Estimated Quadratic Inference Function for Correlated Failure Time Data

Summary:

An estimated quadratic inference function method is proposed for correlated failure time data with auxiliary covariates. The proposed method makes efficient use of the auxiliary information for the incomplete exposure covariates, and preserves the property of the quadratic inference function method that requires the covariates to be completely observed. It can improve the estimation efficiency and easily deal with the situation when the cluster size is large. The proposed estimator which minimizes the estimated quadratic inference function is shown to be consistent and asymptotically normal. A chi-squared test based on the estimated quadratic inference function is proposed to test hypotheses about the regression parameters. The small-sample performance of the proposed method is investigated through extensive simulation studies. The proposed method is then applied to analyze the SOLVD data as an illustration.

1. Introduction

Multivariate failure time data are common in survival analysis when recording of times to two or more distinct events or failures on each subject, and have been extensively investigated, see Wei, Lin, and Weissfeld (1989), Cai and Prentice (1995, 1997), Xue, Wang, and Qu (2010), Yan, Zhou, and Cai (2017), and Prentice and Zhao (2019), among others. The marginal proportional hazard model inherits many advantages of the well-known Cox model (Cox, 1972), and has been used widely in modeling multivariate failure time data, in which the exposure covariates are usually assumed to be complete. One issue that arises frequently in survival analysis is that some expensive variables could be only observed for a subset of the entire cohort due to financial limitations or technology difficulties. However, in many cases there exist other auxiliary variables related to the incomplete ones that may be cheap to measure and are available for the entire cohort. For example, in the Study of Left Ventricular Dysfunction (SOLVD) prevention trial, it is of interest to assess the effects of risk factors on the times to heart failure and the first myocardial infarction. One of the important risk factors is patient’s ejection fraction, the standard measurement of which was to use a standardized radionucleotide technique. Because the technique is too expensive to be used on every patient, only 108 among the total of 4228 patients have their EF measured (LVEF), however, a less precise but cheaper measurement of EF was ascertained for all the patients using a nonstandardized technique. EF was considered as an auxiliary variable of LVEF. Statistical studies have shown that using the auxiliary variables in place of the variables of interest directly could yield bias in the parameter estimation for the variables of interest. Some methods have been developed to correct the bias and increase efficiency in multivariate failure time analysis in the presence of auxiliary covariates. Fan and Wang (2009) used the local linear approximation method to estimate the induced relative risk function, and proposed an estimated partial likelihood estimator for the marginal hazard model. Liu, Zhou, and Cai (2009) and Liu, Wu, and Zhou (2010) proposed an estimated pseudo-partial likelihood method for marginal hazard model with distinguishable baseline hazard function with discrete and continuous auxiliary covariates, respectively. Liu et al. (2012) worked on this problem under the marginal hazard model with common baseline hazard function. Yan et al. (2017) proposed a class of updated-estimators to improve the estimation efficiency by making use of the covariate information, including the auxiliary information, under the case-cohort design.

However, all the aforementioned studies on multivariate failure time data used independent working correlation matrix in their estimation procedures, which could result in loss of efficiency as pointed out by Cai and Prentice (1995, 1997) and Xue et al. (2010). To improve the estimation efficiency, Cai and Prentice (1995, 1997) incorporated a weight matrix to the partial likelihood score equation and found that their methods perform well when the pairwise dependence within clusters is strong and censoring is not severe. However, their methods are hard to implement when the cluster size is large due to the computation burden of weighting correlation matrix. Xue et al. (2010) proposed to apply quadratic inference function (QIF) method which was first proposed by Qu, Lindsay, and Li (2000) as an extension of the generalized estimating equations (GEE). Their method is easy to implement and is more robust against outlying observations.

In this article, we extend the QIF method by Xue et al. (2010) to analyze multivariate failure time data when the primary covariate is only available in a randomly selected subset and a discrete auxiliary variable for the primary variable is available for the entire cohort. We use the auxiliary information to empirically estimate the QIF to obtain the estimate of the regression parameters and refer to our method as EQIF. In addition, a chi-squared test based on the EQIF is developed for hypothesis testing. Our method inherits the advantage of QIF in improving estimation efficiency and easy implementation.

The remainder of the article is organized as follows. In Section 2, we setup the notation and propose the EQIF method. In Section 3, we establish the asymptotic properties of the proposed procedure and propose a chi-squared test for model parameters. The finite-sample performance of the proposed procedures is assessed through extensive simulation studies, and the simulation results are presented in Section 4. Section 5 illustrates the proposed method through the analysis of a real data from the SOLVD study. Some concluding remarks are given in Section 6 and the technical proofs are outlined in the Appendix.

2. Estimated Quadratic Inference Function

Suppose that the entire cohort consists of n independent clusters, and each cluster contains K correlated subjects. Let (i,k) denote the kth subject in the ith cluster. Let ${\tilde{T}}_{ik}$ be the failure time, C_ik denote the censoring time for ${\tilde{T}}_{ik}$, and $T_{ik}=\min \left({\tilde{T}}_{ik},C_{ik}\right)$ be the observed failure time. Let ${\mathrm{\Delta }}_{ik}=I\left({\tilde{T}}_{ik}\hspace{0.17em}⩽\hspace{0.17em}{C}_{ik}\right)$ be the failure indicator. Let ${\tilde{Z}}_{ik}\left(t\right)$ be a p-vector of possibly time-dependent covariates and interaction terms between them. We assume that the censoring times C_i = (C_i1,··· ,C_iK) and the failure times ${\tilde{\mathit{T}}}_i=\left({\tilde{T}}_{i1},\cdots ,{\tilde{T}}_{iK}\right)$ are independent, conditional on ${\tilde{\mathit{Z}}}_i^{\text{T}}\left(t\right)=\left({\tilde{Z}}_{i1}\left(t\right),\dots ,{\tilde{Z}}_{iK}\left(t\right)\right)$. Consider the marginal proportional hazard model with the conditional hazard function for (i,k) taking the following form:

$${\lambda }_{ik}\left(t\right)={\lambda }_{0k}\left(t\right)\exp \left\{{\beta }^{\text{T}}{\tilde{Z}}_{ik}\left(t\right)\right\},$$ (1)

where λ_0k(t) is an unspecified marginal baseline hazard function, and β is a p-vector of unknown regression parameters to be estimated. It is worth noting that model (1) includes the failure-type-specific model (Wei et al., 1989; Greene and Cai, 2004) ${\lambda }_{ik}\left(t\right)={\lambda }_{0k}\left(t\right)\exp \left\{{\beta }_k^{\text{T}}{\tilde{Z}}_{ik}\left(t\right)\right\}$, which allows for different covariate effect for different k, as a special case by defining $\beta ={\left({\beta }_1^{\text{T}},\dots ,{\beta }_k^{\text{T}},\dots ,{\beta }_K^{\text{T}}\right)}^{\text{T}}$ and ${\tilde{Z}}_{ik}^{\ast }\left(t\right)={\left({\mathbf{0}}^{\text{T}},\dots ,{\tilde{Z}}_{ik}^{\text{T}}\left(t\right),\dots ,{\mathbf{0}}^{\text{T}}\right)}^{\text{T}}$.

Let $N_{ik}\left(t\right)={\mathrm{\Delta }}_{ik}I\left(T_{ik}\hspace{0.33em}⩽\hspace{0.33em}t\right)$ and $Y_{ik}\left(t\right)=I\left(T_{ik}\hspace{0.33em}⩾\hspace{0.33em}t\right)$ be the observed counting process and the at-risk indicator process, respectively. Let ${\Lambda }_{0k}\left(t\right)={\int }_0^t{\lambda }_{0k}\left(u\right)\text{d}u$ be the marginal cumulative baseline hazard function for the kth failure type. Given β, Breslow (1972) proposed an estimator for Λ_0k(t) as

$${\widehat{\mathrm{\Lambda }}}_{0k}\left(t;\beta \right)={\int }_0^t\frac{{\sum }_{i=1}^n\text{d}{N}_{ik}\left(u\right)}{{\sum }_{i=1}^n{Y}_{ik}\left(u\right)\exp \left\{{\beta }^{\text{T}}{\tilde{Z}}_{ik}\left(u\right)\right\}}.$$ (2)

Let

$$M_{ik}\left(t;{\beta }_0\right)=N_{ik}\left(t\right)-{\int }_0^t{Y}_{ik}\left(u\right){\lambda }_{0k}\left(u\right)\exp \left\{{\beta }_0^{\text{T}}{\tilde{Z}}_{ik}\left(u\right)\right\}\text{d}u$$ (3)

be the marginal martingale process, where β₀ is the true parameter. Replace Λ_0k(t) in (3) with its estimate, we have the estimate of M_ik(t;β₀) as

$${\widehat{M}}_{ik}\left(t;{\beta }_0\right)=N_{ik}\left(t\right)-{\int }_0^t{Y}_{ik}\left(u\right)\exp \left\{{\beta }_0^{\text{T}}{\tilde{Z}}_{ik}\left(u\right)\right\}{\widehat{\mathrm{\Lambda }}}_{0k}\left(\text{d}t;{\beta }_0\right).$$

Write ${\widehat{\mathit{M}}}_i\left(t;\beta \right)={\left({\widehat{M}}_{i1}\left(t;\beta \right),\dots ,{\widehat{M}}_{iK}\left(t;\beta \right)\right)}^{\text{T}}$. To improve the estimation efficiency, Cai and Prentice (1995) added a weight matrix based on the inverse of correlation matrix of marginal martingales into the partial likelihood score equation. However, their method is computation intensive when the cluster size is large because of the need to estimate the correlation parameters and the calculation of an inverse of high dimension matrix. To reduce the computation burden, following the idea of Liang and Zeger (1986), the parameters can be estimated by solving the following generalized estimating equation:

$$\sum _{i=1}^n{\int }_0^{\tau }{\tilde{\mathit{Z}}}_i^{\text{T}}\left(t\right){\Xi }_i^{1/2}\left(\beta ,t\right){\mathbf{\Sigma }}^{-1}{\Xi }_i^{-1/2}\left(\beta ,t\right){\widehat{\mathit{M}}}_i\left(\text{d}t;\beta \right)=0,$$ (4)

where Ξ_i(β,t) = diag{λ_i1(t),…,λ_iK(t)}, and Σ is the common working correlation matrix. To further reduce the computation burden, Qu et al. (2000) introduced the quadratic inference function (QIF) by approximating the inverse of the working correlation by a linear combination of several pre-specified symmetric basis matrices, namely, ${\mathbf{\Sigma }}^{-1}\approx {\sum }_{j=1}^{m_0}{\alpha }_j{B}_j$, where B₁ is the identity matrix, B_j (j = 2,··· ,m₀) are known symmetric matrices.

