Semiparametric regression analysis of case-cohort studies with multiple interval-censored disease outcomes

Summary

Interval-censored failure time data commonly arise in epidemiological and biomedical studies where the occurrence of an event or a disease is determined via periodic examinations. Subject to interval-censoring, available information on the failure time can be quite limited. Cost-effective sampling designs are desirable to enhance the study power, especially when the disease rate is low and the covariates are expensive to obtain. In this work, we formulate the case-cohort design with multiple interval-censored disease outcomes and also generalize it to non-rare diseases where only a portion of diseased subjects are sampled. We develop a marginal sieve weighted likelihood approach, which assumes that the failure times marginally follow the proportional hazards model. We consider two types of weights to account for the sampling bias, and adopt a sieve method with Bernstein polynomials to handle the unknown baseline functions. We employ a weighted bootstrap procedure to obtain a variance estimate that is robust to the dependence structure between failure times. The proposed method is examined via simulation studies and illustrated with a dataset on incident diabetes and hypertension from the Atherosclerosis Risk in Communities (ARIC) study.

1 ∣. INTRODUCTION

Epidemiological and biomedical studies often encounter interval-censored failure time data, where time to the occurrence of an event or a disease is observed only to fall within some interval rather than known exactly.¹ One area that commonly produces interval-censored data is the AIDS clinical trials. In this case, investigators may be interested in times to AIDS for HIV infected subjects. The determination of AIDS onset is usually based on blood testing which can be performed only periodically instead of continuously. In consequence, only interval-censored data are available for AIDS onset times. Similarly, in HIV preventive vaccine trials, individuals are usually tested for the presence of HIV at discrete clinic visits and thus time to HIV infection is known only to fall between the last negative and first positive test dates. A more specific example arises from the Atherosclerosis Risk in Communities (ARIC) study, a longitudinal epidemiological study, where the participants were examined for various diseases every three years. In this study, the occurrence of a disease such as diabetes and hypertension was known only between two consecutive examinations, yielding interval-censored data for time to disease. When the examination visits are infrequent or the disease rate is low, available information on time to disease can be quite limited and a large sample is usually needed to reach a desired study power. Nevertheless, when the covariate measurements are difficult or expensive to obtain (e.g., antibody responses in HIV vaccine trials), assembling these covariates for a large sample can be prohibitive for investigators with limited budget. Cost-effective sampling schemes for studies that concern interval-censored failure time data are therefore much needed.

The case-cohort study design is a well-known cost-effective sampling strategy for large cohort studies where rare diseases and expensive covariates are of interest. Under this design, the expensive covariate measurements are ascertained only for a random sample of the cohort, called subcohort, and for all cases, that is, subjects who have developed the disease of interest during the follow-up period. The case-cohort design was originally proposed by Prentice.² Since its proposal, a large amount of research has been devoted. For example, Prentice² and Self and Prentice³ proposed a pseudo-likelihood approach for inference; Chen and Lo⁴ developed more efficient estimators based on a class of estimating equations; Borgan et al.⁵ considered selecting the subcohort via stratified sampling to improve efficiency; Cai and Zeng⁶ studied the case-cohort design with non-rare events; Kang and Cai⁷ and Kim et al.⁸ developed estimating equation methods for case-cohort studies with multiple outcomes. A more comprehensive review of the existing case-cohort designs and inference methods can be found in Ding et al.⁹ All of these designs and methods were developed for traditional right-censored data, where the failure time of interest is either exactly observed or right-censored, and they are not directly applicable to studies that yield interval-censored data.

In response to the need of real studies such as those mentioned above, several authors have considered the case-cohort design with interval-censored failure time data. Among others, Gilbert et al.¹⁰ employed the original case-cohort design proposed by Prentice² in a phase 3 HIV vaccine trial, where the interval-censored time-to-HIV data were treated as traditional right-censored data by approximating the true time to HIV infection with the midpoint of the observed censoring interval. Later, Li et al.¹¹ and Li and Nan¹² developed the case-cohort design and inference procedure for grouped failure time data and current status data, respectively, which are special cases of interval-censored data. More recently, Zhou et al.¹³ studied the case-cohort design under general interval-censoring and proposed a semiparametric inference procedure. All the aforementioned work considered the case-cohort study with a single disease of interest. To our knowledge, there is no cost-effective design and method available for studies that concern multiple interval-censored disease outcomes. One key feature of the case-cohort design is that the same subcohort can be used for multiple diseases so that study costs can be further reduced and simultaneous inference for multiple diseases can be performed. In this paper, we formulate the case-cohort design with multiple interval-censored disease outcomes and also generalize it to non-rare diseases where only a portion of diseased subjects are sampled. We develop an inference procedure, based on the case-cohort sampled multivariate interval-censored data, that allows for the comparison of covariate effects on different diseases.

Several methods have been proposed for regression analysis of multivariate interval-censored failure time data. Among others, Goggins and Finkelstein,¹⁴ Chen et al.¹⁵ and Chen et al.¹⁶ proposed marginal approaches under the proportional hazards model, the proportional odds model and the linear transformation model, respectively. These methods assumed that all subjects are examined at a common set of time points. More recently, Wen and Chen,¹⁷ Zhou et al.¹⁸ and Zeng et al.¹⁹ developed frailty model approaches under general interval-censoring. Although these frailty-based methods improved efficiency over the marginal approach, they rely on the parametric assumption of the frailty distribution. Another common approach to analyzing multivariate data is to use the copula models. For example, Wang et al.²⁰ and Hu et al.²¹ developed copula-based methods for bivariate current status data, which is a special case of interval-censored data. As with frailty-based methods, the copula approach usually makes assumptions on the association between failure times.

In this paper, we propose a marginal likelihood approach that handles general interval-censored data and does not pose any assumption on the dependence structure between failure times. In particular, we assume that the failure times marginally follow the proportional hazards model. We construct a pseudo-likelihood function by assuming working independence between failure times. We use a sieve method with Bernstein polynomials to handle the unknown cumulative baseline hazard functions. Unlike the previous marginal approaches,¹⁴ our approach allows different subjects to have distinct sets of examination time points. On the other hand, under the case-cohort design, the observed data forms a biased sample. To account for sampling bias, we employ the inverse probability weighting techinque. We propose two types of weights in our approach and compare their performance through simulation studies. Specifically, one type of weights is constructed based on single disease information while the other type incorporates information from multiple diseases. In addition, we provide a weighted bootstrap procedure to obtain a variance estimate that is robust to the dependence structure between failure times.

The remainder of this paper is organized as follows. In Section 2, we describe models, designs and data structures. In particular, we introduce the case-cohort design for rare events and the generalized case-cohort design for non-rare events when multiple interval-censored event times are concerned. In Section 3, we present a marginal sieve weighted likelihood approach with two types of weights to analyze the resulting data. We obtain a consistent variance estimate using a weighted bootstrap procedure. Section 4 investigates how the proposed method performs through simulation studies. Section 5 includes an illustrative example from the ARIC Study and Section 6 gives some final remarks.

2 ∣. MODEL, DESIGN AND DATA STRUCTURE

Suppose that there are K diseases of interest in a cohort study. For k = 1, 2, … , K, let T_k denote the time to first incidence of the kth disease which has the following event permanence: any assessment made before T_k will show that the disease has never occurred yet, and any assessment after T_k will show that the disease has already occurred. Let Z_k denote the p-dimensional covariate vector for the kth disease, k = 1, 2, … , K. Assume that T_k marginally follows the proportional hazards model with the cumulative hazard function conditional on Z_k given by

$${\Lambda }_k(t\mid Z_k)={\Lambda }_k(t)\text{exp}({\beta }^{\prime }{Z}_k),$$ (1)

where Λ_k(t) is an unspecified disease-specific cumulative baseline hazard function and β is a p-dimensional vector of regression parameters. Model (1) can incorporate disease-specific covariate effects by defining $\beta =[{\beta }_1^{\prime },\dots ,{\beta }_k^{\prime },\dots ,{\beta }_K^{\prime }{]}^{\prime }$ and $Z_k=[0_1^{\prime },\dots ,0_{k-1},(Z_k^{\ast }{)}^{\prime },0_{k+1}^{\prime },\dots ,0_K^{\prime }{]}^{\prime }$ such that ${\beta }^{\prime }{Z}_k={\beta }_k^{\prime }{Z}_k^{\ast }$.

2.1 ∣. Case-Cohort Design

The case-cohort design can be considered as a two-phase sampling scheme. Suppose that there are n subjects in the study cohort. At the first phase, we observe interval-censored data on the K diseases of interest for all n cohort members:

$$Y_{ki}=\{U_{lki},{\Delta }_{lki}=I(U_{l-1,ki}<T_{ki}\le U_{lki}):l=1,\dots ,M_{ki}\},k=1,\dots ,K,i=1,\dots ,n,$$

where 0 = U_0ki < U_1ki < U_2ki < ⋯ < U_Mki,ki < U_Mki+1,ki = ∞ are a sequence of random examination times for T_ki and are assumed to be independent of T_ki conditional on Z_ki, and the number of examination times M_ki is a positive random integer. Note that ${\sum }_{l=1}^{M_{ki}}{\Delta }_{lki}=1$ suggests that the ith subject has developed the kth disease. We select a simple random sample, called subcohort, by Bernoulli sampling with probability p_s from the study cohort. Let ξ_i indicate whether the ith subject is selected into the subcohort, i = 1, … , n.

At the second phase, we obtain the expensive covariate measurements for the subcohort (i.e., subjects with ξ_i = 1) and for all cohort members with the diseases of interest (i.e., subjects with ${\sum }_{l=1}^{M_{ki}}{\Delta }_{lki}=1$ for some k). The observed data under the case-cohort design can then be written as:

$$\begin{gathered} \text{First-phase data}:\left\{Y_{ki},{\xi }_i:k=1,\dots ,K,i=1,\dots ,n\right\}; \\ \text{Second-phase data}:\left\{Z_{ki}\mid {\xi }_i=1\text{or}\sum _{l=1}^{M_{ki}}{\Delta }_{lki}=1:k=1,\dots ,K,i=1,\dots ,n\right\}. \end{gathered}$$

2.2 ∣. Generalized Case-Cohort Design

For non-rare or not-so-rare diseases, it may not be feasible or necessary to measure the expensive covariates for all diseased subjects as designed in the case-cohort study. We therefore generalize the case-cohort design by allowing one to select only a portion of the diseased subjects for the ascertainment of expensive covariate measurements. This is often referred to as the generalized case-cohort design in the literature.^6,7 In particular, we select a simple random sample by Bernoulli sampling with probability p_ck from the subjects who have the kth disease but are not included the subcohort. Let η_ki indicate whether the ith subject is selected into this sample, k = 1, … , K, i = 1, … , n. We assemble the covariate measurements for subjects with ξ_i = 1 or η_ki = 1 for some k. The observed data under the generalized case-cohort design can be summarized as:

$$\begin{gathered} \text{First-phase data}:\left\{Y_{ki},{\xi }_i,{\eta }_{ki}:k=1,\dots ,K,i=1,\dots ,n\right\}; \\ \text{Second-phase data}:\left\{Z_{ki}\mid {\xi }_i=1\text{or}{\eta }_{ki}=1:k=1,\dots ,K,i=1,\dots ,n\right\}. \end{gathered}$$

If ξ_i = 1, then η_ki = 0; if η_ki = 1, then ξ_i = 0 and ${\sum }_{l=1}^{M_{ki}}{\Delta }_{lki}=1$. Figure 1 illustrates the data structure under the generalized case-cohort design.

