards model. To be specific, 12

13 we may consider the following covariance-type test statistics: P n = n 1/2 h(x 1, x 2 )dm i1 {x 1, ˆβ 1, G, ˆΛ 1 (, ˆβ 1, G )}dm i2 {x 2, ˆβ 2, G, ˆΛ 2 (, ˆβ 2, G )}, where h(x 1, x 2 ) is a positive weight function. In addition, we only considered deterministic weight functions h(x) for constructing the test statistics in the paper. However, to improve the power of the test, data-dependent predictable and bounded weight function h n (x) may also be used, where h n (x) converges almost surely to some deterministic function as n goes to. The derivation of the theoretical properties for the covariancetype tests and the tests using data-dependent weight functions, and the estimation of the asymptotic variance of the corresponding test statistics via the resampling method becomes more complicated and needs further investigation. Acknowledgement The author would like to thank the two referees for their insightful and constructive comments. The author is also grateful to Professor Tsiatis for helpful discussions and suggestions. The research was supported in part by NSF grant DMS Appendix To avoid delicate technical issues associated with smoothness and tail fluctuation, we assume that related functions are sufficiently smooth and impose regularity conditions similar to Conditions 1-5 of Ying (1993). Proof of Theorem 1 in (2) can be rewritten as It is easy to see that the Gehan-weight estimating function defined U k, G (β, Λ) = S (0) k (x, β){z ik Z k (x, β)}dm ik (x, β, Λ), k = 1, 2. Note that E{M ik (x, β 0k, Λ 0k )} = 0. Using some empirical process approximation techniques, we can show that n 1/2 U k, G (β 0k, Λ 0k ) = n 1/2 U ik, G + o p (1), (A.1) 13

14 where U ik, G = and z k (x, β) = s (1) k s (0) k (x, β 0k){Z ik z k (x, β 0k )}dm ik (x, β 0k, Λ 0k ), (x, β)/s(0) (x, β) with s(r) (x, β) = lim n E{S (r) (x, β)}, r = 0, 1, k = k k 1, 2. Then it follows from the arguments of Jin et al. (2003) that ˆβ k, G a.s β 0k and k n 1/2 ( ˆβ k, G β 0k ) = A 1 k, G n 1/2 U ik, G + o p (1), (A.2) where A k, G = lim n n 1 E s (0) k (x, β 0k){Z ik z k (x, β 0k )} 2 {d log λ k (x)/dx}dn ik (x, β k ), and λ k (x) = dλ k (x)/dx, a 2 = aa. In addition, it can be shown (Park and Wei, 2003) that there exists a deterministic vector C k, G (x, β) (k = 1, 2) such that, n 1/2 {ˆΛ k (x, ˆβ k, G ) ˆΛ k (x, β 0k )} = C k, G(x, β 0k )n 1/2 ( ˆβ k, G β 0k ) + o p (1). (A.3) Let g(β 1, β 2, Λ 1, Λ 2 ) = lim n n 1 n E{ h(x)dm i(x, β 1, β 2, Λ 1, Λ 2 )}. Then applying the techniques of Tsiatis (1990) and Ying (2003) for deriving linearity of the weighted long-rank estimates as well as some empirical process approximation techniques, we can show that where H n = n 1/2 + h(x)dm i {x, β 01, β 02, ˆΛ 1 (, β 01 ), ˆΛ 2 (, β 02 )} B k, Gn 1/2 ( ˆβ k, G β 0k ) + o p (1), B k, G = g(β 1, β 2, Λ 1, Λ 2 ) β1 =β β 01,β 1 =β 01,Λ 1 =Λ 01,Λ 2 =Λ 02 k h(x)s (0) (x, β 01, β 02 )dc k, G (x, β 0k ) 14