The advantage of QIF approach is that it does not require estimation of nuisance parameters α_js, because the generalized estimating equation (4) is an approximate linear combination of elements in the following estimating function

$$\begin{array}{l}{\mathit{G}}_n\left(\beta \right)=\frac{1}{n}\sum _{i=1}^n{\mathit{g}}_i\left(\beta \right)\\ =\frac{1}{n}\sum _{i=1}^n\left(\begin{array}{c}{\int }_0^{\tau }{\tilde{\mathit{Z}}}_i^T\left(t\right){\Xi }_i^{1/2}\left(\beta ,t\right)B_1{\Xi }_i^{-1/2}\left(\beta ,t\right){\widehat{\mathit{M}}}_i\left(\text{d}t;\beta \right)\\ ⋮\\ {\int }_0^{\tau }{\tilde{\mathit{Z}}}_i^{\text{T}}\left(t\right){\Xi }_i^{1/2}\left(\beta ,t\right)B_{m_0}{\Xi }_i^{-1/2}\left(\beta ,t\right){\widehat{\mathit{M}}}_i\left(\text{d}t;\beta \right)\end{array}\right).\end{array}$$ (5)

However, β cannot be estimated by setting each component in (5) to be zero because the number of the estimating equations in (5) is greater than the dimension of the unknown parameters. Instead, Xue et al. (2010) proposed to estimate β by minimizing the following QIF function,

$${\mathit{Q}}_n\left(\beta \right)={\mathit{G}}_n^{\text{T}}\left(\beta \right){\left\{n^{-2}\sum _{i=1}^n{\mathit{g}}_i\left(\beta \right){\mathit{g}}_i^{\text{T}}\left(\beta \right)\right\}}^{-1}{\mathit{G}}_n\left(\beta \right).$$ (6)

An important aspect in the implementation of QIF is that the diagonal matrix Ξ_i(β,t) involves the unknown baseline hazard function λ_0k(t). Xue et al. (2010) suggested a kernel smoothed estimator ${\widehat{\lambda }}_{0k}\left(t;\beta \right)$ as follows,

$${\widehat{\lambda }}_{0k}\left(t;\beta \right)=\frac{1}{{\nu }_k}\sum _{i=1}^n\kappa \left(\frac{t-T_{ik}}{{\nu }_k}\right)\mathrm{\Delta }{\widehat{\mathrm{\Lambda }}}_{0k}\left(T_{ik};\beta \right),$$ (7)

where κ(·) is the Epanechnikov kernel function with ν_k being the rule-of-thumb bandwidth, and $\mathrm{\Delta }{\widehat{\mathrm{\Lambda }}}_{0k}\left(t;\beta \right)={\widehat{\mathrm{\Lambda }}}_{0k}\left(t;\beta \right)-{\widehat{\mathrm{\Lambda }}}_{0k}\left(t-;\beta \right)$ with ${\widehat{\mathrm{\Lambda }}}_{0k}\left(t;\beta \right)$ being the Breslow estimator given in formula (2).

In this article, we are interested in the situation that the primary covariate can only be ascertained in a validation subset. Let ${\tilde{Z}}_{ik}\left(t\right)={\left(X_{ik}^{\text{T}}\left(t\right),Z_{ik}^{\text{T}}\left(t\right)\right)}^{\text{T}}$, where X_ik(t) is the covariate that can only be observed in a validation subset and Z_ik(t) is the vector of remaining covariates that can be precisely measured in the entire cohort. Accordingly, write the true parameter as β₀ = (β₁₀,β₂₀) with β₁₀ and β₂₀ pertaining to X_ik(t) and Z_ik(t), respectively. Suppose that there exists a time-dependent auxiliary variable A(t) for the primary covariate X(t), and A(t) can be observed for entire cohort. As an auxiliary variable to X(t), A(t) provides no additional information to the regression model given X(t), i.e.,

$${\lambda }_{ik}\left(t;Z_{ik}\left(t\right),X_{ik}\left(t\right),A_{ik}\left(t\right)\right)\equiv {\lambda }_{ik}\left(t;Z_{ik}\left(t\right),X_{ik}\left(t\right)\right).$$

Let η_ik = 1 or 0 indicate whether the subject (i,k) is in the validation set or not. Denote the kth marginal validation set by V_k = {i : η_ik = 1}, and the non-validation set by ${\overline{V}}_k=\left\{i:{\eta }_{ik}=0\right\}$. When η_ik = 0, only {T_ik,∆_ik,Y_ik(t),Z_ik(t),A_ik(t)} is observed and the induced hazards function given Z_ik(t),A_ik(t) (Prentice, 1982) is

$$\begin{gathered} {\lambda }_{ik}\left(t;Z_{ik}\left(t\right),A_{ik}\left(t\right)\right)={\lambda }_{0k}\left(t\right)e^{{\beta }_{\left(2\right)}^{\text{T}}{Z}_{ik}\left(t\right)}E\left\{e^{{\beta }_{\left(1\right)}^{\text{T}}{X}_{ik}\left(t\right)}|Y_{ik}\left(t\right)=1,\hspace{0.33em}{Z}_{ik}\left(t\right),A_{ik}\left(t\right)\right\} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} ={\lambda }_{0k}\left(t\right)e^{{\beta }_{\left(2\right)}^{\text{T}}{Z}_{ik}\left(t\right)}E\left\{e^{{\beta }_{\left(1\right)}^{\text{T}}{X}_{ik}\left(t\right)}|Y_{ik}\left(t\right)=1,\hspace{0.33em}{A}_{ik}^{\ast }\left(t\right)\right\}, \end{gathered}$$

where $A_{ik}^{\ast }\left(t\right)$ denotes all the auxiliary information, which may include A_ik(t) and the part from Z_ik(t).

For simplicity, we write

$${\psi }_{ik}\left(\beta ,t\right)=\exp \left\{{\beta }_{\left(2\right)}^{\text{T}}{Z}_{ik}\left(t\right)\right\}E\left\{e^{{\beta }_{\left(1\right)}^{\text{T}}{X}_{ik}\left(t\right)}|Y_{ik}\left(t\right)=1,\hspace{0.33em}{A}_{ik}^{\ast }\left(t\right)\right\},$$

$${\phi }_{ik}\left(\beta ,t\right)=\exp \left\{{\beta }_{\left(1\right)}^{\text{T}}{X}_{ik}\left(t\right)+{\beta }_{\left(2\right)}^{\text{T}}{Z}_{ik}\left(t\right)\right\}.$$

Then the relative risk function can be written in general as r_ik(β,t) = φ_ik(β,t)η_ik+ψ_ik(β,t)(1− η_ik). We consider the situation that A^∗ are categorical variables so that ψ_ik(β,t) can be empirically estimated by

$${\widehat{\psi }}_{ik}\left(\beta ,t\right)=\frac{{\sum }_{j\in V_k}{Y}_{jk}\left(t\right)I\left\{A_{jk}^{\ast }\left(t\right)=A_{ik}^{\ast }\left(t\right)\right\}\exp \left\{{\beta }_{\left(1\right)}^{\text{T}}{X}_{jk}\left(t\right)\right\}}{{\sum }_{j\in V_k}{Y}_{jk}\left(t\right)I\left\{A_{jk}^{\ast }\left(t\right)=A_{ik}^{\ast }\left(t\right)\right\}}{e}^{{\beta }_{\left(2\right)}^{\text{T}}{Z}_{ik}\left(t\right)},$$ (8)

hence the estimated relative risk function is ${\widehat{r}}_{ik}\left(\beta ,t\right)={\phi }_{ik}\left(\beta ,t\right){\eta }_{ik}+{\widehat{\psi }}_{ik}\left(\beta ,t\right)\left(1-{\eta }_{ik}\right)$.

Let ${\widehat{r}}_{ik}^{\left(a\right)}\left(\beta ,t\right)\left(a=0,1,2\right)$ denote the ath derivative of ${\widehat{r}}_{ik}\left(\beta ,t\right)$ with respect to β. Replacing $\exp \left\{{\beta }^{\text{T}}{\tilde{Z}}_{ik}\left(t\right)\right\}$ by ${\widehat{r}}_{ik}\left(t;\beta \right)$ and ${\tilde{Z}}_{ik}\left(t\right)$ by ${\widehat{r}}_{ik}^{\left(1\right)}\left(\beta ,t\right)/{\widehat{r}}_{ik}\left(\beta ,t\right)$ in the notations in (2)–(7), we obtain estimated versions of Λ_0k(t), M_ik(t;β), g_i(β), G_n(β), Q_n(β) and λ_0k(t), denoted as ${\tilde{\mathrm{\Lambda }}}_{0k}\left(t\right)$, ${\tilde{M}}_{ik}\left(t;\beta \right)$, ${\widehat{\mathit{g}}}_i\left(\beta \right)$, ${\widehat{\mathit{G}}}_n\left(\beta \right)$, ${\widehat{\mathit{G}}}_n\left(\beta \right)$, ${\widehat{\mathit{Q}}}_n\left(\beta \right)$ and ${\tilde{\lambda }}_{0k}\left(t\right)$. Then β can be estimated by minimizing ${\widehat{\mathit{Q}}}_n\left(\beta \right)$. We referred ${\widehat{\beta }}_Q$ as EQIF estimator.

The proposed EQIF method inherits the merit of the QIF method to provide an inference function for testing of β. Specifically, let β be partitioned into ξ and ζ, where ξ is the vector of parameters of interest with dimension q, and ζ is a vector of nuisance parameters with dimension p − q. Suppose we are interested in testing

$$H_0:\xi ={\xi }_0\hspace{1em}\text{versus}\hspace{1em}{H}_1:\xi \ne {\xi }_0,$$ (9)

we propose a test statistic

$$\mathcal{T}={\widehat{\mathit{Q}}}_n\left({\xi }_0,\tilde{\zeta }\right)-{\widehat{\mathit{Q}}}_n\left(\widehat{\xi },\widehat{\zeta }\right),$$ (10)

where $\tilde{\zeta }=\underset{\zeta }{\text{argmin}}{\widehat{\mathit{Q}}}_n\left({\xi }_0,\zeta \right)$, $\left(\widehat{\xi },\widehat{\zeta }\right)=\underset{\left(\xi ,\zeta \right)}{\text{argmin}}{\widehat{\mathit{Q}}}_n\left(\xi ,\zeta \right)$.

3. Asymptotic Properties

In this section, we will study the asymptotic properties of EQIF estimator. Theorem 1 and Theorem 2 establish the consistency and asymptotic normality of the resulting estimator. Theorem 3 shows the asymptotic distribution of test statistic $\mathcal{T}$.