FIGURE 1.

An illustration of the generalized case-cohort design

3 ∣. ESTIMATION AND INFERENCE PROCEDURE

For both designs described above, the covariates can be considered as missing at random. To account for missingness, we employ the inverse probability weighting (IPW) method. In particular, under the working independence assumption,^22,14 the inverse probability weighted log-pseudolikelihood function has the form

$$\begin{gathered} l_n(\beta ,{\Lambda }_1,\dots ,{\Lambda }_K)=\sum _{i=1}^n\sum _{k=1}^K{w}_{ki}{l}_k(\beta ,{\Lambda }_k\mid O_{ki}) \\ =\sum _{i=1}^n\sum _{k=1}^K{w}_{ki}\left\{\sum _{l=1}^{M_{ki}+1}{\Delta }_{lki}\log \left(\text{exp}\left\{-{\Lambda }_k(U_{l-1,ki})\text{exp}({\beta }^{\prime }{Z}_{ki})\right\}-\text{exp}\left\{-{\Lambda }_k(U_{lki})\text{exp}({\beta }^{\prime }{Z}_{ki})\right\}\right)\right\} \\ =\sum _{i=1}^n\sum _{k=1}^K{w}_{ki}\left\{\log \left(\text{exp}\left\{-{\Lambda }_k(L_{ki})\text{exp}({\beta }^{\prime }{Z}_{ki})\right\}-\text{exp}\left\{-{\Lambda }_k(R_{ki})\text{exp}({\beta }^{\prime }{Z}_{ki})\right\}\right)\right\} \end{gathered}$$ (2)

where O_ki = {Y_ki, Z_ki}, ${\Delta }_{M_{ki}+1,ki}=1-{\sum }_{l=1}^{M_{ki}}{\Delta }_{lki}$, L_ki = max{U_lki : U_lki < T_ki, l = 0, … , M_ki} and R_ki min{U_lki : U_lki ≥ T_ki, l = 1, … , M_ki + 1}. We consider two methods of constructing the weights w_ki.

Method I (IPW-S) uses only information from a single disease: the weights are defined as

$$w_{ki}=\left(\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right)+\left(1-\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right){\xi }_i{p}_s^{-1}$$

for the case-cohort design, and defined as

$$w_{ki}=\left(\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right)(1-{\xi }_i){\eta }_{ki}{p}_{ck}^{-1}+\left(\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right){\xi }_i+\left(1-\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right){\xi }_i{p}_s^{-1}$$

for the generalized case-cohort design. This method is simple but it ignores the additional covariate measurements collected on subjects with the other diseases.

Method II (IPW-M) uses information from multiple diseases: the weights are defined as

$$w_{ki}=\left\{1-\prod _{k=1}^K\left(1-\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right)\right\}+\left\{\prod _{k=1}^K\left(1-\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right)\right\}{\xi }_i{p}_s^{-1}$$

for the case-cohort design, and defined as

$$\begin{gathered} w_{ki}=\left\{\sum _{k=1}^2\left(\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right){\eta }_{ki}{p}_{ck}^{-1}-\prod _{k=1}^2\left(\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right){\eta }_{ki}{p}_{ck}^{-1}\right\}(1-{\xi }_i) \\ +\left\{1-\prod _{k=1}^2\left(1-\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right)\right\}{\xi }_i+\left\{\prod _{k=1}^2\left(1-\sum _{l=1}^{M_{ki}}{\Delta }_{lki}\right)\right\}{\xi }_i{p}_s^{-1} \end{gathered}$$

for the generalized case-cohort design with K = 2. For simplicity, we only present the weight for K = 2. The generalization of this type of weight to K > 2 for the generalized case-cohort design can be defined similarly as in Kim et al.²³

The inverse probability weighting (IPW) method is commonly used in the literature to account for the sampling bias.^24,25,7 If the sampling probability (under which the sample is drawn from the target population) is known, then the inverse of this probability is used to weight the observations. Our weights are constructed based on the proposed sampling schemes. Take the IPW-S weight as an example. Under the case-cohort design, all cases have a weight of 1 and noncases in the subcohort have a weight of 1/p_s, where p_s is the sampling probability for the subcohort; under the generalized case-cohort design, all cases being selected at the second phase (except for those in the subcohort) have a weight of 1/p_ck, cases in the subcohort have a weight of 1, and noncases in the subcohort have a weight of 1/p_s, where p_ck is the sampling probability of cases for disease k who are not included in the subcohort, k = 1, … , K. The IPW-M weight can be constructed similarly except that we treat subjects with any disease as cases when assigning the weight for disease k. On the other hand, one may be interested in the comparison of the proposed method with the separate analyses where the weighted likelihood method for one disease is applied to each disease separately. In fact, when all the covariate effects are disease-specific, the proposed method with the IPW-S weight is equivalent to the separate analyses. If the IPW-M weight is used, the proposed method usually yields more efficient results than the separate analyses since the IPW-M weight can utilize information from multiple diseases. This can be seen from the simulation results below that show the efficiency gain of using the IPW-M weight compared to using the IPW-S weight.

The unknown parameters in the weighted log-pseudolikelihood function (2) include the regression coefficients and the cumulative baseline hazard functions. Our main interest is to estimate the regression coefficients β. However, different from handling right-censored data, there is no tool like the partial likelihood method that can be used to avoid estimating the baseline functions. Following Zhou et al.,¹⁸ we employ a sieve method with Bernstein polynomials to handle the unknown cumulative baseline functions {Λ₁, … , Λ_K}. In particular, let

$$\Theta =\left\{\theta =(\beta ,{\Lambda }_1,\dots ,{\Lambda }_K)\in \mathcal{B}\otimes {\mathcal{M}}_1\otimes \cdots \otimes {\mathcal{M}}_K\right\}$$

denote the parameter space, where $\mathcal{B}=\{\beta \in R^p:\Vert \beta \Vert \le M\}$ with M being a positive constant, and ${\mathcal{M}}_k$ is the collection of all continuous nonnegative and nondecreasing functions over the interval [σ_k, γ_k], k = 1, … , K. Here σ_k and γ_k are known constants usually taken in practice to be the lower and upper bounds of all examination times for T_k. We define the sieve parameter space as

$${\Theta }_n=\left\{{\theta }_n=(\beta ,{\Lambda }_{1n},\dots ,{\Lambda }_{Kn})\in \mathcal{B}\otimes {\mathcal{M}}_{1n}\otimes \cdots \otimes {\mathcal{M}}_{Kn}\right\},$$

where $\mathcal{B}$ is given above and, for k = 1, … , K,

$${\mathcal{M}}_{kn}=\left\{{\Lambda }_{kn}(t)=\sum _{l=0}^m{\varphi }_{kl}{B}_l(t,m,{\sigma }_k,{\gamma }_k),0\le {\varphi }_{k0}\le {\varphi }_{k1}\le \dots \le {\varphi }_{km}<\infty ,\sum _{l=0}^m\mid {\varphi }_{kl}\mid \le H_n\right\}$$

with B_l(t, m, σ_k, γ_k) being the Bernstein basis polynomials of degree m = o(n^ν) for some ν ∈ (0, 1), given by

$$B_l(t,m,{\sigma }_k,{\gamma }_k)=\left(\begin{array}{c}\hfill m\hfill \\ \hfill l\hfill \end{array}\right){\left(\frac{t-{\sigma }_k}{{\gamma }_k-{\sigma }_k}\right)}^l{\left(1-\frac{t-{\sigma }_k}{{\gamma }_k-{\sigma }_k}\right)}^{m-1},l=0,\dots ,m,$$

and H_n = O(n^a) for some a > 0. The constraints on the Bernstein coefficients ϕ_kl’s are imposed to guarantee that the estimates of the cumulative baseline hazard functions are nonnegative and nondecreasing. We define our estimate ${\widehat{\theta }}_n=\{{\widehat{\beta }}_n,{\widehat{\Lambda }}_{1n},\dots ,{\widehat{\Lambda }}_{Kn}\}$ to be the value of θ = {β, Λ₁, … , Λ_K} that maximizes the weighted log-pseudolikelihood function (2) over the sieve parameter space Θ_n.

We now establish the large sample properties of ${\widehat{\theta }}_n$. The forthcoming theorems will hold for both designs and both types of weights defined above. The proofs will go through under all four scenarios, noting that all weights are bounded and do not depend on θ, and satisfy E{w_ki∣Δ_1ki, … , Δ_Mki,ki} = 1. For any ${\theta }^1=({\beta }^1,{\Lambda }_1^1,\dots ,{\Lambda }_K^1)$ and ${\theta }^2=({\beta }^2,{\Lambda }_1^2,\dots ,{\Lambda }_K^2)$ in the parameter space $\Theta =\mathcal{B}\otimes {\mathcal{M}}_1\otimes \cdots \otimes {\mathcal{M}}_K$, define a distance:

$$d({\theta }^1,{\theta }^2)=\Vert {\beta }^1-{\beta }^2\Vert +\Vert {\Lambda }_1^1-{\Lambda }_1^2{\Vert }_2+\cdots +\Vert {\Lambda }_K^1-{\Lambda }_K^2{\Vert }_2,$$

where ∥v∥ denotes the Euclidean norm for a vector v and $\Vert {\Lambda }_k^1-{\Lambda }_k^2{\Vert }_2={\left[{\int }_{{\sigma }_k}^{{\gamma }_k}({\Lambda }_k^1(t)-{\Lambda }_k^2(t){)}^{2}dt\right]}^{1∕2}$. Let θ₀ = (β₀, Λ₁₀, … , Λ_K0) denote the true value of θ. The following theorems give the strong consistency, rate of convergence and asymptotic normality of the proposed estimator ${\widehat{\theta }}_n$ when n → ∞. The proofs of these theorems and the regularity conditions needed for them are given in the Appendix.

Theorem 1. Assume that Conditions (C1) - (C5) given in the Appendix hold. Then $d({\widehat{\theta }}_n,{\theta }_0)\to 0$ almost surely and $d({\widehat{\theta }}_n,{\theta }_0)=O_p(n^{-\text{min}\{(1-v)∕2,vr∕2\}})$, where ν ∈ (0, 1) such that m = o(n^ν) and r is defined in Condition (C4).