15 and s (0) (x, β 1, β 2 ) = lim n E{S (0) (x, β 1, β 2 )}. Then it follows that H n = n 1/2 n 1/2 = n 1/2 n 1/2 h(x)dm i {x, β 01, β 02, Λ 01, Λ 02 ) B k, GA 1 k, G n 1/2 U ik, G h(x)i{ W i (β 01, β 02 ) x}d{ˆλ k (x, β 0k ) Λ 01 (x)} + o p (1) h(x)dm i {x, β 01, β 02, Λ 01, Λ 02 ) = n 1/2 ψ i + o p (1). Thus, the results established in Theorem 1 hold. B k, GA 1 k, G n 1/2 h(x) s(0) (x, β 01, β 02 ) s (0) k (x, β dm ik (x, β 0k, Λ 0k ) + o p (1) 0k) U ik, G Proof of Theorem 2 From the arguments given in the appendix of Jin et al. (2006), we have that where U k, G( ˆβ k, G ) = ˆβ k, G a.s β 0k (k = 1, 2) and n 1/2 ( ˆβ k, G ˆβ k, G ) = A 1 k, G n 1/2 Uk, G( ˆβ k, G ) + o p (1), (A.4) S (0) k (x, ˆβ k, G ){Z ik Z k (x, ˆβ k, G )}dm ik {x, ˆβ k, G, ˆΛ k (, ˆβ k, G )}(V i 1). Furthermore, by the similar derivation given in Appendix A, we can show that H n H n = Then it follows from (A.4) that H n H n = n 1/2 [ B k, Gn 1/2 ( ˆβ k, G ˆβ k, G ) + o p (1). B k, GA 1 k, G S (0) k (x, ˆβ k, G ){Z ik Z k (x, ˆβ k, G )}dm ik {x, ˆβ k, G, ˆΛ k (, ˆβ ] k, G )} (V i 1) + o p (1). 15

16 Now we have Q n = n 1/2 [ h(x)dm i {x, ˆβ 1, G, ˆβ 2, G, ˆΛ 1 (, ˆβ 1, G ), ˆΛ 2 (, ˆβ 2, G )} S (0) k (x, ˆβ k, G ){Z ik Z k (x, ˆβ k, G )}dm ik {x, ˆβ k, G, ˆΛ k (, ˆβ k, G )} B k, GA 1 k, G h(x) S(0) (x, ˆβ 1, G, ˆβ 2, G ) S (0) k (x, ˆβ dm ik {x, ˆβ k, G, ˆΛ k (, ˆβ ] k, G )} (V i 1) + o p (1) k, G ) n 1/2 ˆψ i (V i 1) + o p (1). Note that given the observed data, ˆψi is a constant. Thus, as n goes to, the conditional distribution of Q n given the observed data converges almost surely to a normal distribution with mean 0 and variance σ 2 Q lim n (1/n) n ˆψ 2 i. In addition, under H 0, σ 2 Q = V ar(ψ 1) = σ 2. It completes the proof of Theorem 2. Proof of Theorem 3 Following the similar steps used in the proofs of Theorems 1 and 2, we can also show the results established in Theorem 3. The details are omitted here. References Clayton DG (1978) A Model for Association in Bivariate Life Tables and Its Application in Epidemiological Studies of Familial Tendency in Chronic Disease Incidence. Biometrika 65: Cox DR (1972) Regression models and life tables (with Discussion). J Roy Stat Soc Ser B 34: Cox DR, Oakes D (1984) Analysis of Survival Data. Chapman & Hall, New York, NY. Ding AA, Wang W (2004) Testing Independence for Bivariate Current Status Data. J Am Stat Assoc 99: Fygenson M, Ritov Y (1994) Monotone Estimating Equations for Censored Data. Ann Stat 22:

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19 Table 1 Simulation Results for Bivariate Events Data under Bivariate Normal Model n θ Censoring Mean SE SEE POW % % % % % % % % % % % % NOTE: Mean and SE present the mean and sample standard error of the test statistics H n. SEE is the mean of the estimated standard errors using the resampling method, and POW is the power of the test or the type I error under the null hypothesis. 19

20 Table 2 Simulation Results for Bivariate Events Data under Clayton-Oakes Model (n = 100) Error Distribution θ Censoring Mean SE SEE POW Extreme Value % % % % % % Logistic % % % % % % Table 3 Simulation Results for Bivariate Events Data under Clayton-Oakes Model (n = 200) Error Distribution θ Censoring Mean SE SEE POW Extreme Value % % % % % % Logistic % % % % % %

21 Table 4 Simulation Results for Bivariate Clustered Failure Time Data under Bivariate Normal Model (n = 100) θ Censoring Mean SE SEE POW 0 25% % % % % % Table 5 Simulation Results for Bivariate Clustered Failure Time Data under Clayton-Oakes Model (n = 100) Error Distribution θ Censoring Mean SE SEE POW Extreme Value % % % % % % Logistic % % % % % %