Theorem 1: Under the regularity conditions (C1)–(C8) in Appendix, there exists a local minimizer ${\widehat{\beta }}_Q$ of object function ${\widehat{\mathit{Q}}}_n\left(\beta \right)$ satisfying that ${\widehat{\beta }}_Q$ is a consistent estimator of the true parameter β₀.

Theorem 2: Under the regularity conditions (C1)–(C8) in Appendix, $\sqrt{n}\left({\widehat{\beta }}_Q-{\beta }_0\right)$ is asymptotically normally distributed with mean zero and variance matrix ${\mathbf{\Sigma }}_{EQIF}\left({\beta }_0\right)={\left({\mathit{J}}_0^{\text{T}}{\mathit{W}}_0^{-1}{\mathit{J}}_0\right)}^{-1}$, where W₀ is defined as in (A.2) in Appendix, and J₀ = J(β₀) with

$$\mathit{J}\left({\beta }_0\right)=\left(\begin{array}{c}\Gamma \left({\beta }_0,B_1\right)\\ ⋮\\ \Gamma \left({\beta }_0,B_{m_0}\right)\end{array}\right),$$

where Γ(β₀, B) is given as in (A.1) in Appendix.

The asymptotic covariance Σ_EQIF (β₀) can be consistently estimated. We describe the details in Appendix.

It is obvious that if the working correlation structure is taken to be identity matrix, the proposed estimator ${\widehat{\beta }}_Q$ is the same as the EPPL estimator proposed by Liu et al. (2009). Moreover, when the validation fractions ρ_k = 1, namely, all the exposure covariates are complete, the EQIF estimator is equivalent to the QIF estimator in Xue et al. (2010).

Theorem 3: Under the regularity conditions (C1)–(C8) given in Appendix, $\mathcal{T}$ in (10) asymptotically follows chi-squared distribution with q degrees of freedom under the null hypothesis in (9).

The proof of Theorem 3 utilizes the results in Xue et al. (2010). According to Xue et al. (2010), an appropriate test statistic for (9) is ${\mathcal{T}}^{\ast }={\mathit{Q}}_n\left({\xi }_0,\widetilde{{\zeta }^{\ast }}\right)-{\mathit{Q}}_n\left(\widehat{{\xi }^{\ast }},\widehat{{\zeta }^{\ast }}\right)$, where $\widetilde{{\zeta }^{\ast }}=\underset{\zeta }{\text{argmin}}{\mathit{Q}}_n\left({\xi }_0,\zeta \right)$ and $\left(\widehat{{\xi }^{\ast }},\widehat{{\zeta }^{\ast }}\right)=\underset{\left(\xi ,\zeta \right)}{\text{argmin}}{\mathit{Q}}_n\left(\xi ,\zeta \right)$. They showed that ${\mathcal{T}}^{\ast }$ as an asymptotical chi-squared distribution with degree of freedom being q. We further showed in the Appendix that $\mathcal{T}={\mathcal{T}}^{\ast }+o_p\left(1\right)$. Hence, $\mathcal{T}$ asymptotically follows ${\chi }_q^2$ under H₀.

4. Simulation Studies

We examine the finite-sample properties of the EQIF method and the proposed chi-squared test statistic via simulation studies. The EQIF method is compared with the EPPL method (Liu et al., 2009) and the QIF method based only on the validation set. When applying QIF method, we take the exchangeable working correlation.

The multivariate failure times are generated from Clayton and Cuzick (1985) model, where the joint survival distribution function takes the form

$$S\left(t_1,\dots ,t_K;{\tilde{Z}}_1,\dots ,{\tilde{Z}}_K\right)={\left\{\sum _{k=1}^K\exp \left({\theta }^{-1}{\lambda }_{0k}{t}_k{e}^{{\beta }_k^{\text{T}}{\tilde{Z}}_k}\right)-\left(K-1\right)\right\}}^{-\theta },$$

where θ > 0 characterizes the within-cluster dependence of failure times, with a decreasing value of θ corresponding to an increasing positive correlation. The generated failure times satisfied the marginal hazards model λ_ik(t) = λ_0k(t)exp{β_1kX_ik + β_2kZ_ik}.

We set the baseline hazard function λ_0k = 1, the dependence parameter θ is 0.2, 0.5 or 2, which represents a varying level of dependence between failure times within a cluster. We consider the situations for exposures which have equal or different effect for different failure types. The censoring times are uniform variates on (0,c), where c is chosen to yield different censoring rates.

The auxiliary covariate A_k is generated as follows. We first generate ${\tilde{A}}_k=X_k+{\epsilon }_k$, where ε_k follows a normal distribution N(0,σ²), the positive parameter σ controls the strength of association between ${\tilde{A}}_k$ and X_k. We set σ = 0.1 or 1 to represent the situations where A_k is a strong or weak auxiliary variable. The auxiliary covariate A_k then takes the value 1, 2, 3, or 4 based on whether ${\tilde{A}}_k$ is in the interval (−∞,q₁], (q₁,q₂], (q₂,q₃], (q₃,∞), where q₁, q₂, q₃ are the quartiles of ${\tilde{A}}_k$. The validation set V_k is randomly sampled from the entire cohort with equal probability. Moreover, we set the validation fraction ρ_k = 0.5. For each setting, we simulate 1000 replicates. The following simulation settings are considered.

Simulation 1.

We simulate K=4 and 8 failure types. The number of independent clusters n = 200. The effects of X and Z pertaining to failure type k are set as β₁₁ = ··· = β_1K = β₍₁₎ = 0.5 and β₂₁ = ··· = β_2K = β₍₂₎ = 0.2. The partly observed covariates $X_{ik}\text{s}$ are generated from standard normal distribution. The fully observed covariates $Z_{ik}\text{s}$ are generated from Bernoulli distribution with success probability 0.5.

Simulation 2.

Set n = 300 and 600. We simulate K=2 failure types. The covariate effects vary with failure type. Set (β₁₁,β₂₁)^T = (ln(2),−ln(1.3))^T, (β₁₂,β₂₂)^T = (0.5,−0.1)^T. Both $X_{ik}\text{s}$ and $Z_{ik}\text{s}$ are generated from standard normal distribution.

The estimate of ψ_ik(β,t) using (8) could be zero when there are no subjects with auxiliary variable equaling A_ik being left in the kth marginal validation set. Under this circumstance, we replace the missing value of X_ik with $X_{i^{\ast }k}$, where subject (i^∗,k) is the one who has the same value of auxiliary variable as subject (i,k) and has the largest observed failure time in the kth marginal validation set. In addition, we assume exchangeable working correlation for estimation equations in G_n(β) and ${\widehat{\mathit{G}}}_n\left(\beta \right)$, that is, m₀ = 2 and the basis matrix B₂ is 0 on the diagonal and 1 off the diagonal. Furthermore, since the Epanechnikov kernel function is of bounded support, ${\tilde{\lambda }}_{0k}\left(t\right)$ could be zero. In this situation, we replace ${\tilde{\lambda }}_{0k}\left(t\right)$ by ${\tilde{\mathrm{\Lambda }}}_{0k}\left(\tau \right)/n$.

To assess the estimation performance, we report the absolute value of empirical bias (|Bias|), the sample standard deviation (SD), the average of estimated standard errors (SE), the coverage rate of the nominal 95% confidence intervals (CR), and the sample relative efficiency (RE), which is the ratio of the empirical variance of the EPPL estimator to those of the other estimators.

Tables 1–2 report the simulation results for Simulation 1 when censoring rate is around 20% and 60%, respectively. We make the following observations. (i) The estimates obtained from all considered methods are approximately unbiased. In addition, the estimators of the asymptotic standard errors are approximately equal to the empirical standard deviations. The corresponding 95% confidence intervals calculated by the estimated standard errors provide reasonable coverage rates. This suggests that the estimates of asymptotic standard errors for all methods work well. (ii) The EPPL estimator and the EQIF estimator, which utilize the auxiliary information, gain more efficiency than the QIF estimator based only on the validation set. When σ is larger (e.g. σ = 1), which means A is less informative about X, the efficiency gains from both EPPL and EQIF is smaller. (iii) The proposed EQIF method is more efficient than EPPL in all the considered settings. With the increasing of the degree of within-cluster correlation (i.e. the decreasing of θ) or the decreasing of censoring rate, the relative efficiency gain of EQIF to EPPL (REs) increases. (iv) As the cluster size K increases, the empirical standard deviations (SDs) decrease. That is reasonable because of the increase in the total size of data.

Table 1.

Simulation results under marginal model λ_ik(t) = λ_0k(t)exp{0.5X_ik + 0.2Z_ik}, the censoring rate is 20%.

			β(1) = 0.5					β(2) = 0.2
K	θ	Method	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE
			σ = 0.1
4	0.2	EPPL	0.30	0.52	0.51	0.94	–	0.03	0.80	0.81	0.95	–
		QIF_V	0.32	0.62	0.58	0.93	0.72	0.16	1.05	0.98	0.93	0.58
		EQIF	0.28	0.39	0.37	0.93	1.81	0.02	0.47	0.48	0.96	2.91
	0.5	EPPL	0.33	0.52	0.49	0.93	–	0.00	0.82	0.81	0.94	–
		QIF_V	0.36	0.62	0.58	0.93	0.69	0.11	1.09	1.02	0.93	0.56
		EQIF	0.31	0.41	0.39	0.94	1.56	0.00	0.59	0.57	0.94	1.92
	2	EPPL	0.40	0.48	0.47	0.94	–	0.01	0.83	0.81	0.94	–
		QIF_V	0.33	0.63	0.59	0.94	0.59	0.15	1.16	1.09	0.93	0.50
		EQIF	0.38	0.45	0.43	0.94	1.14	0.04	0.77	0.74	0.94	1.14
8	0.2	EPPL	0.37	0.41	0.40	0.94	–	0.00	0.55	0.58	0.96	–
		QIF_V	0.22	0.45	0.44	0.94	0.83	0.00	0.71	0.67	0.93	0.62
		EQIF	0.15	0.32	0.31	0.94	1.67	0.03	0.31	0.32	0.96	3.14
	0.5	EPPL	0.29	0.39	0.38	0.93	–	0.02	0.55	0.57	0.95	–
		QIF_V	0.36	0.44	0.43	0.95	0.78	0.06	0.74	0.70	0.94	0.55
		EQIF	0.02	0.31	0.30	0.95	1.56	0.02	0.37	0.38	0.95	2.22
	2	EPPL	0.24	0.34	0.34	0.95	–	0.07	0.55	0.57	0.96	–
		QIF_V	0.39	0.43	0.42	0.94	0.63	0.06	0.81	0.76	0.94	0.45
		EQIF	0.02	0.31	0.31	0.95	1.21	0.07	0.49	0.50	0.96	1.24
			σ = 1
4	0.2	EPPL	2.19	0.58	0.56	0.91	–	0.87	0.80	0.84	0.96	–
		QIF_V	0.32	0.62	0.58	0.93	0.89	0.16	1.05	0.98	0.93	0.58
		EQIF	2.31	0.42	0.41	0.88	1.90	0.87	0.49	0.54	0.96	2.64
	0.5	EPPL	2.18	0.58	0.54	0.90	–	0.92	0.82	0.84	0.95	–
		QIF_V	0.36	0.62	0.58	0.93	0.86	0.11	1.09	1.02	0.93	0.56
		EQIF	2.24	0.46	0.43	0.89	1.60	0.91	0.60	0.62	0.95	1.85
	2	EPPL	2.20	0.56	0.53	0.90	–	0.92	0.83	0.84	0.95	–
		QIF_V	0.33	0.63	0.59	0.94	0.79	0.15	1.16	1.09	0.93	0.50
		EQIF	2.19	0.52	0.49	0.90	1.14	0.99	0.77	0.77	0.95	1.15
8	0.2	EPPL	2.46	0.46	0.43	0.88	–	0.85	0.56	0.59	0.96	–
		QIF_V	0.22	0.45	0.44	0.94	1.01	0.00	0.71	0.67	0.93	0.62
		EQIF	2.31	0.34	0.32	0.86	1.80	0.79	0.34	0.36	0.96	2.69
	0.5	EPPL	2.35	0.43	0.41	0.89	–	0.80	0.56	0.59	0.96	–
		QIF_V	0.36	0.44	0.43	0.95	0.99	0.06	0.74	0.70	0.94	0.56
		EQIF	2.12	0.34	0.32	0.88	1.65	0.78	0.39	0.41	0.95	2.01
	2	EPPL	2.30	0.39	0.38	0.89	–	0.85	0.55	0.59	0.96	–
		QIF_V	0.39	0.43	0.42	0.94	0.84	0.06	0.81	0.76	0.94	0.45
		EQIF	2.10	0.36	0.35	0.88	1.18	0.84	0.50	0.53	0.96	1.22