Theorem 2. Assume that Conditions (C1) - (C5) given in the Appendix hold. If 1/2r < ν < 1/2, then

$$n^{1∕2}({\widehat{\beta }}_n-{\beta }_0)={\left\{\sum _{k=1}^K{I}_k({\beta }_0)\right\}}^{-1}\left\{n^{-1∕2}\sum _{i=1}^n\sum _{k=1}^K{w}_{ki}{l}_k^{\ast }({\beta }_0,{\Lambda }_{k0};O_{ki})\right\}+o_p(1)\to N(0,\Sigma )$$

in distribution, where

$$\Sigma ={\left\{\sum _{k=1}^K{I}_k({\beta }_0)\right\}}^{-1}E\left\{{\left\{\sum _{k=1}^K{w}_k{l}_k^{\ast }({\beta }_0,{\Lambda }_{k0};O_k)\right\}}^{\otimes 2}\right\}{\left\{\sum _{k=1}^K{I}_k({\beta }_0)\right\}}^{-1}$$

with v^⨂2 = vv′ for a vector v, and I_k(β) and $l_k^{\ast }(\beta ,{\Lambda }_k;O_k)$ being the information and efficient score for β, respectively, based on the complete observation O_k = {Y_k, Z_k} for the kth disease, which will be given in the Appendix.

For variance estimation of ${\widehat{\beta }}_n$, one may use the finite sample version of Σ by treating l_n(θ) as a function of the finite-dimensional parameters {β, ϕ_kl : k = 1, … , K, l = 0, … , m}. Based on our simulation studies, this estimator performs reasonably well in practical settings. Nevertheless, since it involves calculating the inverse of the observed information matrix, it may not perform stably when the number of parameters is large. To provide a more stable variance estimator, we suggest a simple weighted bootstrap procedure proposed by Ma and Kosorok,²⁶ which is easy to implement and works well in our setting. In particular, let {u₁, … , u_n} denote n independent realizations of a bounded positive random variable u satisfying E(u) = 1 and var(u) = ϵ₀ < ∞. Define the new weights $w_{ki}^{\ast }=u_i{w}_{ki}$, i = 1, … , n. Let ${\widehat{\theta }}_n^{\ast }=\{{\widehat{\beta }}_n^{\ast },{\widehat{\Lambda }}_{1n}^{\ast },\dots ,{\widehat{\Lambda }}_{Kn}^{\ast }\}$ be the estimator that maximizes the weighted log-pseudolikelihood function with new weights $w_{ki}^{\ast }$. If we generate B samples of {u₁, … , u_n} and obtain the corresponding ${\widehat{\beta }}_n^{\ast }$, then the sample variance of these ${\widehat{\beta }}_n^{\ast }$’s rescaled by ϵ₀ can be used to estimate the variance of ${\widehat{\beta }}_n$. Although the likelihood function was derived under the working independence assumption, as shown in the simulation studies below, this variance estimator performs well regardless of the true dependence structure between failure times.

Although there are positivity and monotonicity constraints imposed on the coefficents of Bernstein polynomials, we can easily remove those constraints by reparametrization. For example, we may reparametrize the coefficients {ϕ_k0, … , ϕ_km} of the Bernstein polynomial ${\Lambda }_{kn}(t)={\sum }_{l=0}^m{\varphi }_{kl}{\mathit{B}}_l(t,m,{\sigma }_k,{\gamma }_k)$ as the cumulative sums of {$\text{exp}({\varphi }_{k0}^{\ast }),\dots ,\text{exp}({\varphi }_{km}^{\ast })$}, where the new parameters {${\varphi }_{k0}^{\ast },\dots ,{\varphi }_{km}^{\ast }$} do not have any constraints, k = 1, … , K. Thus, we only need to deal with the unconstrained optimization problem which can be solved by many existing algorithms. In our numerical studies, we employ the Quasi-Newton algorithm built in fminunc in Matlab. To implement the proposed method, one also needs to specify the degree of Bernstein polynomials m, which controls the smoothness of the approximation. For this, we suggest to consider several different values of m and choose the one that minimizes

$$AIC=-2l_n({\widehat{\theta }}_n)+2(p+Km+K).$$ (3)

More discussion about the choice of m will be given below.

4 ∣. SIMULATION STUDIES

We now conduct some simulation studies to evaluate the performance of the proposed method in practical settings. We consider two diseases and assume that their failure times T₁ and T₂ marginally follow the proportional hazards model: for k = 1, 2,

$${\Lambda }_k(t\mid Z)={\Lambda }_k(t)\text{exp}({\beta }_{k}Z),$$

where β₁ = log 1.5, β₂ = log 2, Λ₁(t) = log(1 + t/10), Λ₂(t) = 0.15t^0.8, and Z ~ N(0, 1). Moreover, we assume that the joint survival function of T₁ and T₂ is given by

$$S(t_1,t_2\mid Z)=C_{\theta }(S_1(t_1\mid Z),S_2(t_2\mid Z)),$$

where S_k(t∣Z) = exp(−Λ_k(t∣Z)) is the marginal survival function of T_k, k = 1, 2, and C_θ(u, v) = (u^−1/θ + v^−1/θ − 1)^−θ is the Clayton copula function with the association parameter θ that is related to Kendall’s tau as τ = 1/(2θ + 1). We consider two values of Kendall’s tau that represent weak and strong associations between T₁ and T₂: τ = 0.25 and 0.75, corresponding to θ = 1.5 and 0.1667, respectively.

To simulate interval-censored failure time data, we first generate a sequence of examination times U_1k < U_2k < ⋯ < U_Mk,k over the study period [0, ζ] as the cumulative sums of independent and identically distributed uniform random variables, that is, we keep sampling the uniform random variables until the cumulative sum is greater ζ. Let U_0k = 0 and U_Mk+1,k = ∞. We then define L_k = max{U_lk : U_lk < T_k, l = 0, … , M_k} and R_k = min{U_lk : U_lk ≥ T_k, l = 1, … , M_k + 1}. In particular, when generating the examination times, we make the uniform random variable dependent on the covariate Z by setting its mean equal to 0.05 I(∣Z∣ > 1)+0.1 I(∣Z∣ ≤ 1), where I(·) denotes an indicator function. We consider three values of ζ: 0.35, 0.55 and 1.35, yielding the proportion of cases in the cohort (i.e., disease rate) being (p_d1, p_d2) = (0.03, 0.07), (0.05, 0.10) and (0.12, 0.20) for T₁ and T₂, respectively, and yielding the average number of examination times (4,4), (7,7) and (17,17) for T₁ and T₂, respectively.

To simulate the (generalized) case-cohort sample, we first generate a cohort of size n = 500 or 1000 with the interval-censored observations Y_ki = {U_lki, Δ_lki = I(U_l−1,ki < T_ki ≤ U_lki) : l = 1, … , M_ki}, k = 1, 2, i = 1, … , n, where 0 = U_0ki < U_1ki < U_2ki < ⋯ < U_Mki,ki < U_Mki+1,ki = ∞ are a sequence of examination times generated as described above. We then select a subcohort using independent Bernoulli sampling with the success rate p_s = 0.2. Under the case-cohort design, the covariates are measured for the subcohort and all cases (i.e., all subjects with ${\sum }_{l=1}^{M_{ki}}{\Delta }_{lki}=1$ for k = 1 or 2), corresponding to (p_c1, p_c2) = (1, 1). Under the generalized case-cohort design, the covariate measurements are ascertained for the subcohort and a portion of cases selected using independent Bernoulli sampling with the success rates (p_c1, p_c2) = (0.5, 0.5).

We compare four methods: (i) the maximum likelihood method based only on the subcohort, denoted by ${\widehat{\beta }}_{sub}$; (ii) the maximum likelihood method based on a simple random sample (SRS) of the same size as the (generalized) case-cohort sample, denoted by ${\widehat{\beta }}_{srs}$; (iii) the proposed method with the IPW-S weight constructed based on single disease information, denoted by ${\widehat{\beta }}_{IPW-S}$; (iv) the proposed method with the IPW-M weight constructed based on information from multiple diseases, denoted by ${\widehat{\beta }}_{IPW-M}$. The degree of Bernstein polynomials is taken as m = 3. In the weighted bootstrap procedure for variance estimation, 200 bootstrap samples are used. The simulation results based on 1000 replicates are presented in Table 1&2 for the cohort size n = 500 and 1000, respectively. In the tables, “Bias" is the average regression parameter estimate minus the true value, “SSD" is the sample standard deviation of the parameter estimates,“ESE" is the average estimated standard error from the weighted bootstrap procedure, and “CP" is the coverage proportion of the 95% confidence interval based on the normal approximation.

TABLE 1.

Estimation results of the regression parameters β₁ and β₂ for two diseases, respectively, when one covariate is considered and the cohort size is n = 500

				β1 = log(1.5)				β2 = log(2)
(pd1, pd2)	(pc1, pc2)	Kendall’s τ		Bias	SSD	ESE	CP	Bias	SSD	ESE	CP
(0.03, 0.07)	(1,1)	0.25	${\widehat{\beta }}_{sub}$	0.103	1.014	0.478	0.72	0.054	0.506	0.398	0.90
			${\widehat{\beta }}_{srs}$	−0.006	0.780	0.427	0.80	0.029	0.396	0.343	0.91
			${\widehat{\beta }}_{IPW-S}$	0.004	0.293	0.281	0.94	0.032	0.231	0.219	0.93
			${\widehat{\beta }}_{IPW-M}$	−0.000	0.285	0.270	0.93	0.031	0.227	0.215	0.93
		0.75	${\widehat{\beta }}_{sub}$	0.079	0.922	0.487	0.74	0.048	0.622	0.409	0.89
			${\widehat{\beta }}_{srs}$	0.030	0.743	0.421	0.80	0.026	0.393	0.347	0.92
			${\widehat{\beta }}_{IPW-S}$	0.020	0.298	0.285	0.94	0.023	0.234	0.219	0.93
			${\widehat{\beta }}_{IPW-M}$	0.012	0.288	0.274	0.93	0.023	0.233	0.218	0.92
(0.05, 0.10)	(1,1)	0.25	${\widehat{\beta }}_{sub}$	0.028	0.768	0.418	0.84	0.041	0.409	0.346	0.92
			${\widehat{\beta }}_{srs}$	0.017	0.393	0.343	0.89	0.029	0.293	0.266	0.93
			${\widehat{\beta }}_{IPW-S}$	0.004	0.234	0.233	0.95	0.017	0.197	0.190	0.94
			${\widehat{\beta }}_{IPW-M}$	−0.000	0.225	0.221	0.94	0.015	0.193	0.185	0.94
		0.75	${\widehat{\beta }}_{sub}$	0.051	0.617	0.408	0.83	0.042	0.411	0.342	0.91
			${\widehat{\beta }}_{srs}$	0.037	0.418	0.349	0.90	0.024	0.308	0.274	0.93
			${\widehat{\beta }}_{IPW-S}$	0.024	0.237	0.236	0.95	0.033	0.216	0.192	0.92
			${\widehat{\beta }}_{IPW-M}$	0.018	0.227	0.224	0.95	0.033	0.217	0.191	0.92
(0.12, 0.20)	(0.5,0.5)	0.25	${\widehat{\beta }}_{sub}$	0.003	0.337	0.291	0.92	0.023	0.256	0.246	0.94
			${\widehat{\beta }}_{srs}$	0.005	0.233	0.229	0.95	0.006	0.192	0.188	0.95
			${\widehat{\beta }}_{IPW-S}$	0.008	0.206	0.200	0.94	0.013	0.181	0.172	0.93
			${\widehat{\beta }}_{IPW-M}$	−0.003	0.193	0.187	0.95	0.011	0.176	0.167	0.93
		0.75	${\widehat{\beta }}_{sub}$	0.012	0.331	0.297	0.93	0.034	0.262	0.249	0.94
			${\widehat{\beta }}_{srs}$	0.024	0.251	0.233	0.93	0.018	0.201	0.194	0.94
			${\widehat{\beta }}_{IPW-S}$	0.015	0.221	0.200	0.92	0.018	0.179	0.171	0.93
			${\widehat{\beta }}_{IPW-M}$	0.003	0.212	0.189	0.93	0.014	0.178	0.171	0.93

TABLE 2.

Estimation results of the regression parameters β₁ and β₂ for two diseases, respectively, when one covariate is considered and the cohort size is n = 1000

				β1 = log(1.5)				β2 = log(2)
(pd1, pd2)	(pc1, pc2)	Kendall’s τ		Bias	SSD	ESE	CP	Bias	SSD	ESE	CP
(0.03, 0.07)	(1,1)	0.25	${\widehat{\beta }}_{sub}$	0.062	0.540	0.356	0.85	0.053	0.292	0.277	0.94
			${\widehat{\beta }}_{srs}$	0.019	0.375	0.320	0.89	0.016	0.247	0.235	0.93
			${\widehat{\beta }}_{IPW-S}$	0.023	0.200	0.196	0.94	0.032	0.153	0.154	0.95
			${\widehat{\beta }}_{IPW-M}$	0.022	0.195	0.190	0.94	0.031	0.151	0.152	0.95
		0.75	${\widehat{\beta }}_{sub}$	0.021	0.514	0.380	0.84	0.025	0.307	0.278	0.92
			${\widehat{\beta }}_{srs}$	0.007	0.374	0.326	0.90	0.031	0.262	0.241	0.94
			${\widehat{\beta }}_{IPW-S}$	0.010	0.205	0.198	0.95	0.024	0.159	0.154	0.94
			${\widehat{\beta }}_{IPW-M}$	0.007	0.202	0.192	0.95	0.024	0.159	0.153	0.94
(0.05, 0.10)	(1,1)	0.25	${\widehat{\beta }}_{sub}$	0.018	0.337	0.305	0.92	0.023	0.243	0.234	0.93
			${\widehat{\beta }}_{srs}$	−0.001	0.270	0.244	0.93	−0.003	0.183	0.184	0.95
			${\widehat{\beta }}_{IPW-S}$	0.008	0.164	0.162	0.95	0.011	0.132	0.133	0.95
			${\widehat{\beta }}_{IPW-M}$	0.004	0.159	0.155	0.94	0.009	0.129	0.131	0.95
		0.75	${\widehat{\beta }}_{sub}$	0.014	0.333	0.298	0.91	0.008	0.246	0.231	0.94
			${\widehat{\beta }}_{srs}$	0.002	0.262	0.250	0.93	0.021	0.196	0.188	0.95
			${\widehat{\beta }}_{IPW-S}$	0.003	0.166	0.162	0.94	0.011	0.137	0.133	0.95
			${\widehat{\beta }}_{IPW-M}$	−0.000	0.159	0.155	0.94	0.011	0.136	0.133	0.95
(0.12, 0.20)	(0.5,0.5)	0.25	${\widehat{\beta }}_{sub}$	0.008	0.223	0.206	0.93	0.011	0.170	0.170	0.95
			${\widehat{\beta }}_{srs}$	−0.001	0.166	0.162	0.94	0.002	0.134	0.132	0.95
			${\widehat{\beta }}_{IPW-S}$	0.010	0.139	0.140	0.95	0.009	0.130	0.121	0.92
			${\widehat{\beta }}_{IPW-M}$	0.006	0.135	0.132	0.94	0.008	0.125	0.118	0.93
		0.75	${\widehat{\beta }}_{sub}$	0.002	0.205	0.204	0.94	0.017	0.178	0.169	0.94
			${\widehat{\beta }}_{srs}$	−0.003	0.167	0.163	0.94	0.001	0.138	0.136	0.94
			${\widehat{\beta }}_{IPW-S}$	0.001	0.144	0.139	0.95	0.007	0.128	0.121	0.94
			${\widehat{\beta }}_{IPW-M}$	−0.004	0.141	0.133	0.94	0.007	0.126	0.121	0.94

One can see from Table 1&2 that (i) the proposed method using either weight yields virtually unbiased estimates of the regression parameters; (ii) the variance estimates based on the weighted bootstrap procedure reflect the true variabilities; (iii) the coverage proportions of the 95% confidence intervals based on the normal approximation are close to the nominal level. For all scenarios considered, the proposed method using either weight is more efficient than the methods based on the subcohort or on a SRS of the same size as the (generalized) case-cohort sample. In terms of comparing the two types of weights for the proposed method, we have found that (i) the IPW-M weight that uses multiple diseases information generally yields more efficient estimates than the IPW-S weight that uses only single disease information; (ii) the efficiency gain achieved by using the IPW-M weight increases with the disease rates; in particular, the estimation of β₁ (β₂) gains more efficiency when the rate of T₂ (T₁) is higher, because it has more information to borrow from T₂ (T₁). Lastly, when the cohort size n increases from 500 to 1000, the performance of the proposed method improves as expected. In addition, for the degree of Bernstein polynomials, we have tried several other values (e.g., m = 4, 5 or 6) and obtained similar results. It seems that the proposed method is robust to the choice of m. Moreover, for the weighted bootstrap procedure, we examined different numbers of bootstrap samples, including 10, 20, … , 200. We found that the standard error estimates stabilize at around 50 bootstrap samples.

We also carry out a simulation study by considering two covariates. We assume that the failure times T₁ and T₂ marginally follow the proportional hazards model: for k = 1, 2,

$${\Lambda }_k(t\mid Z_1,Z_2)={\Lambda }_k(t)\text{exp}({\beta }_{k1}{Z}_1+{\beta }_{k2}{Z}_2),$$

where (β₁₁, β₁₂) = (−0.5, 0.5), (β₂₁, β₂₂) = (0.2, 0.4), Λ₁(t) = log(1+t/10), Λ₂(t) = 0.15t^0.8, Z₁ ~ N(0, 1), and Z₂ ~ Ber(0.5). We simulate interval-censored failure time data in the same way as before. Now we make the uniform random variable, which is used to generate the examination times, dependent on the covariates (Z₁, Z₂) by setting its mean equal to 0.05 I(∣Z₁∣ > 1 and Z₂ = 1) + 0.1 I(∣Z₁∣ ≤ 1 or Z₂ = 0). We consider two values of ζ: 0.55 and 1.05, yielding the disease rates (p_d1, p_d2) = (0.07, 0.10) and (0.13, 0.17) and the average number of examination times (6,6) and (12,12) for T₁ and T₂, respectively. We consider the cohort size n = 500 and the sampling probability for cases (p_c1, p_c2) = (1, 1) and (0.5, 0.5). The other setups are the same as before. The simulation results are given in Table 3. The proposed method performs well similarly as in the simulation study with one covariate.

TABLE 3.

Estimation results of the regression parameters (β₁₁, β₁₂) and (β₂₁, β₂₂) for two diseases, respectively, when two covariates are considered and the cohort size is n = 500

				β11 = −0.5				β12 = 0.5
(pd1, pd2)	(pc1, pc2)	Kendall’s τ		Bias	SSD	ESE	CP	Bias	SSD	ESE	CP
(0.07, 0.10)	(1,1)	0.25	${\widehat{\beta }}_{sub}$	−0.031	0.618	0.403	0.89	1.233	6.175	1.359	0.91
			${\widehat{\beta }}_{srs}$	0.000	0.344	0.304	0.91	0.388	2.941	0.821	0.97
			${\widehat{\beta }}_{IPW-S}$	−0.014	0.221	0.218	0.95	0.031	0.439	0.433	0.95
			${\widehat{\beta }}_{IPW-M}$	−0.014	0.216	0.213	0.95	0.030	0.437	0.427	0.95
		0.75	${\widehat{\beta }}_{sub}$	−0.025	0.533	0.414	0.93	0.949	6.022	1.343	0.92
			${\widehat{\beta }}_{srs}$	−0.004	0.359	0.320	0.92	0.400	3.397	0.859	0.97
			${\widehat{\beta }}_{IPW-S}$	−0.030	0.221	0.219	0.94	0.028	0.439	0.433	0.95
			${\widehat{\beta }}_{IPW-M}$	−0.030	0.218	0.216	0.94	0.024	0.434	0.429	0.95
(0.13, 0.17)	(0.5,0.5)	0.25	${\widehat{\beta }}_{sub}$	−0.028	0.310	0.303	0.94	0.272	2.388	0.738	0.97
			${\widehat{\beta }}_{srs}$	−0.008	0.246	0.234	0.93	0.053	0.862	0.510	0.98
			${\widehat{\beta }}_{IPW-S}$	−0.022	0.207	0.204	0.95	0.033	0.400	0.408	0.95
			${\widehat{\beta }}_{IPW-M}$	−0.023	0.206	0.199	0.94	0.031	0.392	0.403	0.96
		0.75	${\widehat{\beta }}_{sub}$	−0.017	0.325	0.305	0.94	0.134	1.925	0.708	0.97
			${\widehat{\beta }}_{srs}$	0.000	0.255	0.238	0.93	0.034	0.507	0.518	0.97
			${\widehat{\beta }}_{IPW-S}$	−0.025	0.217	0.204	0.93	0.022	0.408	0.408	0.95
			${\widehat{\beta }}_{IPW-M}$	−0.025	0.215	0.202	0.93	0.019	0.394	0.404	0.96
				β21 = log(1.5)				β22 = log(2)
(pd1, pd2)	(pc1, pc2)	Kendall’s τ		Bias	SSD	ESE	CP	Bias	SSD	ESE	CP
(0.07, 0.10)	(1,1)	0.25	${\widehat{\beta }}_{sub}$	0.019	0.368	0.329	0.93	0.180	2.559	0.867	0.96
			${\widehat{\beta }}_{srs}$	0.016	0.270	0.249	0.93	0.080	1.089	0.581	0.97
			${\widehat{\beta }}_{IPW-S}$	0.019	0.184	0.181	0.96	0.028	0.371	0.356	0.94
			${\widehat{\beta }}_{IPW-M}$	0.019	0.181	0.177	0.95	0.027	0.369	0.353	0.94
		0.75	${\widehat{\beta }}_{sub}$	0.015	0.358	0.325	0.91	0.114	1.747	0.828	0.98
			${\widehat{\beta }}_{srs}$	0.022	0.273	0.255	0.92	0.121	1.505	0.610	0.97
			${\widehat{\beta }}_{IPW-S}$	0.008	0.182	0.179	0.95	0.009	0.359	0.355	0.95
			${\widehat{\beta }}_{IPW-M}$	0.009	0.182	0.178	0.95	0.009	0.356	0.353	0.95
(0.13, 0.17)	(0.5,0.5)	0.25	${\widehat{\beta }}_{sub}$	0.018	0.263	0.252	0.94	0.021	0.557	0.559	0.98
			${\widehat{\beta }}_{srs}$	0.005	0.194	0.198	0.95	0.027	0.423	0.424	0.96
			${\widehat{\beta }}_{IPW-S}$	0.013	0.178	0.176	0.95	0.038	0.357	0.353	0.95
			${\widehat{\beta }}_{IPW-M}$	0.011	0.172	0.173	0.95	0.037	0.351	0.348	0.95
		0.75	${\widehat{\beta }}_{sub}$	−0.003	0.258	0.258	0.95	0.059	0.565	0.558	0.97
			${\widehat{\beta }}_{srs}$	0.011	0.206	0.200	0.94	0.012	0.450	0.428	0.96
			${\widehat{\beta }}_{IPW-S}$	−0.000	0.175	0.178	0.95	0.016	0.352	0.352	0.96
			${\widehat{\beta }}_{IPW-M}$	0.005	0.174	0.175	0.95	0.019	0.354	0.350	0.94