Table 2.

Simulation results under marginal model λ_ik(t) = λ_0k(t)exp{0.5X_ik + 0.2Z_ik}, the censoring rate is 60%.

			β(1) = 0.5					β(2) = 0.2
K	θ	Method	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE
			σ = 0.1
4	0.2	EPPL	0.06	0.70	0.66	0.93	–	0.15	1.15	1.13	0.95	–
		QIF_V	0.77	0.88	0.80	0.92	0.63	0.20	1.62	1.46	0.93	0.50
		EQIF	0.13	0.57	0.53	0.93	1.50	0.06	0.89	0.85	0.94	1.65
	0.5	EPPL	0.07	0.67	0.64	0.93	–	0.16	1.19	1.13	0.94	–
		QIF_V	0.64	0.86	0.80	0.93	0.60	0.21	1.66	1.51	0.92	0.51
		EQIF	0.01	0.58	0.56	0.93	1.31	0.18	1.04	0.97	0.93	1.31
	2	EPPL	0.08	0.65	0.62	0.92	–	0.13	1.16	1.13	0.95	–
		QIF_V	0.71	0.85	0.81	0.93	0.59	0.05	1.71	1.56	0.94	0.46
		EQIF	0.11	0.64	0.60	0.92	1.05	0.20	1.16	1.09	0.94	1.02
8	0.2	EPPL	0.03	0.51	0.50	0.95	–	0.11	0.82	0.81	0.95	–
		QIF_V	0.72	0.60	0.58	0.95	0.73	0.18	1.09	1.00	0.92	0.57
		EQIF	0.07	0.41	0.41	0.95	1.52	0.15	0.59	0.57	0.94	1.92
	0.5	EPPL	0.19	0.47	0.47	0.94	–	0.07	0.84	0.80	0.94	–
		QIF_V	0.42	0.58	0.57	0.93	0.65	0.26	1.11	1.05	0.94	0.57
		EQIF	0.10	0.40	0.40	0.94	1.36	0.13	0.71	0.65	0.94	1.41
	2	EPPL	0.33	0.45	0.44	0.95	–	0.10	0.82	0.80	0.95	–
		QIF_V	0.19	0.60	0.57	0.93	0.57	0.07	1.14	1.10	0.94	0.52
		EQIF	0.24	0.43	0.42	0.94	1.08	0.11	0.79	0.76	0.95	1.08
			σ = 1
4	0.2	EPPL	1.52	0.76	0.73	0.93	–	0.57	1.15	1.16	0.95	–
		QIF_V	0.77	0.88	0.80	0.92	0.75	0.20	1.62	1.46	0.93	0.51
		EQIF	1.45	0.62	0.59	0.93	1.49	0.40	0.90	0.89	0.95	1.63
	0.5	EPPL	1.61	0.73	0.71	0.94	–	0.58	1.19	1.16	0.94	–
		QIF_V	0.64	0.86	0.80	0.93	0.72	0.21	1.66	1.51	0.92	0.51
		EQIF	1.57	0.65	0.63	0.93	1.29	0.49	1.04	1.00	0.94	1.30
	2	EPPL	1.36	0.71	0.70	0.94	–	0.54	1.17	1.16	0.94	–
		QIF_V	0.71	0.85	0.81	0.93	0.69	0.05	1.71	1.56	0.94	0.47
		EQIF	1.38	0.69	0.68	0.93	1.04	0.54	1.16	1.12	0.95	1.01
8	0.2	EPPL	1.35	0.56	0.55	0.93	–	0.46	0.82	0.82	0.96	–
		QIF_V	0.72	0.60	0.58	0.95	0.88	0.18	1.09	1.00	0.92	0.57
		EQIF	1.29	0.44	0.43	0.93	1.65	0.48	0.60	0.60	0.95	1.85
	0.5	EPPL	1.48	0.52	0.52	0.93	–	0.39	0.84	0.82	0.95	–
		QIF_V	0.42	0.58	0.57	0.93	0.81	0.26	1.11	1.05	0.94	0.56
		EQIF	1.41	0.44	0.44	0.93	1.38	0.42	0.71	0.68	0.94	1.37
	2	EPPL	1.60	0.50	0.50	0.94	–	0.41	0.82	0.82	0.96	–
		QIF_V	0.19	0.60	0.57	0.93	0.70	0.07	1.14	1.10	0.94	0.52
		EQIF	1.51	0.48	0.48	0.94	1.09	0.41	0.79	0.78	0.94	1.07

Table 3 summarizes the results for Simulation 2 when censoring rate is around 60%. We omitted the results of the QIF estimator due to the space limitations, but the observations made on QIF in Simulation 1 still hold in Simulation 2. From Table 3, similar findings can be made for EPPL and EQIF as in Simulation 1, except for when θ = 2 where EPPL and EQIF performed similarly. This is reasonable since the within-cluster correlation is weak when θ = 2 and EQIF benefitted less from further considering the correlation structure.

Table 3.

Simulation results under marginal model λ_ik(t) = λ_0k(t) exp{β_1kX_ik + β_2kZ_ik}, the censoring rate is 60%.

			β11 = ln(2)					β21 = −ln(1.3)					β12 = 0.5					β22 = ‒0.1
N	θ	Method	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE
			σ = 0.1
300	0.2	EPPL	0.54	1.16	1.07	0.93	–	0.10	0.96	0.94	0.94	–	0.17	1.04	1.03	0.95	–	0.08	0.92	0.94	0.95	–
		EQIF	0.32	1.02	0.92	0.92	1.28	0.18	0.82	0.79	0.93	1.39	0.17	0.92	0.88	0.94	1.26	0.28	0.78	0.78	0.95	1.39
	0.5	EPPL	0.54	1.16	1.07	0.93	–	0.10	0.96	0.94	0.94	–	0.22	1.03	1.03	0.95	–	0.02	0.92	0.94	0.95	–
		EQIF	0.42	1.10	0.99	0.92	1.11	0.05	0.89	0.86	0.94	1.16	0.17	0.98	0.95	0.94	1.10	0.09	0.86	0.85	0.95	1.14
	2	EPPL	0.54	1.16	1.07	0.93	–	0.10	0.96	0.94	0.94	–	0.25	1.06	1.03	0.93	–	0.08	0.92	0.94	0.95	–
		EQIF	0.58	1.17	1.04	0.92	0.99	0.11	0.96	0.92	0.94	1.00	0.24	1.06	1.00	0.93	1.00	0.05	0.93	0.91	0.94	0.99
			σ = 1
	0.2	EPPL	0.65	1.31	1.23	0.93	–	0.79	0.96	0.98	0.95	–	0.27	1.21	1.18	0.95	–	0.30	0.92	0.96	0.95	–
		EQIF	1.10	1.16	1.06	0.93	1.27	1.07	0.81	0.84	0.95	1.39	0.46	1.09	1.02	0.94	1.24	0.51	0.79	0.82	0.95	1.35
	0.5	EPPL	0.65	1.31	1.23	0.93	–	0.79	0.96	0.98	0.95	–	0.01	1.20	1.18	0.95	–	0.18	0.93	0.96	0.95	–
		EQIF	0.96	1.24	1.13	0.92	1.10	0.94	0.88	0.90	0.95	1.18	0.14	1.15	1.09	0.93	1.09	0.32	0.87	0.88	0.95	1.14
	2	EPPL	0.65	1.31	1.23	0.93	–	0.79	0.96	0.98	0.95	–	0.00	1.26	1.18	0.94	–	0.11	0.93	0.96	0.96	–
		EQIF	0.77	1.30	1.19	0.92	1.01	1.02	0.95	0.95	0.95	1.01	0.05	1.27	1.15	0.92	0.98	0.27	0.93	0.94	0.95	1.00
			σ = 0.1
600	0.2	EPPL	0.25	0.77	0.76	0.95	–	0.19	0.68	0.66	0.94	–	0.20	0.73	0.72	0.94	–	0.14	0.65	0.66	0.96	–
		EQIF	0.16	0.68	0.66	0.95	1.31	0.20	0.57	0.56	0.94	1.40	0.24	0.63	0.62	0.95	1.34	0.27	0.54	0.55	0.96	1.42
	0.5	EPPL	0.25	0.77	0.76	0.95	–	0.19	0.68	0.66	0.94	–	0.05	0.71	0.72	0.95	–	0.19	0.64	0.66	0.96	–
		EQIF	0.12	0.73	0.70	0.94	1.12	0.21	0.63	0.61	0.93	1.15	0.09	0.67	0.67	0.96	1.14	0.29	0.59	0.60	0.95	1.15
	2	EPPL	0.25	0.77	0.76	0.95	–	0.19	0.68	0.66	0.94	–	0.06	0.71	0.72	0.95	–	0.27	0.63	0.66	0.96	–
		EQIF	0.19	0.77	0.75	0.94	1.00	0.18	0.68	0.65	0.94	0.99	0.06	0.71	0.71	0.94	1.00	0.34	0.63	0.65	0.96	1.00
			σ = 1
	0.2	EPPL	0.19	0.93	0.88	0.94	–	0.51	0.68	0.69	0.95	–	0.05	0.85	0.83	0.94	–	0.28	0.65	0.68	0.96	–
		EQIF	0.40	0.82	0.76	0.93	1.27	0.53	0.57	0.59	0.96	1.42	0.12	0.75	0.72	0.94	1.30	0.36	0.56	0.58	0.96	1.37
	0.5	EPPL	0.19	0.93	0.88	0.94	–	0.51	0.68	0.69	0.95	–	0.23	0.81	0.83	0.95	–	0.33	0.64	0.67	0.96	–
		EQIF	0.38	0.87	0.81	0.93	1.12	0.52	0.63	0.64	0.95	1.16	0.26	0.76	0.77	0.95	1.13	0.40	0.60	0.63	0.96	1.14
	2	EPPL	0.19	0.93	0.88	0.94	–	0.51	0.68	0.69	0.95	–	0.41	0.81	0.83	0.95	–	0.39	0.63	0.67	0.97	–
		EQIF	0.30	0.93	0.86	0.93	1.00	0.53	0.68	0.68	0.95	1.00	0.43	0.81	0.81	0.94	0.99	0.45	0.63	0.66	0.96	1.00