To assess the performance of our proposed estimator under sampling without replacement in the (generalized) case-cohort study, we consider the same simulation setup as in Table 1 except that sampling without replacement is used for selecting the subchort and cases. The results are presented in Table 4 and suggest that the proposed estimator still performs well in this situation. More discussion on the (generalized) case-cohort studies under sampling without replacement are given in Section 6.

TABLE 4.

Estimation results of the regression parameters β₁ and β₂ for two diseases, respectively, under the same setup as in Table 1 except that sampling without replacement is used in the (generalized) case-cohort study

				β1 = log(1.5)				β2 = log(2)
(pd1, pd2)	(pc1, pc2)	Kendall’s τ		Bias	SSD	ESE	CP	Bias	SSD	ESE	CP
(0.03, 0.07)	(1,1)	0.25	${\widehat{\beta }}_{sub}$	0.049	1.033	0.465	0.74	0.032	0.485	0.400	0.89
			${\widehat{\beta }}_{srs}$	0.046	0.720	0.417	0.81	0.007	0.376	0.330	0.92
			${\widehat{\beta }}_{IPW-S}$	0.009	0.317	0.288	0.92	0.024	0.240	0.220	0.92
			${\widehat{\beta }}_{IPW-M}$	−0.001	0.303	0.273	0.93	0.022	0.235	0.217	0.92
		0.75	${\widehat{\beta }}_{sub}$	0.077	1.021	0.468	0.71	0.024	0.525	0.401	0.91
			${\widehat{\beta }}_{srs}$	0.022	0.663	0.420	0.81	0.026	0.406	0.338	0.90
			${\widehat{\beta }}_{IPW-S}$	−0.012	0.286	0.283	0.94	0.016	0.233	0.220	0.93
			${\widehat{\beta }}_{IPW-M}$	−0.018	0.275	0.272	0.94	0.015	0.232	0.219	0.94
(0.05, 0.10)	(1,1)	0.25	${\widehat{\beta }}_{sub}$	0.067	0.587	0.424	0.85	0.043	0.396	0.343	0.92
			${\widehat{\beta }}_{srs}$	0.019	0.406	0.335	0.89	0.034	0.279	0.267	0.94
			${\widehat{\beta }}_{IPW-S}$	0.018	0.246	0.234	0.94	0.024	0.199	0.191	0.94
			${\widehat{\beta }}_{IPW-M}$	0.011	0.233	0.221	0.93	0.022	0.194	0.186	0.94
		0.75	${\widehat{\beta }}_{sub}$	0.015	0.573	0.409	0.84	0.047	0.390	0.343	0.93
			${\widehat{\beta }}_{srs}$	0.022	0.428	0.346	0.88	0.026	0.307	0.275	0.94
			${\widehat{\beta }}_{IPW-S}$	0.007	0.243	0.232	0.94	0.030	0.205	0.190	0.94
			${\widehat{\beta }}_{IPW-M}$	0.003	0.233	0.221	0.94	0.030	0.204	0.189	0.93
(0.12, 0.20)	(0.5,0.5)	0.25	${\widehat{\beta }}_{sub}$	0.013	0.319	0.297	0.94	0.022	0.258	0.245	0.93
			${\widehat{\beta }}_{srs}$	0.025	0.251	0.231	0.93	0.013	0.194	0.190	0.95
			${\widehat{\beta }}_{IPW-S}$	0.005	0.204	0.200	0.95	0.014	0.181	0.172	0.93
			${\widehat{\beta }}_{IPW-M}$	−0.001	0.201	0.189	0.93	0.011	0.175	0.166	0.93
		0.75	${\widehat{\beta }}_{sub}$	0.007	0.315	0.295	0.94	0.030	0.249	0.245	0.94
			${\widehat{\beta }}_{srs}$	0.012	0.233	0.233	0.94	0.016	0.192	0.194	0.96
			${\widehat{\beta }}_{IPW-S}$	0.014	0.203	0.200	0.94	0.027	0.181	0.171	0.93
			${\widehat{\beta }}_{IPW-M}$	0.011	0.193	0.191	0.95	0.024	0.179	0.171	0.93

5 ∣. APPLICATION TO THE ARIC STUDY

We now illustrate the proposed method using a dataset on incident diabetes and hypertension from the Atherosclerosis Risk in Communities (ARIC) study, a longitudinal epidemiological observational study conducted in four U.S. communities.²⁷ The cohort component of the ARIC study began in 1987 and the field center at each community randomly selected and recruited approximately 4,000 men and women aged 45-64 years. In particular, the four field centers are Forsyth County, NC (Center-F), Jackson, MS (Center-J), Minneapolis Suburbs, MN (Center-M) and Washington County, MD (Center-W). Minneapolis Suburbs and Washington County include white participants, Jackson has African American participants, and Forsyth County includes both white and African American participants. Every participant received extensive examinations, including medical, social and demographic data at enrollment, and then had follow-up examinations on average every three years with the first (baseline) occurring in 1987-1989, the second in 1990-1992, the third in 1993-1995, and the fourth in 1996-1998.

Diabetes was defined as a fasting glucose level of 126 mg/dL or above, a non-fasting glucose level of 200 mg/dL or above, self-reported physician diagnosis of diabetes, or use of diabetic medications. Hypertension was defined as systolic blood pressure ≥ 140 mm Hg, diastolic blood pressure ≥ 90 mm Hg, or self-reported use of antihypertensive medications. Sitting blood pressure was measured 3 times at each examination visit and the average of the last 2 readings was used. Since the participants were only periodically examined instead of being continuously monitored, the incident diabetes and hypertension were known only to occur between two examinations and thereby only interval-censored failure time data were obtained. We are interested in evaluating the effect of high-density lipoprotein (HDL) cholesterol level on the risks of incident diabetes and hypertension in white men younger than 55 years. The model also adjusts for age, smoking status, total cholesterol level, body mass index (BMI) and indicators of field centers.

The cohort of interest consists of 1743 individuals. During the study period, 172 developed diabetes, 384 had hypertension and 58 experienced both diseases. We first select a subcohort using independent Bernoulli sampling with a success rate of 0.2. The resulting subcohort has 334 individuals. Under the case-cohort design, the total sample size is 730. Under the generalized case-cohort design, we select some cases outside the subcohort for diabetes and hypertension, respectively, using independent Bernoulli sampling with a success rate of 0.9. The resulting sample includes 699 individuals in total. We analyze the data from each design and compare four methods: (i) the maximum likelihood method based only on the subcohort, denoted by ${\widehat{\beta }}_{sub}$; (ii) the proposed method with the IPW-S weight constructed based only on single disease information, denoted by ${\widehat{\beta }}_{IPW-S}$; (iii) the proposed method with the IPW-M weight constructed based on information from multiple diseases, denoted by ${\widehat{\beta }}_{IPW-M}$; (iv) the maximum likelihood method based on the full cohort, denoted by ${\widehat{\beta }}_{full}$. Since the full cohort data are available in this illustration, we consider the analysis based on full cohort to see if the proposed method gives reasonable results.

Regarding the degree of Bernstein polynomials used in the proposed method, we consider the m values ranging from 3 to 8 and m = 7 is selected according to the AIC criterion (3). In the weighted bootstrap procedure for variance estimation, 200 bootstrap samples are used. The estimation results for the regression parameters are summarized in Table 5. As seen from Table 5, under both case-cohort and generalized case-cohort designs, the proposed method using either weight reaches the same conclusion as the full cohort method: higher HDL cholesterol level and lower BMI are associated with lower risk of diabetes, while lower total cholesterol level and lower BMI are associated with lower risk of hypertension. In addition, under both designs, ${\widehat{\beta }}_{IPW-M}$ is generally more efficient than ${\widehat{\beta }}_{IPW-S}$ and both of them are more efficient than the MLE based on the subcohort only.

TABLE 5.

Analysis results for ARIC data on diabetes and hypertension

			Generalized Case-cohort Study				Case-cohort Study
	${\widehat{\beta }}_{sub}$		${\widehat{\beta }}_{IPW-S}$		${\widehat{\beta }}_{IPW-M}$		${\widehat{\beta }}_{IPW-S}$		${\widehat{\beta }}_{IPW-M}$		${\widehat{\beta }}_{full}$
Variables	Est.	SE	Est.	SE	Est.	SE	Est.	SE	Est.	SE	Est.	SE
Diabetes
Age	−0.112	0.072	0.008	0.039	0.012	0.036	0.013	0.037	0.017	0.035	0.025	0.028
BMI	0.154*	0.052	0.190*	0.033	0.137*	0.023	0.179*	0.031	0.133*	0.022	0.122*	0.016
HDL Cholesterol	−0.028	0.022	−0.026*	0.013	−0.032*	0.011	−0.029*	0.013	−0.032*	0.011	−0.031*	0.009
Total Cholesterol	0.003	0.004	0.002	0.003	0.004	0.002	0.002	0.003	0.003	0.002	0.001	0.002
Current Smoking	0.232	0.451	0.161	0.245	0.034	0.228	0.116	0.235	0.014	0.243	−0.030	0.179
Center-F	0.145	0.470	0.004	0.248	−0.055	0.237	0.086	0.268	0.035	0.222	0.024	0.191
Center-W	0.441	0.458	0.255	0.249	0.210	0.244	0.239	0.246	0.228	0.220	0.025	0.183
Hypertension
Age	0.009	0.041	0.017	0.027	0.005	0.026	0.012	0.028	0.001	0.025	0.005	0.018
BMI	0.048	0.034	0.075*	0.023	0.071*	0.018	0.077*	0.021	0.074*	0.019	0.063*	0.014
HDL Cholesterol	0.001	0.011	−0.005	0.007	−0.002	0.007	−0.004	0.006	−0.003	0.006	−0.001	0.005
Total Cholesterol	0.006*	0.003	0.005*	0.002	0.004*	0.002	0.004*	0.002	0.004*	0.002	0.003*	0.001
Current Smoking	−0.145	0.271	−0.043	0.176	0.002	0.175	−0.029	0.173	−0.008	0.168	0.036	0.117
Center-F	−0.069	0.285	−0.038	0.182	0.025	0.192	0.030	0.166	0.047	0.158	0.033	0.127
Center-W	0.339	0.265	0.270	0.181	0.245	0.170	0.286	0.172	0.262	0.171	0.078	0.123

^“*"indicates that the test for β = 0 based on the normal approximation yields a significant result at the level of 0.05.