Next, we conduct additional simulation studies to examine the type I error rate of the proposed chi-squared test and compare it with the Z-test by EPPL method in terms of the empirical power. The null hypothesis H₀ : β₍₁₎ = 0.5. The data are generated under the same model as in Simulation 1 with K = 4 and θ = 0.5. The sample size is n = 200. Since the dimension of β₍₁₎ under H₀ is 1, the test statistic asymptotically follows ${\chi }_1^2$. Figure 1 provides Q-Q plots under 10% and 70% censoring rates and illustrates that under H₀ the empirical quantiles follow the theoretical quantiles of the ${\chi }_1^2$ distribution quite well. We next examine the empirical type I error rate when β₍₁₎ = 0.5 and the power of the proposed chi-squared test when β₍₁₎ deviates from 0.5. Under H₀, the empirical type I error rates for chi-squared test by EQIF method and the Z-test by EPPL method are 0.063 and 0.067 when censoring rate is 10%, and 0.059 and 0.048 when censoring rate is 70%. The powers with significance level 0.05 are calculated when β₍₁₎ takes values in (0.5,0.7]. Figure 2 illustrates the type I error/power function curves. It can be seen that when β₍₁₎ is equal to 0.5, the empirical type I error rates for both methods are approximately 0.05, and when β₍₁₎ reaches 0.7, the powers for both methods are close to 1. In all the considered settings, the chi-squared test by EQIF method is more powerful than the Z-test by EPPL method. It shows that both test methods have proper type I error and the proposed chi-squared test provides a higher probability of correctly rejecting the null hypothesis when the null hypothesis is false. However, when the censoring rate is high (e.g. at 70% censoring), the difference between the powers of the two test methods is getting smaller.

Figure 1.

Q-Q plots for the test statistic versus ${\chi }_1^2$ under H₀ from 1000 replications.

Figure 2.

Empirical type I error/power function curves for chi-squared test by EQIF method and Z-test by EPPL method.

Furthermore, we expand the simulation to the situation closer to the SOLVD data, to which we will apply our proposed method in the next section. We set n = 4000, K = 4, validation rate ρ_k = 0.025, σ = 0.1, and the censoring rate is around 80%. The results based on 500 replicates are summarized in Table 4. All the considered estimates are approximately unbiased. The estimates of the asymptotic standard error are close to the empirical standard deviations. The EPPL estimator and the EQIF estimator gain noticeable efficiency compared to the QIF estimator based only on the validation set. The relative efficiency of EQIF estimate vs EPPL estimate increases when the degree of within-cluster dependence increases (i.e. θ decreases). These results imply that the proposed approach is adequate for the settings similar to the real data.

Table 4.

Simulation results under marginal model λ_ik(t) = λ_0k(t)exp{0.5X_ik + 0.2Z_ik}, n = 4000, K = 4, ρ_k = 0.025, σ = 0.1, and the censoring rate is 80%.

		β(1) = 0.5					β(1) = 0.2
θ	Method	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE	|Bias| (×102)	SD (×10)	SE (×10)	CR	RE
0.2	EPPL	0.53	0.27	0.26	0.93	–	0.39	0.36	0.43	0.96	–
	QIF_V	0.53	1.13	1.14	0.95	0.05	0.74	2.40	2.25	0.94	0.02
	EQIF	0.52	0.24	0.24	0.94	1.21	0.45	0.30	0.39	0.98	1.39
0.5	EPPL	0.51	0.27	0.26	0.93	–	0.41	0.34	0.43	0.98	–
	QIF_V	0.29	1.15	1.14	0.94	0.05	1.70	2.37	2.25	0.94	0.02
	EQIF	0.50	0.25	0.25	0.93	1.09	0.50	0.31	0.42	0.98	1.14
2	EPPL	0.45	0.26	0.25	0.93	–	0.45	0.34	0.43	0.98	–
	QIF_V	0.78	1.14	1.13	0.95	0.05	2.87	2.36	2.25	0.94	0.02
	EQIF	0.44	0.26	0.25	0.93	1.00	0.50	0.34	0.43	0.98	0.99

In addition, it is worth noting that although the proposed EQIF method assumes that the auxiliary covariates are categorical, it can also be applied to the situation where the auxiliary covariates are continuous by discretizing them first. We conduct additional simulation studies to evaluate the performance of the discretized version of the proposed EQIF method when the auxiliary covariates are continuous. In this situation, a naive approach would be to regress the true variables on the auxiliary variables and then do multiple imputation or calibration. Another choice is to apply the QIF approach to the validation set. We compare the discretized version of the proposed approach with these two approaches and display the simulation results in Tables S2 and S3 in Web Appendix 2. The naive estimates for both regression parameters and the variance of parameter estimates are biased. The proposed EQIF estimates and the QIF estimate based on validation set are approximately unbiased. The estimated standard errors are close to the empirical standard deviations, and the 95% confidence interval coverage rates are close to the nominal level. In addition, the EQIF estimate is more efficient than the QIF estimator based on validation set.

5. Real Data Analysis

In this section, we illustrate the application of the proposed method through the analysis of the dataset from the Left Ventricular Dysfunction (SOLVD) study. The SOLVD study was a randomized, double-masked, placebo-controlled trial, which was conducted at 83 hospitals linked to 23 centers in the United States, Canada, and Belgium from 1986 to 1991. The trial had a three-year recruitment and a two-year follow-up. A total of 4228 patients were monitored for the heart failure and the first nonfatal myocardial infarction (MI) throughout the study period. In the SOLVD study, along with the treatment indicator (TRT, 1 for enalapril and 0 for placebo), three potential covariates were considered: patient’s gender (SEX, 1 for male and 0 for female), patient’s age (AGE, in years), and left ventricular ejection fraction. Ejection fraction is a number between 0 and 100 that measures the efficiency of the heart in ejecting blood. However, only 108 among the total of 4228 patients have their ejection fraction accurately measured using a standardized radionucleotide technique (LVEF), while a related nonstandardized measure (EF) was ascertained for all the patients. Therefore, the nonstandardized measure (EF) is a surrogate measure for the standardized measure (LVEF) in this case. Both LVEF and EF were measured in percentage. The data has been analyzed by Liu et al. (2009).

According to the conclusion of Liu et al. (2009), the categorical auxiliary covariate AEF is created with values 1, 2, 3, or 4 being assigned depending on whether the EF is in the interval [min(EF),q₁], (q₁,q₂], (q₂,q₃], or (q₃,max(EF)], where q₁, q₂, q₃ are the quartiles of EF. They also show that the LVEF is conditional independent of other covariates given AEF, thus we take A^∗ = A in (8).

In terms of the notation in the previous sections, we have X = LV EF, A = AEF, Z = (TRT,SEX,AGE)^T. Let k denote failure type with k = 1 for heart failure and k = 2 for nonfatal MI and i denote the patient with i = 1,…,4228, we fit the failure-type-specific model ${\lambda }_{ik}\left(t\right)={\lambda }_{0k}\left(t\right)\exp \left\{{\beta }_{1k}{X}_{ik}+{\beta }_{2k}^{\text{T}}{Z}_{ik}\right\}$ to the SOLVD data.

The results of the analysis are presented in Table 5. From Table 5, we observe the following. The SEs from the proposed EQIF method are smaller than those from the EPPL method. Consequently, the proposed EQIF method provides tighter 95% confidence intervals, e.g., the 95% confidence interval for TRT is (−0.590,−0.218) for the EQIF method, while it is (−0.691,−0.193) for the EPPL method. For these two methods, the p-values indicate that LVEF and TRT are statistically significant for heart failure and only TRT is significant for nonfatal MI at the significance level of 0.05. Meanwhile, AGE is significant for heart failure from the EQIF method.

Table 5.

Analysis results of SOLVD data.

	EQIF method			EPPL method
Covariate	Coef	SE	P-value	Coef	SE	P-value
For heart failure
LVEF	−0.068	0.012	< 0.001	−0.075	0.013	< 0.001
TRT	−0.404	0.095	< 0.001	−0.442	0.127	< 0.001
SEX	−0.317	0.175	0.119	−0.317	0.229	0.167
AGE	0.030	0.006	0.039	0.023	0.020	0.254
For nonfatal MI
LVEF	−0.016	0.014	0.415	−0.009	0.016	0.587
TRT	−0.418	0.121	< 0.001	−0.392	0.129	0.002
SEX	0.013	0.196	0.861	0.036	0.205	0.861
AGE	0.004	0.006	0.741	0.004	0.006	0.546

6. Concluding Remarks

In this article, we propose an EQIF method for the multivariate failure time data with discrete auxiliary covariates. This method utilizes the auxiliary information nonparametrically and no assumption is needed for the association between auxiliary covariates and the true exposure. The proposed method further takes the intra-cluster correlation into the estimation procedure without explicitly estimating the correlation parameters, which decreases the computation burden. This advantage becomes important especially when the cluster size is large. Another advantage of the EQIF approach is that the inference function has an explicit asymptotic form, which allows us to test whether coefficients are zero or non-zero for regression models. Simulation studies show that EQIF method gains efficiency over EPPL method by using the within-cluster correlation information. Although we assume that the auxiliary covariates are discrete, our proposed method can be used for the continuous auxiliary covariates by discretizing them first.