It should be noted that there is some limitation of using interval-censoring methods to assess diabetes or hypertension incidence. For example, it is possible that a person is hypertensive at one assessment but becomes normotensive at the next without use of medications. Since there is no cure yet for hypertension and hypertensive patients usually rely on medications to keep the blood pressure under control, it is believed that such situation rarely occurs. Further note that the exact time to hypertension incidence cannot be observed. Thus, interval-censoring methods are often used in the literature as a decent approximative approach to assessing hypertension incidence. Among others, Schroeder et al.²⁸ and Zeng et al.¹⁹ employed interval-censoring methods to study the time to hypertension incidence based on data from the ARIC study. Nevertheless, the limitation of using interval-censoring methods in this situation should be recognized.

6 ∣. FINAL REMARKS

We consider the case-cohort designs for studies that concern multiple interval-censored failure times. We propose a marginal sieve weighted likelihood approach for regression analysis of case-cohort sampled multivariate interval-censored data under the proportional hazards model. We handle the unknown baseline functions via a sieve method that eases the computation burden and enjoys good large sample properties. We employ a weighted bootstrap procedure to obtain the variance estimate that is robust to the dependence structure between failure times. In addition, we develop two types of weights in our approach. The simulation results show that the proposed method using either weight performs well in practical settings, and the IPW-M weight that utilizes information from multiple diseases generally yields more efficient results than the IPW-S weight that uses only single disease information.

If the disease prevalence is high (say, > 40%), oversampling cases via the (generalized) case-cohort design may not help improve the study power as desired in comparison to simple random sampling. In this work, we are mainly concerned about the case-cohort design with rare diseases (say, prevalence < 10%) and the generalized case-cohort design with non-rare or not-so-rare diseases (say, prevalence < 30%). As shown in our simulation studies, when the diseases are rare or not-so-rare, the proposed method based on the (generalized) case-cohort sample is more efficient than the maximum likelihood method based on a simple random sample of the same size as the (generalized) case-cohort sample.

We focus on the (generalized) case-cohort designs under independent Bernoulli sampling for the subcohort and cases in the derivations of asymptotic properties of our proposed estimator. In practice, sampling without replacement is often used to select random samples. We conducted simulation studies to examine the performance of our proposed estimator when sampling without replacement is used. The results show that the proposed estimator performed well in the situations we considered. Based on the simulation results, we recommend our proposed method in practice for (generalized) case-cohort studies under sampling without replacement.

There are a few extensions or future research directions. One extension is to consider stratified sampling for the subcohort selection. Stratification can ensure proper representation of certain subgroups in the subcohort and improve the estimation of stratum specific quantities. It should not take much effort to extend our method to stratified sampling. Another extension of this work is to consider other models than the proportional hazards model. In some applications, the proportional hazards assumption may not be appropriate, or investigators may be interested in a different form of association between risk factors and disease outcomes. Alternative models such as the additive hazards model, proportional odds model, accelarated failure time model, and semiparametric transformation model could be of interest. Extending our method to these models warrants further research.

Moreover, one may notice that using the IPW-M weight does not seem to gain much efficiency compared to using the IPW-S weight, particularly when the diseases of concern are rare. The main reason is that there is only limited information to borrow from each other via the IPW-M weight when the disease rates are low. The marginal approach may also contribute to the limited gain of efficiency in using the IPW-M weight. Joint modeling approaches (e.g., copula or frailty-based methods) may help further improve the estimation efficiency. For example, by accounting for the dependence between failure times, the copula method is expected to be more efficient than the marginal approach that assumes working independence. Nevertheless, the copula method usually imposes some parametric assumption on the copula function and thus only allows for a specific type of dependence structure between failure times. Investigating the copula method, especially on more flexible copula models, would an interesting direction for future research.^20,21

APPENDIX

A PROOFS OF THEOREMS 1 AND 2

In this Appendix, we sketch the proofs of Theorems 1&2. Let $O^{\xi }=\{O_k^{\xi }=\{Y_k,{\tilde{\xi }}_k{Z}_k,{\tilde{\xi }}_k\}:k=1,\dots ,K\}$ denote a single observation, where Y_k = {U_lk, Δ_lk = I(U_l−1,k < T_k ≤ U_lk): l = 1, … , M_k} and ${\tilde{\xi }}_k$ indicates whether Z_k is observed. Before proving the theorems, we first describe the regularity conditions needed as follows: for k = 1, … , K,

(C1) The number of examination times M_k is positive with E(M_k) < ∞. There exists c_k > 0 such that P(min_0≤l≤Mk (U_l+1,k − U_lk) ≥ c_k ∣ M_k, Z_k) = 1. The union of the supports of {U_lk : l = 1, … , M_k} is contained in the interval [σ_k, γ_k], where 0 < σ_k < γ_k < ∞ and 0 < Λ_k0 (σ_k) < Λ_k0(γ_k) < ∞.

(C2) The distribution of Z_k has a bounded support in R^p and is not concentrated on any proper subspace of R^p. Also E{var(Z_k∣U_lk, M_k)}, l = 1, … , M_k, are positive definite.

(C3) β₀ lies in the interior of a compact set $\mathcal{B}\subset R^p$.

(C4) The first derivative of Λ_k0(·), denoted by ${\Lambda }_{k0}^{(1)}(\cdot )$, is strictly positive on [σ_k, γ_k] and Hölder continuous with exponent a ∈ (0, 1], that is, there exists some constant K > 0 such that $\mid {\Lambda }_{k0}^{(1)}(t_1)-{\Lambda }_{k0}^{(1)}(t_2)\mid \le K\mid t_1-t_2{\mid }^a$ for all t₁, t₂ ∈ [σ_k, γ_k]. Let r = 1 + a, which is related to the rate of convergence in Theorem 1.

(C5) The conditional densities of (U_lk, U_l+1,k) given M_k and Z_k, denoted by g_kl(u, v) for l = 0, … , M_k, have continuous second-order partial derivatives with respect to u and v when v − u ≥ c_k, and are continuously differentiable with respect to Z_k.

These conditions are mild and commonly used in the studies of interval-censored data.^29,18,19 We will prove Theorems 1&2 under these conditions by employing the empirical process theory and some nonparametric techniques. For the proofs, let P_n denote the empirical measure based on n independent observations and P denote the true probability measure. We define the covering number of the class ${\mathcal{L}}_n=\{l(\theta ,O^{\xi }):\theta \in {\Theta }_n\}$, where l(θ, O^ξ) is the weighted log-pseudolikelihood function based on a single observation O^ξ. In particular, l(θ, O^ξ) is given by

$$\begin{gathered} l(\theta ,O^{\xi })=\sum _{k=1}^K{l}_k(\theta ,O_k^{\xi })=\sum _{k=1}^K{w}_k{l}_k(\theta ,O_k) \\ =\sum _{k=1}^K{w}_k\left\{\sum _{l=1}^{M_k+1}{\Delta }_{lk}\log \left(\text{exp}\left\{-{\Lambda }_k(U_{l-1,k})\text{exp}({\beta }^{\prime }{Z}_k)\right\}-\text{exp}\left\{-{\Lambda }_k(U_{lk})\text{exp}({\beta }^{\prime }{Z}_k)\right\}\right)\right\} \\ =\sum _{k=1}^K{w}_k\left\{\log \left(\text{exp}\left\{-{\Lambda }_k(L_k)\text{exp}({\beta }^{\prime }{Z}_k)\right\}-\text{exp}\left\{-{\Lambda }_k(R_k)\text{exp}({\beta }^{\prime }{Z}_k)\right\}\right)\right\} \end{gathered}$$

where w_k is the weight, O_k = {Y_k, Z_k} is the complete observation for the kth disease, L_k = max{U_lk : U_lk < T_k, l = 0, … , M_k} and R_k = min{U_lk : U_lk ≥ T_k, l = 1, … , M_k + 1}. For any ϵ > 0, define the covering number $N\left(ϵ,{\mathcal{L}}_n,L_1(P_n)\right)$ as the smallest positive integer κ for which there exists {θ⁽¹⁾, … , θ^(κ)} such that

$$\underset{j\in \{1,\cdots ,\kappa \}}{\text{min}}\frac{1}{n}\sum _{i=1}^n\mid l(\theta ,O_i^{\xi })-l({\theta }^{(j)},O_i^{\xi })\mid <ϵ$$

for all θ ∈ Θ_n, where {$O_1^{\xi },\cdots ,O_n^{\xi }$} represent the observed data and for j = 1, … , κ, ${\theta }^{(j)}=({\beta }^{(j)},{\Lambda }_1^{(j)},\dots ,{\Lambda }_K^{(j)})\in {\Theta }_n$. If no such κ exists, define $N\left(ϵ,{\mathcal{L}}_n,L_1(P_n)\right)=\infty$.