Appendix

In the following, we use notation $\stackrel{p}{\to }$, $\stackrel{a.s.}{\to }$ and $\stackrel{d}{\to }$ to denote the convergence in probability, convergence almost surely, and convergence in distribution, respectively. Unless otherwise stated, all the limits are taken as n → ∞.

For a vector a = (a_i), define ||a|| = sup_i|a_i|. For a matrix A = (a_ij), define ||A|| = sup_i,j|a_ij|. For a = 0,1,2, $r_{ik}^{\left(a\right)}\left(\beta ,t\right)$ and ${\gamma }_{ik}^{\left(a\right)}\left({\beta }_{\left(1\right)},t\right)$ denote the ath derivative of r_ik(β,t) with respect to β and the ath derivative of γ_ik(β₍₁₎,t) with respect to β₍₁₎. Let $R_{lm}\left(\beta ,t\right)=r_{lm}^{\left(1\right)}\left(\beta ,t\right)/r_{lm}\left(\beta ,t\right)$, l = 1,…,n, m = 1,…,K. Let ϕ_lmk(m,k = 1,…,K) denote the (m,k)-th component of matrix ${\Xi }_l^{1/2}\left(\beta ,t\right)B{\Xi }_l^{-1/2}\left(\beta ,t\right)$. For k = 1,…,K, define $S_k^{\left(d\right)}\left(\beta ,t\right)=n^{-1}{\sum }_{l=1}^n{Y}_{lk}\left(t\right)r_{lk}^{\left(d\right)}\left(\beta ,t\right)\left(d=0,1\right)$, $S_k^{\left(d\right)}\left(\beta ,t,B\right)=n^{-1}{\sum }_{l=1}^n{\sum }_{m=1}^K{Y}_{lk}\left(t\right)R_{lm}\left(\beta ,t\right){\varphi }_{lmk}\left(\beta ,t,B\right)r_{lk}^{\left(d-2\right)}\left(\beta ,t\right)\left(d=2,3\right)$, $S_k^{\left(4\right)}\left(\beta ,t,B\right)=n^{-1}{\sum }_{l=1}^n{\sum }_{m=1}^K{Y}_{lk}\left(t\right)R_{lm}\left(\beta ,t\right)\left\{\partial {\varphi }_{lmk}\left(\beta ,t,B\right)/\partial {\beta }^{\text{T}}\right\}{r}_{lk}\left(\beta ,t\right)$, $S_k^{\left(5\right)}\left(\beta ,t,B\right)=n^{-1}{\sum }_{l=1}^n{\sum }_{m=1}^K{Y}_{lk}\left(t\right)\left\{\partial R_{lm}\left(\beta ,t\right)/\partial {\beta }^{\text{T}}\right\}{\varphi }_{lmk}\left(\beta ,t,B\right)r_{lk}\left(\beta ,t\right)$. Define ${\widehat{S}}_k^{\left(d\right)}\left(d=0,\dots ,5\right)$ by replacing r_lk(β,t), R_lm(β,t), and ϕ_lmk(β,t,B) with their estimators in $S_k^{\left(d\right)}$. We also define ${\mathit{V}}_k\left(\beta ,t,B\right)=S_k^{\left(3\right)}\left(\beta ,t,B\right)/S_k^{\left(0\right)}\left(\beta ,t\right)-S_k^{\left(2\right)}\left(\beta ,t,B\right){\left\{S_k^{\left(1\right)}\left(\beta ,t\right)\right\}}^{\text{T}}/{\left\{S_k^{\left(0\right)}\left(\beta ,t\right)\right\}}^2$, ${\mathit{E}}_k\left(\beta ,t,B\right)=S_k^{\left(2\right)}\left(\beta ,t,B\right)/S_k^{\left(0\right)}\left(\beta ,t\right)$.

The asymptotic results rely on the following regularity conditions. Some conditions can also be found in Liu et al. (2009).

• (C1)For $k=1,\dots ,K$, ${\mathrm{\Lambda }}_{0k}\left(\tau \right)={\int }_0^{\tau }{\lambda }_{0k}\left(t\right)\text{d}t<\infty$.
• (C2)For i = 1,…,n, k = 1,…,K, and $m=1,\dots ,L$, $Pr\left\{Y_{ik}\left(t\right)=1|A_{ik}^{\ast }=a_m\right\}>0$.
• (C3)There exist compact sets $B_1$, $B_2$ containing β₁₀ and β₂₀ as interior points respectively, such that the elements (∂²/∂β_j∂β_l)ψ_ik(β,t) exist and are uniformly continuous on $B=B_1\times B_2$.
• (C4)
  For d = 0,1,2, and $k=1,\dots K$, $\underset{t\in \left[0,\tau \right]}{\sup }\left|U_k^{\left(d\right)}\left(t\right)\right|=O_p\left(1\right)$, where

  $$\begin{gathered} U_k^{\left(d\right)}\left(t\right)=\sqrt{n_k}\left[\frac{1}{n_k}\sum _{j\in V_k}I\left\{Y_{jk}\left(t\right)=1,A_{jk}^{\ast }\left(t\right)=a\right\}{\gamma }_{jk}^{\left(d\right)}\left({\beta }_{\left(1\right)},t\right)\right. \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left.-E\left\{I\left(Y_{ik}\left(t\right)=1,A_{ik}^{*}\left(t\right)=a\right){\gamma }_{ik}^{\left(d\right)}\left({\beta }_{\left(1\right)},t\right)\right\}\right], \end{gathered}$$

  with ${\gamma }_{ik}\left({\beta }_{\left(1\right)},t\right)=\exp \left\{{\beta }_{\left(1\right)}^{\text{T}}{X}_{ik}\left(t\right)\right\}$, n_k being the sample size of validation set V_k.
• (C5)There exist scalar, vector and matrix functions $s_k^{\left(d\right)}\left(\beta ,t\right)\hspace{0.33em}\left(d=0,1\right)$, $s_k^{\left(d\right)}\left(\beta ,t,B\right)\hspace{0.33em}\left(d=2,\dots ,5\right)$, such that $\underset{\beta \in B,t\in \left[0,\tau \right]}{\sup }‖S_k^{\left(d\right)}-s_k^{\left(d\right)}‖\stackrel{p}{\to }0$, for all k = 1,…,K and all constant matrix B.
• (C6)
  Let ${\mathit{v}}_k\left(\beta ,t,B\right)=s_k^{\left(3\right)}\left(\beta ,t,B\right)/s_k^{\left(0\right)}\left(\beta ,t\right)-s_k^{\left(2\right)}\left(\beta ,t,B\right){\left\{s_k^{\left(1\right)}\left(\beta ,t\right)\right\}}^{\text{T}}/{\left\{s_k^{\left(0\right)},\left(\beta ,t\right)\right\}}^2$, ${\mathit{e}}_k\left(\beta ,t,B\right)=s_k^{\left(2\right)}\left(\beta ,t,B\right)/s_k^{\left(0\right)}\left(\beta ,t\right)$. Then for all $\beta \in B$, $t\in \left[0,\tau \right]$, and $k=1,\dots ,K$, $s_k^{\left(1\right)}\left(\beta ,t\right)=\partial s_k^{\left(0\right)}\left(\beta ,t\right)/\partial \beta$, $s_k^{\left(3\right)}\left(\beta ,t,B\right)=\partial s_k^{\left(2\right)}\left(\beta ,t,B\right)/\partial \beta -s_k^{\left(4\right)}\left(\beta ,t,B\right)-s_k^{\left(5\right)}\left(\beta ,t,B\right)$, and $s_k^{\left(0\right)}\left(\beta ,t\right)$ is bounded away from zero on $B\times \left[0,\tau \right]$. For all basis matrix B, the matrix

  $$\mathrm{\Gamma }\left({\beta }_0,B\right)=-\sum _{k=1}^K{\int }_0^{\tau }{\mathit{v}}_k\left({\beta }_0,t,B\right)s_k^{\left(0\right)}\left({\beta }_0,t\right){\lambda }_{0k}\left(t\right)\text{d}t$$ (A.1)

  is negative definite.
• (C7)
  There exists a matrix function ω(·,·,·), such that for any K × K constant matrices B₁, B₂, $n^{-1}{\sum }_{i=1}^n{\sum }_{k=1}^K{\int }_0^{\tau }{\mathit{h}}_{ik}\left(\beta ,t,B_1\right){\mathit{h}}_{ik}^{\text{T}}\left(\beta ,t,B_2\right)s_k^{\left(0\right)}\left(\beta ,t\right){\lambda }_{0k}\left(t\right)\text{d}t\stackrel{p}{\to }\mathit{\omega }\left(\beta ,B_1,B_2\right)$, uniformly for $\beta \in B$, where ${\mathit{h}}_{ik}\left(\beta ,t,B\right)={\sum }_{m=1}^K{R}_{im}\left(\beta ,t\right){\varphi }_{imk}\left(\beta ,t,B\right)-{\mathit{e}}_k\left(\beta ,t,B\right)$. Furthermore, for any set of basis matrices {B_j,j = 1,…,m₀}, the matrix

  $${\mathit{W}}_0=\mathit{W}\left({\beta }_0\right)={\left\{\mathit{\omega }\left({\beta }_0,B_j,B_{j^{\prime }}\right)\right\}}_{j,j^{\prime }=1}^{m_0}$$ (A.2)

  is positive definite.
• (C8)The baseline hazard rates {λ_0k(·),k = 1,…,K} are twice continuously differentiable on $\left[0,\tau \right]$.

Proof of Theorem 1. Based on the extension of Xue et al. (2010) and Qu and Li (2006), one can show that ${\widehat{\beta }}_Q$ is consistent for β₀ provided that:

1. $\partial {\widehat{\mathit{G}}}_n\left(\beta \right)/\partial {\beta }^{\text{T}}$ exists and is continuous, and it converges in probability to a fixed function, say J(β), uniformly for $\beta \in B$;

2. ${\widehat{\mathit{G}}}_n\left({\beta }_0\right)\to 0$ in probability;

3. $n{\mathbf{\Omega }}_n\left(\beta \right)=n^{-1}{\sum }_{i=1}^n{\widehat{\mathit{g}}}_i\left(\beta \right){\widehat{\mathit{g}}}_i^{\text{T}}\left(\beta \right)$converges in probability to a constant matrix W(β) uniformly for $\beta \in B$;

4. nΩ_n(β₀) is positive definite with probability going to 1 as n → ∞.