A.1 Proof of Theorem 1

We first prove the strong consistency of ${\widehat{\theta }}_n$. Note that the weight w_k is bounded and does not depend on θ, and E{w_k∣O_k} = 1. Following the proof of Lemma 1 in Zhou et al.,¹⁸ we can show that the covering number of ${\mathcal{L}}_n$ satisfies

$$N\left(ϵ,{\mathcal{L}}_n,L_1(P_n)\right)\le \tilde{K}{H}_n^{K(m+1)}{ϵ}^{-(p+Km+K)}.$$

for some constant $\tilde{K}$, where m = o(n^ν) with ν ∈ (0, 1) is the degree of Bernstein polynomials, H_n = O(n^a) with a > 0 controls the size of the sieve space Θ_n, and p is the dimension of β. Furthermore, similarly as the proof of Lemma 2 in Zhou et al.,¹⁸ we can prove that

$$\underset{\theta \in {\Theta }_n}{\text{sup}}\mid P_n\{l(\theta ,O^{\xi })\}-P\{l(\theta ,O^{\xi })\}\mid \to 0\text{almost surely}.$$ (A.1)

Note that E(w_k∣O_k) = 1, then $P\{l(\theta ,O^{\xi })\}={\sum }_{k=1}^{K}P\{w_k{l}_k(\theta ,O_k)\}={\sum }_{k=1}^{K}P\{l_k(\theta ,O_k)\}$, where l_k(θ, O_k) is the log-likelihood function based on the complete observation O_k for the kth disease, k = 1, … , K. The true parameter value θ₀ maximizes P{l_k(θ, O_k)} and thus also maximizes P{l(θ, O^ξ)}. Let M(θ, O^ξ) = −l(θ, O^ξ). Define K_ϵ = {θ : d(θ, θ₀) ≥ ϵ, θ ∈ Θ_n} for ϵ > 0,

$${\zeta }_{1n}=\underset{\theta \in {\Theta }_n}{\text{sup}}\mid P_n\{M(\theta ,O^{\xi })\}-P\{M(\theta ,O^{\xi })\}\mid \text{and}{\zeta }_{2n}=P_n\{M({\theta }_0,O^{\xi })\}-P\{M({\theta }_0,O^{\xi })\}.$$

Then

$$\underset{K_{ϵ}}{\text{inf}}P\{M(\theta ,O^{\xi })\}=\underset{K_{ϵ}}{\text{inf}}\left\{P\{M(\theta ,O^{\xi })\}-P_n\{M(\theta ,O^{\xi })\}+P_n\{M(\theta ,O^{\xi })\}\right\}\le {\zeta }_{1n}+\underset{K_{ϵ}}{\text{inf}}{P}_n\{M({\theta }_0,O^{\xi })\}.$$ (A.2)

If ${\widehat{\theta }}_n\in K_{ϵ}$, then we have

$$\underset{K_{ϵ}}{\text{inf}}{P}_n\{M(\theta ,O^{\xi })\}=P_n\{M({\widehat{\theta }}_n,O^{\xi })\}\le P_n\{M({\theta }_0,O^{\xi })\}={\zeta }_{2n}+P\{M({\theta }_0,O^{\xi })\}.$$ (A.3)

Define δ_ϵ = inf_Kϵ P{M (θ, O^ξ)} − P{M (θ₀, O^ξ)}. Under Condition (C2), using similar arguments as those in Zhang et al.,²⁹ we can prove δ_ϵ > 0. It follows from (A.2) and (A.3) that

$$\underset{K_{ϵ}}{\text{inf}}P\{M(\theta ,O^{\xi })\}\le {\zeta }_{1n}+{\zeta }_{2n}+P\{M({\theta }_0,O^{\xi })\}={\zeta }_n+P\{M({\theta }_0,O^{\xi })\}$$

with ζ_n = ζ_1n + ζ_2n, and hence ζ_n ≥ δ_ϵ. This gives $\{{\widehat{\theta }}_n\in K_{ϵ}\}\subseteq \{{\zeta }_n\ge {\delta }_{ϵ}\}$, and by (A.1) and the strong law of large numbers, we have both ζ_1n → 0 and ζ_2n → 0 almost surely. Therefore, ${\cap }_{k=1}^{\infty }{\cup }_{n=k}^{\infty }\{\widehat{\theta }\in K_{ϵ}\}\subseteq {\cap }_{k=1}^{\infty }{\cup }_{n=k}^{\infty }\{{\zeta }_n\ge {\delta }_{ϵ}\}$, which proves that $d({\widehat{\theta }}_n,{\theta }_0)\to 0$ almost surely.

Now we will show the convergence rate of ${\widehat{\theta }}_n$ by using Theorem 3.4.1 of van der Vaart and Wellner.³⁰ Below we use $\tilde{K}$ to denote a universal positive constant which may differ from place to place. First note from Theorem 1.6.2 of Lorentz³¹ that there exists a Bernstein polynomial Λ_kn0 such that ∥Λ_kn0 − Λ_k0∥_∞ = O(m^−r/2), k = 1, … , K. Define θ_n0 = (β₀, Λ_1n0, … , Λ_Kn0). Then we have d(θ_n0, θ₀) = O(n^−rν/2). Let ${\rho }_n=\tilde{K}{n}^{-rv∕2}$. For any ρ > ρ_n, define the class of functions ${\mathcal{F}}_{\rho }=\{l(\theta ,O^{\xi })-l({\theta }_{n0},O^{\xi }):\theta \in {\Theta }_n$, ρ/2 < d(θ, θ_n0) ≤ ρ} for a given single observation O^ξ. One can easily show that $P\{l({\theta }_0,O^{\xi })-l({\theta }_{n0},O^{\xi })\}\le \tilde{K}d({\theta }_0,{\theta }_{n0})\le \tilde{K}{n}^{-rv∕2}$. Also under Condition (C2), using similar arguments as those in Zhang et al.,²⁹ we obtain $P\{l({\theta }_0,O^{\xi })-l(\theta ,O^{\xi })\}\ge \tilde{K}{d}^2({\theta }_0,\theta )$. Thus, for large n, we have $P\{l(\theta ,O^{\xi })-l({\theta }_{n0},O^{\xi })\}=P\{l(\theta ,O^{\xi })-l({\theta }_0,O^{\xi })\}+P\{l({\theta }_0,O^{\xi })-l({\theta }_{n0},O^{\xi })\}\le -\tilde{K}{\rho }^2+\tilde{K}{n}^{-rv∕2}=-\tilde{K}{\rho }^2$, for any $l(\theta ,O^{\xi })-l({\theta }_{n0},O^{\xi })\in {\mathcal{F}}_{\rho }$.

Following the calculations in Shen and Wong,³² we can establish that for 0 < ε < ρ, $\log N_{[]}(\epsilon ,{\mathcal{F}}_{\rho },L_2(P))\le \tilde{K}\tilde{N}\log (\rho ∕\epsilon )$ with $\tilde{N}=K(m+1)$, where $N_{[]}(ϵ,{\mathcal{F}}_{\rho },L_2(P))$ represents the bracketing number defined as the minimum number of ϵ-brackets under the L₂(P) norm needed to cover ${\mathcal{F}}_{\rho }$.³⁰ Moreover, some algebraic manipulations yield that $P\{l(\theta ,O^{\xi })-l({\theta }_{n0},O^{\xi }){\}}^2\le \tilde{K}{\rho }^2$ for any $l(\theta ,O^{\xi })-l({\theta }_{n0},O^{\xi })\in {\mathcal{F}}_{\rho }$. Under Conditions (C1)-(C5), it is easy to see that ${\mathcal{F}}_{\rho }$ is uniformly bounded. Therefore, by Lemma 3.4.2 of van der Vaart and Wellner,³⁰ we obtain

$$E_P{\Vert n^{1∕2}(P_n-P)\Vert }_{{\mathcal{F}}_{\rho }}\le \tilde{K}{J}_{[]}\left(\rho ,{\mathcal{F}}_{\rho },L_2(P)\right)\left[1+\frac{J_{[]}\left(\rho ,{\mathcal{F}}_{\rho },L_2(P)\right)}{{\rho }^2{n}^{1∕2}}\right]$$

where $J_{[]}\left(\rho ,{\mathcal{F}}_{\rho },L_2(P)\right)={\int }_0^{\rho }{\left[1+\log N_{[]}\left(\epsilon ,{\mathcal{F}}_{\rho },L_2(P)\right)\right]}^{1∕2}d\epsilon \le \tilde{K}{\tilde{N}}^{1∕2}\rho$. This yields ${\varphi }_n(\rho )={\tilde{N}}^{1∕2}\rho +\tilde{N}∕n^{1∕2}$. It is easy to see that ϕ_n(ρ)/ρ is decreasing in ρ. Let r_n = n^{min{rν/2,(1−ν)/2}}. Then $r_n\le \tilde{K}{\rho }_n^{-1}$ and $r_n^2{\varphi }_n(1∕r_n)\le \tilde{K}{n}^{1∕2}$.

Finally note that $P_n\left\{l({\widehat{\theta }}_n,O^{\xi })-l({\theta }_{n0},O^{\xi })\right\}\ge 0$ and $d({\widehat{\theta }}_n,{\theta }_{n0})\le d({\widehat{\theta }}_n,{\theta }_0)+d({\theta }_0,{\theta }_{n0})\to 0$ in probability. Thus, by applying Theorem 3.4.1 of van der Vaart and Wellner,³⁰ we have $r_{n}d({\widehat{\theta }}_n,{\theta }_{n0})=O_p(1)$. This together with d(θ_n0, θ₀) = O(n^−rν/2) yields that $r_{n}d({\widehat{\theta }}_n,{\theta }_0)=O_p(1)$, which completes the proof.

A.2 Proof of Theorem 2

Now we establish the asymptotic normality of ${\widehat{\beta }}_n$. Recall that the weighted log-pseudolikelihood function based on a single observation O^ξ is given by

$$l(\beta ,\Lambda ;O^{\xi })=\sum _{k=1}^K{w}_k{l}_k(\beta ,{\Lambda }_k;O_k),$$

where θ = (β, Λ), Λ = (Λ₁, … , Λ_K), l_k(β, Λ_k; O_k) is the log-likelihood function based on the complete observation O_k for the kth disease, the weight w_k is bounded and does not depend on θ, and E{w_k∣O_k} = 1, k = 1, … , K. The score function for β is

$${\stackrel{.}{l}}_{\beta }(\beta ,\Lambda ;O^{\xi })=\sum _{k=1}^K{w}_k{\stackrel{.}{l}}_{k,\beta }(\beta ,{\Lambda }_k;O_k),$$

where i_k,β(β, Λ_k; O_k) is the derivative of l_k(β, Λ_k; O_k) with respect to β. To obtain the score operator for Λ = (Λ₁, … , Λ_K), we consider a one-dimensional submodel Λ_ϵ,h, where h = (h₁, … , h_K)^T is a vector of functions with h_k ∈ L₂[σ_k, γ_k]. In particular, the submodel specifies that Λ_k,ϵ,hk = (1 + ϵh_k)Λ_k, k = 1, … , K. The score function for Λ along this submodel is

$$\begin{gathered} {\stackrel{.}{l}}_{\Lambda }(\beta ,\Lambda ;O^{\xi })(h)={\phantom{\mid }\frac{\partial }{\partial ϵ}l(\beta ,{\Lambda }_{1,ϵ,h_1},\dots ,{\Lambda }_{K,ϵ,h_K};O^{\xi })\mid }_{ϵ=0} \\ ={\phantom{\mid }\sum _{k=1}^K{w}_k\frac{\partial }{\partial ϵ}l(\beta ,{\Lambda }_{k,ϵ,h_k};O_k)\mid }_{ϵ=0} \\ =\sum _{k=1}^K{w}_k{\stackrel{.}{l}}_{k,{\Lambda }_k}(\beta ,{\Lambda }_k;O_k)(h_k), \end{gathered}$$