Denote L = {1,…,L}, for m ∈ L, let $f_k\left(1,a_m,t\right)=Pr\left\{Y_{ik}\left(t\right)=1,\hspace{0.33em}{A}_{ik}^{\ast }\left(t\right)=a_m\right\}$, and ${\mathrm{\Phi }}_n^k\left({\beta }_{\left(1\right)},t,a_m\right)=n_k^{-1}{\sum }_{j\in V_k}I\left\{Y_{jk}\left(t\right)=1,A_{jk}^{\ast }\left(t\right)=a_m\right\}\exp \left\{{\beta }_{\left(1\right)}^{\text{T}}{X}_{ik}\left(t\right)\right\}$. By similar techniques used in Liu et al. (2009), we can prove that sup $\underset{{\Theta }_k}{\sup }\left|{\mathrm{\Phi }}_n^k\left(0,t,a_m\right)-f_k\left(1,a_m,t\right)\right|\stackrel{p}{\to }0$. Consequently, it follows that

$$\underset{{\Theta }_k}{\sup }\left|\frac{{\mathrm{\Phi }}_n^k\left({\beta }_{\left(1\right)},t,a_m\right)}{{\mathrm{\Phi }}_n^k\left(0,t,a_m\right)}-E\left\{e^{{\beta }_{\left(1\right)}^{\text{T}}{X}_{ik}\left(t\right)}|Y_{ik}\left(t\right)=1,A_{ik}^{\ast }\left(t\right)=a_m\right\}\right|\stackrel{p}{\to }0.$$

By the definition of ${\widehat{\psi }}_{ik}\left(\beta ,t\right)$, that ${\widehat{\psi }}_{ik}\left(\beta ,t\right)=\exp \left\{{\beta }_{\left(2\right)}^{\text{T}}{Z}_{ik}\left(t\right)\right\}{\mathrm{\Phi }}_n^k\left({\beta }_{\left(1\right)},t,A_{ik}^{\ast }\left(t\right)\right)/{\mathrm{\Phi }}_n^k\left(0,t,A_{ik}^{\ast }\left(t\right)\right)$, we have sup $\underset{\beta \in B,t\in \left[0,\tau \right]}{\sup }\left|{\widehat{\psi }}_{ik}\left(\beta ,t\right)-{\psi }_{ik}\left(\beta ,t\right)\right|\stackrel{p}{\to }0$.

Using the same argument, we can prove that $\underset{\beta \in B,t\in \left[0,\tau \right]}{\sup }‖\partial {\widehat{\psi }}_{ik}\left(\beta ,t\right)/\partial \beta -\partial {\psi }_{ik}\left(\beta ,t\right)/\partial \beta ‖\stackrel{p}{\to }0$, and $\underset{\beta \in B,t\in \left[0,\tau \right]}{\sup }‖{\partial }^2{\widehat{\psi }}_{ik}\left(\beta ,t\right)/\partial {\beta }^2-{\partial }^2{\psi }_{ik}\left(\beta ,t\right)/\partial {\beta }^2‖\stackrel{p}{\to }0$. Since ${\tilde{\lambda }}_{0k}\left(t\right)$ is a consistent estimator for λ_0k(t), then, by the definitions of ${\widehat{S}}_k^{\left(d\right)}$ and $S_k^{\left(d\right)}$ and condition (C5), we have sup $\underset{\beta \in B,t\in \left[0,\tau \right]}{\sup }‖{\widehat{S}}_k^{\left(d\right)}-S_k^{\left(d\right)}‖\stackrel{p}{\to }0$, and $\underset{\beta \in B,t\in \left[0,\tau \right]}{\sup }‖{\widehat{S}}_k^{\left(d\right)}-s_k^{\left(d\right)}‖\stackrel{p}{\to }0$, $d=0,\cdots ,5$.

Let ${\tilde{\mathit{M}}}_i\left(t;\beta \right)={\left({\tilde{M}}_{i1}\left(t;\beta \right),\dots ,{\tilde{M}}_{iK}\left(t;\beta \right)\right)}^{\text{T}}$, ${\widehat{\mathit{R}}}_i^{\text{T}}\left(\beta ,t\right)=\left(\frac{{\widehat{r}}_{i1}^{\left(1\right)}\left(\beta ,t\right)}{{\widehat{r}}_{i1}\left(\beta ,t\right)},\dots ,\frac{{\widehat{r}}_{iK}^{\left(1\right)}\left(\beta ,t\right)}{{\widehat{r}}_{iK}\left(\beta ,t\right)}\right)$, and ${\widehat{\Xi }}_i\left(\beta ,t\right)=\text{diag}\left\{{\tilde{\lambda }}_{01}\left(t\right){\widehat{r}}_{i1}\left(t\right),\dots ,{\tilde{\lambda }}_{0K}\left(t\right){\widehat{r}}_{iK}\left(t\right)\right\}$. Define

$${\mathit{H}}_n\left(\beta ,B\right)=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\widehat{\mathit{R}}}_i^{\text{T}}\left(\beta ,t\right)B{\widehat{\Xi }}_i^{-1/2}\left(\beta ,t\right){\tilde{\mathit{M}}}_i\left(\text{d}t;\beta \right),$$

then ${\widehat{\mathit{G}}}_n\left(\beta \right)={\left({\mathit{H}}_n^{\text{T}}\left(\beta ,B_1\right),\dots ,{\mathit{H}}_n^{\text{T}}\left(\beta ,B_{m_0}\right)\right)}^{\text{T}}$. After some algebraic manipulations, we decompose H_n(β,B) as ${\mathit{H}}_n\left(\beta ,B\right)={\mathit{H}}_n^0\left(\beta ,B\right)+{\mathit{H}}_n^{\left(1\right)}\left(\beta ,B\right)$, where

$${\mathit{H}}_n^{\left(0\right)}\left(\beta ,B\right)=\frac{1}{n}\sum _{i=1}^n\sum _{k=1}^K{\int }_0^{\tau }{\widehat{\mathit{h}}}_{ik}\left(\beta ,t,B\right)M_{ik}\left(\text{d}t\right),$$

$${\mathit{H}}_n^{\left(1\right)}\left(\beta ,B\right)=\frac{1}{n}\sum _{i=1}^n\sum _{k=1}^K{\int }_0^{\tau }{\widehat{\mathit{h}}}_{ik}\left(\beta ,t,B\right)Y_{ik}\left(t\right)r_{ik}\left(\beta ,t\right){\lambda }_{0k}\left(t\right)\text{d}t,$$

with ${\widehat{\mathit{h}}}_{ik}\left(\beta ,t,B\right)={\sum }_{m=1}^K{\widehat{R}}_{im}\left(\beta ,t\right){\widehat{\varphi }}_{imk}\left(\beta ,t,B\right)-{\widehat{S}}_k^{\left(2\right)}\left(\beta ,t,B\right)/{\widehat{S}}_k^{\left(0\right)}\left(\beta ,t\right)$. It follows that

$$\frac{\partial {\mathit{H}}_n\left(\beta ,B\right)}{\partial {\beta }^{\text{T}}}=\frac{1}{n}\sum _{i=1}^n\sum _{k=1}^K{\int }_0^{\tau }{\mathit{F}}_{ik}\left(\beta ,t,B\right)M_{ik}\left(\text{d}t\right)+{\mathit{H}}_n^{\left(2\right)}\left(\beta ,B\right),$$ (A.3)

where

$$\begin{gathered} {\mathit{F}}_{ik}\left(\beta ,t,B\right)=\sum _{m=1}^K\frac{\partial {\widehat{R}}_{im}\left(\beta ,t\right){\widehat{\varphi }}_{imk}\left(\beta ,t,B\right)}{\partial {\beta }^{\text{T}}}+\frac{{\widehat{S}}_k^{\left(2\right)}\left(\beta ,t,B\right){\left\{{\widehat{S}}_k^{\left(1\right)}\left(\beta ,t\right)\right\}}^{\text{T}}}{{\left\{{\widehat{S}}_k^{\left(0\right)}\left(\beta ,t\right)\right\}}^2} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\frac{{\widehat{S}}_k^{\left(3\right)}\left(\beta ,t,B\right)+{\widehat{S}}_k^{\left(4\right)}\left(\beta ,t,B\right)+{\widehat{S}}_k^{\left(5\right)}\left(\beta ,t,B\right)}{{\widehat{S}}_k^{\left(0\right)}\left(\beta ,t\right)}, \end{gathered}$$

$${\mathit{H}}_n^{\left(2\right)}\left(\beta ,B\right)=\frac{1}{n}\sum _{i=1}^n\sum _{k=1}^K{\int }_0^{\tau }{\mathit{F}}_{ik}\left(\beta ,t,B\right)Y_{ik}\left(t\right)r_{ik}\left(\beta ,t\right){\lambda }_{0k}\left(t\right)\text{d}t.$$

For any constant matrix B, the first term on the right-hand side of (A.3) is a local square integrable martingale, which converges in probability to zero uniformly for $\beta \in B$ by Lenglart inequality (Anderson et al., 1993, p86). The second term converges uniformly to $\mathrm{\Gamma }\left(\beta ,B\right)=-{\sum }_{k=1}^K{\int }_0^{\tau }{\mathit{v}}_k\left(\beta ,t,B\right)s_k^{\left(0\right)}\left(\beta ,t\right){\lambda }_{0k}\text{d}t$. Hence, ∂H_n(β,B)/∂β^T converges to Γ(β,B) in probability uniformly for $\beta \in B$, and $\partial {\widehat{\mathit{G}}}_n\left(\beta \right)/\partial {\beta }^{\text{T}}$ converges in probability to J(β) uniformly for $\beta \in B$, with J(β) defined in Theorem 2. Obviously, ∂H_n(β,B)/∂β^T is continuous, therefore, (i) is verified.

To prove (ii), we can first prove that $\sqrt{n}{\mathit{H}}_n^{\left(1\right)}\left({\beta }_0,B\right)$ is equivalent to

$$-\frac{1}{\sqrt{n}}\sum _{i=1}^n\sum _{k=1}^K{\int }_0^{\tau }{\mathit{h}}_{ik}\left({\beta }_0,t,B\right)\left\{{\widehat{r}}_{ik}\left({\beta }_0,t\right)-r_{ik}\left({\beta }_0,t\right)\right\}{Y}_{ik}\left(t\right){\lambda }_{0k}\left(t\right)\text{d}t+o_p\left(1\right).$$

Then by the similar arguments in Liu et al. (2009),

$$\sqrt{n}{\mathit{H}}_n^{\left(0\right)}\left({\beta }_0,B\right)=\frac{1}{\sqrt{n}}\sum _{k=1}^K\sum _{i=1}^n{\int }_0^{\tau }{\mathit{h}}_{ik}\left({\beta }_0,t,B\right)M_{ik}\left(\text{d}t\right)+o_p\left(1\right),$$

$$\sqrt{n}{\mathit{H}}_n^{\left(1\right)}\left({\beta }_0,B\right)=-\frac{1}{\sqrt{n}}\sum _{k=1}^K\sum _{i\in V_k}{\mathbf{\Psi }}_{ik}\left({\beta }_0,B\right)+o_p\left(1\right),$$

where ${\mathbf{\Psi }}_{ik}\left(\beta ,B\right)=\left(n-n_k\right)/n_k{\int }_0^{\tau }{\mathit{h}}_{ik}\left(\beta ,t,B\right)\left\{{\phi }_{ik}\left(\beta ,t\right)-{\psi }_{ik}\left(\beta ,t\right)\right\}{Y}_{ik}\left(t\right){\lambda }_{0k}\left(t\right)\text{d}t$. It follows that $\sqrt{n}{\mathit{H}}_n\left({\beta }_0,B\right)=n^{-1/2}{\sum }_{k=1}^K\left\{{\sum }_{i=1}^n{\int }_0^{\tau }{\mathit{h}}_{ik}\left({\beta }_0,t,B\right)M_{ik}\left(\text{d}t\right)-{\sum }_{i\in V_k}{\mathbf{\Psi }}_{ik}\left({\beta }_0,B\right)\right\}+o_p\left(1\right)$. Similar arguments as Liu et al. (2009) show that the expectation of H_n(β₀,B) is 0. Then by the strong law of large numbers, ${\mathit{H}}_n\left({\beta }_0,B\right)\stackrel{a.s.}{\to }0$. Based on the above results, (ii) is satisfied.