where ${\stackrel{.}{l}}_{k,{\Lambda }_k}(\beta ,{\Lambda }_k;O_k)(h_k)$ is the score function for Λ_k along the submodel Λ_k,ϵ,hk = (1 + ϵh_k)Λ_k based on the complete observation O_k for the kth disease. According to Zhang et al.,²⁹ the least favorable direction $h_k^{\ast }$ minimizing $P\Vert {\stackrel{.}{l}}_{k,\beta }({\beta }_0,{\Lambda }_{k0};O_k)-{\stackrel{.}{l}}_{k,{\Lambda }_k}({\beta }_0,{\Lambda }_{k0};O_k)(h_k){\Vert }^2$ exists. Denote the efficient score as

$$l_k^{\ast }(\beta ,\Lambda ;O_k)={\stackrel{.}{l}}_{k,\beta }(\beta ,{\Lambda }_k;O_k)-{\stackrel{.}{l}}_{k,{\Lambda }_k}(\beta ,{\Lambda }_k;O_k)(h_k^{\ast })$$

and the information for β as

$$I_k(\beta )=P\{l_k^{\ast }(\beta ,{\Lambda }_k;O_k{)}^{\otimes 2}\}.$$

In particular, we have

$$I_k({\beta }_0)=-{\ddot{l}}_{k,\beta \beta }({\beta }_0,{\Lambda }_{k0};O_k)+{\ddot{l}}_{k,{\Lambda }_k\beta }({\beta }_0,{\Lambda }_{k0};O_k)(h_k^{\ast }),$$ (A.4)

where ${\ddot{l}}_{k,\beta \beta }$ and ${\ddot{l}}_{k,{\Lambda }_k\beta }(h_k^{\ast })$ are the derivatives of ${\stackrel{.}{l}}_{k,\beta }$ and ${\stackrel{.}{l}}_{k,{\Lambda }_k}(h_k^{\ast })$ with respect to β, respectively. Note that the least favorable direction $h_k^{\ast }$ satisfies

$$P\left\{{\ddot{l}}_{k,\beta {\Lambda }_k}({\beta }_0,{\Lambda }_{k0};O_k)(h_k)-{\ddot{l}}_{k,{\Lambda }_k{\Lambda }_k}({\beta }_0,{\Lambda }_{k0};O_k)(h_k^{\ast },h_k)\right\}=0,$$ (A.5)

for all h_k ∈ L₂[σ_k, γ_k], where ${\ddot{l}}_{k,\beta {\Lambda }_k}(h_k)$ and ${\ddot{l}}_{k,{\Lambda }_k{\Lambda }_k}(h_k^{\ast },h_k)$ are the derivatives of ${\stackrel{.}{l}}_{k,\beta }$ and ${\stackrel{.}{l}}_{k,{\Lambda }_k}(h_k^{\ast })$ along the submodel Λ_k,ϵ,hk = (1 + ϵh_k)Λ_k, respectively.

It is clear that $P_n\{{\stackrel{.}{l}}_{\beta }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })\}=0$. Similarly as the proof of (B1) in Zhang et al.,²⁹ we can show that $P_n\{{\stackrel{.}{l}}_{\Lambda }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })(h^{\ast })\}=o_p(n^{-1∕2})$ given that ν > 1/2r, where $h^{\ast }=(h_1^{\ast },\dots ,h_K^{\ast })$. Since E{w_k∣O_k} = 0, we have

$$P\{{\stackrel{.}{l}}_{\beta }({\beta }_0,{\Lambda }_0;O^{\xi })\}=\sum _{k=1}^{K}P\{w_k{\stackrel{.}{l}}_{k,\beta }({\beta }_0,{\Lambda }_0;O_k)\}=\sum _{k=1}^{K}P\{{\stackrel{.}{l}}_{k,\beta }({\beta }_0,{\Lambda }_0;O_k)\}=0.$$

Similarly, $P\{{\stackrel{.}{l}}_{\Lambda }({\beta }_0,{\Lambda }_0;O^{\xi })(h^{\ast })\}=0$. Hence,

$$\begin{gathered} (P_n-P)\{{\stackrel{.}{l}}_{\beta }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })\}=-[P\{{\stackrel{.}{l}}_{\beta }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })\}-P\{{\stackrel{.}{l}}_{\beta }({\beta }_0,{\Lambda }_0;O^{\xi })\}], \\ (P_n-P)\{{\stackrel{.}{l}}_{\Lambda }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })(h^{\ast })\}=-[P\{{\stackrel{.}{l}}_{\Lambda }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })(h^{\ast })\}-P\{{\stackrel{.}{l}}_{\Lambda }({\beta }_0,{\Lambda }_0;O^{\xi })(h^{\ast })\}]+o_p(n^{-1∕2}) \end{gathered}$$

We apply Taylor series expansions about (β₀, Λ₀) to the right-hand sides of the above two equations. Based on the convergence rate derived in Theorem 1, $d({\widehat{\theta }}_n,{\theta }_0{)}^2=o_p(n^{-1∕2})$ when 1/2r < ν < 1/2. Thus, we obtain

$$(P_n-P)\{{\stackrel{.}{l}}_{\beta }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })\}=-P{\ddot{l}}_{\beta \beta }({\widehat{\beta }}_n-{\beta }_0)-P{\ddot{l}}_{\beta \Lambda }({\widehat{\Lambda }}_n-{\Lambda }_0)+o_p(n^{-1∕2}),$$ (A.6)

$$(P_n-P)\{{\stackrel{.}{l}}_{\Lambda }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })(h^{\ast })\}=-P{\ddot{l}}_{\Lambda \beta }(h^{\ast })({\widehat{\beta }}_n-{\beta }_0)-P{\ddot{l}}_{\Lambda \Lambda }(h^{\ast },{\widehat{\Lambda }}_n-{\Lambda }_0)+o_p(n^{-1∕2}),$$ (A.7)

where ${\ddot{l}}_{\beta \beta }={\sum }_{k=1}^K{w}_k{\ddot{l}}_{k,\beta \beta }$, ${\ddot{l}}_{\beta \Lambda }={\sum }_{k=1}^K{w}_k{\ddot{l}}_{k,\beta {\Lambda }_k}$, ${\ddot{l}}_{\Lambda \beta }={\sum }_{k=1}^K{w}_k{\ddot{l}}_{k,{\Lambda }_k\beta }$, ${\ddot{l}}_{\Lambda \Lambda }={\sum }_{k=1}^K{w}_k{\ddot{l}}_{k,{\Lambda }_k{\Lambda }_k}$, and all of these derivatives are evaluated at (β₀, Λ₀). From (A.5), we have

$$P{\ddot{l}}_{\beta \Lambda }({\widehat{\Lambda }}_n-{\Lambda }_0)=\sum _{k=1}^{K}P{\ddot{l}}_{k,\beta {\Lambda }_k}({\widehat{\Lambda }}_{kn}-{\Lambda }_{k0})=\sum _{k=1}^{K}P{\ddot{l}}_{k,{\Lambda }_k{\Lambda }_k}(h_k^{\ast },{\widehat{\Lambda }}_{kn}-{\Lambda }_{k0})=P{\ddot{l}}_{\Lambda \Lambda }(h^{\ast },{\widehat{\Lambda }}_n-{\Lambda }_0)$$

Therefore, the difference of (A.6) and (A.7) yields

$$(P_n-P)\{{\stackrel{.}{l}}_{\beta }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })-{\stackrel{.}{l}}_{\Lambda }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })(h^{\ast })\}=-P[{\ddot{l}}_{\beta \beta }-{\ddot{l}}_{\Lambda \beta }(h^{\ast })]({\widehat{\beta }}_n-{\beta }_0)+o_p(n^{-1∕2}).$$ (A.8)

Note that

$$\begin{gathered} {\stackrel{.}{l}}_{\beta }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })-{\stackrel{.}{l}}_{\Lambda }({\widehat{\beta }}_n,{\widehat{\Lambda }}_n;O^{\xi })(h^{\ast })=\sum _{k=1}^K{w}_k\{{\stackrel{.}{l}}_{k,\beta }({\widehat{\beta }}_n,{\widehat{\Lambda }}_{kn};O_k)-{\stackrel{.}{l}}_{k,{\Lambda }_k}({\widehat{\beta }}_n,{\widehat{\Lambda }}_{kn};O_k)(h_k^{\ast })\} \\ =\sum _{k=1}^K{w}_k{l}_k^{\ast }({\widehat{\beta }}_n,{\widehat{\Lambda }}_{kn};O_k). \end{gathered}$$

From the proof of Theorem 2 in Zhang et al.,²⁹ we have that $l_k^{\ast }({\widehat{\beta }}_n,{\widehat{\Lambda }}_{kn};O_k)$ belongs to a Donsker class and converges in the L₂(P)-norm to $l_k^{\ast }({\beta }_0,{\Lambda }_{k0};O_k)$. Moreover, from (A.4), $P[{\ddot{l}}_{\beta \beta }-{\ddot{l}}_{\Lambda \beta }(h^{\ast })]={\sum }_{k=1}^{K}P[{\ddot{l}}_{k,\beta \beta }-{\ddot{l}}_{k,{\Lambda }_k\beta }(h_k^{\ast })]=-{\sum }_{k=1}^K{I}_k({\beta }_0)$, where I_k(β₀) is nonsingular. Thus, (A.8) entails $n^{1∕2}({\widehat{\beta }}_n-{\beta }_0)=O_p(1)$ and further yields

$$n^{1∕2}({\widehat{\beta }}_n-{\beta }_0)={\left\{\sum _{k=1}^K{I}_k({\beta }_0)\right\}}^{-1}\left\{n^{-1∕2}\sum _{i=1}^n\sum _{k=1}^K{w}_{ki}{l}_k^{\ast }({\beta }_0,{\Lambda }_{k0};O_{ki})\right\}+o_p(1).$$

Since $E\left\{{\sum }_{k=1}^K{w}_k{l}_k^{\ast }({\beta }_0,{\Lambda }_{k0};O_k)\right\}={\sum }_{k=1}^{K}E\{l_k^{\ast }({\beta }_0,{\Lambda }_{k0};O_k)\}=0$, we have

$$n^{1∕2}({\widehat{\beta }}_n-{\beta }_0)\to N(0,\Sigma )$$

in distribution, with

$$\Sigma ={\left\{\sum _{k=1}^K{I}_k({\beta }_0)\right\}}^{-1}E\left\{{\left\{\sum _{k=1}^K{w}_k{l}_k^{\ast }({\beta }_0,{\Lambda }_{k0};O_k)\right\}}^{\otimes 2}\right\}{\left\{\sum _{k=1}^K{I}_k({\beta }_0)\right\}}^{-1}.$$

This completes the proof of Theorem 2.

DATA AVAILABILITY STATEMENT

The ARIC data used in Section 5 were obtained from the NHLBI Biologic Specimen and Data Repository Information Coordinating Center. The authors are not permitted to supply the ARIC data due to the data use agreement.