The proofs of (iii) and (iv) are similar to those of (iv) and (v) in the Theorem 1 of Xue et al. (2010), hence we omit them here, and the proof of Theorem 1 is done.

Proof of Theorem 2. By Taylor expansion of $n^{-1}\partial {\widehat{\mathit{Q}}}_n\left(\beta \right)/\partial \beta$ around the true parameter β₀, we have $\sqrt{n}\left({\widehat{\beta }}_Q-{\beta }_0\right)=-{\left\{n^{-1}{\partial }^2{\widehat{\mathit{Q}}}_n\left(\tilde{\beta }\right)/\partial {\beta }^2\right\}}^{-1}\left\{n^{-1/2}\partial {\widehat{\mathit{Q}}}_n\left({\beta }_0\right)/\partial \beta \right\}$, where $\tilde{\beta }$ is between β₀ and ${\widehat{\beta }}_Q$.

From the conclusions of Qu et al. (2000) and the previous Theorem 1, we conclude that

$$\frac{\partial {\widehat{\mathit{Q}}}_n\left({\beta }_0\right)}{\partial \beta }=2{\left\{\frac{\partial {\widehat{\mathit{G}}}_n\left({\beta }_0\right)}{\partial \beta }\right\}}^{\text{T}}{\mathbf{\Omega }}_n^{-1}\left({\beta }_0\right){\widehat{\mathit{G}}}_n\left({\beta }_0\right)+o_p\left(1\right),$$

and

$$\frac{{\partial }^2{\widehat{\mathit{Q}}}_n\left(\tilde{\beta }\right)}{\partial {\beta }^2}=2{\left\{\frac{\partial {\widehat{\mathit{G}}}_n\left({\beta }_0\right)}{\partial \beta }\right\}}^{\text{T}}{\mathbf{\Omega }}_n^{-1}\left({\beta }_0\right)\frac{\partial {\widehat{\mathit{G}}}_n\left({\beta }_0\right)}{\partial \beta }+o_p\left(1\right).$$

According to the proof of Theorem 1, we have that $\partial {\widehat{\mathit{G}}}_n\left({\beta }_0\right)/\partial \beta$ and nΩ_n(β₀) converge to J₀ and W₀ in probability, respectively. To show the asymptotic distribution of $\sqrt{n}{\widehat{\mathit{G}}}_n\left({\beta }_0\right)$, we can express each component of it as $\sqrt{n}{\mathit{H}}_n\left({\beta }_0,B\right)=n^{-1/2}{\sum }_{i=1}^n{\mathit{D}}_i\left(B\right)+o_p\left(1\right)$, where ${\mathit{D}}_i\left(B\right)={\sum }_{k=1}^K{\int }_0^{\tau }{\mathit{h}}_{ik}\left({\beta }_0,t,B\right)M_{ik}\left(\text{d}t\right)$. The first term is a summation of independent random variables with mean zero and variance var{D_i(B)}. By the multivariate central limit theorem, we can show that the distribution of $\sqrt{n}{\mathit{H}}_n\left({\beta }_0,B\right)$ converges to a zero-mean normal random vector with variance ω(β₀,B,B), and $\sqrt{n}{\widehat{\mathit{G}}}_n\left({\beta }_0\right)$ converges in distribution to a zero-mean normal random vector with variance W₀. Hence, it follows that $\sqrt{n}\left({\widehat{\beta }}_Q-{\beta }_0\right)\stackrel{d}{\to }N\left(0,{\left({\mathit{J}}_0^{\text{T}}{\mathit{W}}_0^{-1}{\mathit{J}}_0\right)}^{-1}\right)$.

Proof of Theorem 3. The proof of Theorem 3 utilizes the results in Xue et al. (2010). Let ${\mathit{Q}}_n\left(\beta \right)$ be as in (6), define ${\tilde{\zeta }}^{\ast }=\underset{\zeta }{\text{argmin}}{\mathit{Q}}_n\left({\xi }_0,\zeta \right)$ and $\left(\widehat{{\xi }^{\ast }},\widehat{{\zeta }^{\ast }}\right)=\underset{\left(\xi ,\zeta \right)}{\text{argmin}}{\mathit{Q}}_n\left(\xi ,\zeta \right)$, Xue et al. (2010) showed that ${\mathcal{T}}^{\ast }={\mathit{Q}}_n\left({\xi }_0,{\tilde{\zeta }}^{\ast }\right)-{\mathit{Q}}_n\left(\widehat{{\xi }^{\ast }},\widehat{{\zeta }^{\ast }}\right)$ asymptotically follows ${\chi }_q^2$ under H₀. By the arguments in the proof of Theorem 1 above, we have $\mathcal{T}={\mathcal{T}}^{\ast }+o_p\left(1\right)$. Theorem 3 holds by Slutsky Theorem.

Estimation of asymptotic covariance. The asymptotic covariance Σ_EQIF (β₀) can be consistently estimated by ${\widehat{\mathbf{\Sigma }}}_{EQIF}\left({\widehat{\beta }}_Q\right)={\left\{{\widehat{\mathit{J}}}^{\text{T}}\left({\widehat{\beta }}_Q\right){\widehat{\mathit{W}}}^{-1}\widehat{\mathit{J}}\left({\widehat{\beta }}_Q\right)\right\}}^{-1}$, where

$$\widehat{\mathit{J}}\left(\beta \right)=\left(\begin{array}{c}\widehat{\mathrm{\Gamma }}\left(\beta ,B_1\right)\\ ⋮\\ \widehat{\mathrm{\Gamma }}\left(\beta ,B_{m_0}\right)\end{array}\right),$$

and

$$\begin{gathered} \widehat{\mathrm{\Gamma }}\left(\beta ,B\right)=\frac{1}{n}\sum _{i=1}^n\sum _{k=1}^K{\mathrm{\Delta }}_{ik}\left[\sum _{m=1}^K\frac{\partial {\widehat{R}}_{im}\left(\beta ,T_{ik}\right){\widehat{\varphi }}_{imk}\left(\beta ,T_{ik},B\right)}{\partial {\beta }^{\text{T}}}-\frac{{\widehat{S}}_k^{\left(3\right)}\left(\beta ,T_{ik},B\right)}{{\widehat{S}}_k^{\left(0\right)}\left(\beta ,T_{ik}\right)}\right. \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left.-\frac{{\widehat{S}}_k^{\left(4\right)}\left(\beta ,T_{ik},B\right)+{\widehat{S}}_k^{\left(5\right)}\left(\beta ,T_{ik},B\right)}{{\widehat{S}}_k^{\left(0\right)}\left(\beta ,T_{ik}\right)}+\frac{{\widehat{S}}_k^{\left(2\right)}\left(\beta ,T_{ik},B\right){\left\{{\widehat{S}}_k^{\left(1\right)}\left(\beta ,T_{ik}\right)\right\}}^{\text{T}}}{{\left\{{\widehat{S}}_k^{\left(0\right)}\left(\beta ,T_{ik}\right)\right\}}^2}\right], \end{gathered}$$

$$\widehat{\mathit{W}}={\left\{\widehat{\mathit{\omega }}\left({\widehat{\beta }}_Q,B_j,B_{j^{\prime }}\right)\right\}}_{j,j^{\prime }=1}^{m_0},$$

with

$$\widehat{\mathit{\omega }}\left(\beta ,B_j,B_{j^{\prime }}\right)=\frac{1}{n}\sum _{i=1}^n\sum _{k=1}^K\sum _{m=1}^K{\widehat{\mathit{H}}}_{ik}\left(\beta ,B_j\right){\widehat{\mathit{H}}}_{im}^{\text{T}}\left(\beta ,B_{j^{\prime }}\right),$$

$$\begin{gathered} {\widehat{\mathit{H}}}_{ik}\left(\beta ,B\right)={\mathrm{\Delta }}_{ik}{\widehat{\mathit{h}}}_{ik}\left(\beta ,T_{ik},B\right)-\sum _{l=1}^n{\mathrm{\Delta }}_{lk}{\widehat{\mathit{h}}}_{ik}\left(\beta ,T_{lk},B\right)\frac{Y_{ik}\left(T_{lk}\right){\widehat{r}}_{ik}\left(\beta ,T_{lk}\right)}{n{\widehat{S}}_k^{\left(0\right)}\left(\beta ,T_{lk}\right)} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\frac{n-n_k}{n_k}\sum _{l=1}^n{\mathrm{\Delta }}_{lk}{\widehat{\mathit{h}}}_{ik}\left(\beta ,T_{lk},B\right)\frac{Y_{ik}\left(T_{lk}\right)\left\{{\widehat{r}}_{ik}\left(\beta ,T_{lk}\right)-{\widehat{\psi }}_{ik}\left(\beta ,T_{lk}\right)\right\}}{n{\widehat{S}}_k^{\left(0\right)}\left(\beta ,T_{lk}\right)}, \end{gathered}$$

and ${\widehat{\mathit{h}}}_{ik}\left(\beta ,t,B\right)={\sum }_{m=1}^K{\widehat{R}}_{im}\left(\beta ,t\right){\widehat{\varphi }}_{imk}\left(\beta ,t,B\right)-{\widehat{S}}_k^{\left(2\right)}\left(\beta ,t,B\right)/{\widehat{S}}_k^{\left(0\right)}\left(\beta ,t\right)$.

Data Availability Statement

The data that support the findings in this paper are openly available in BioLINCC at https://biolincc.nhlbi.nih.gov/home/, reference number HLB00320404a.