Regression analysis of longitudinal data with correlated censoring and observation times

Abstract

Longitudinal data occur in many fields such as the medical follow-up studies that involve repeated measurements. For their analysis, most existing approaches assume that the observation or follow-up times are independent of the response process either completely or given some covariates. In practice, it is apparent that this may not be true. In this paper, we present a joint analysis approach that allows the possible mutual correlations that can be characterized by time-dependent random effects. Estimating equations are developed for the parameter estimation and the resulted estimators are shown to be consistent and asymptotically normal. The finite sample performance of the proposed estimators is assessed through a simulation study and an illustrative example from a skin cancer study is provided.

Appendices

Appendix 1: Proof of Theorem 1

To derive the asymptotic properties of the proposed estimators $\hat{\beta }$ and $\hat{\eta }$, we need the following regularity conditions analogous to those given by Lin et al. (2000) (Sect. 2):

(C1):

$\{\widetilde{N}_i(\cdot ), Y_i(\cdot ), C_i, \mathbf{Z}_i\}_{i=1}^n$ are independent and identically distributed.

(C2):

There exists a $\tau >0$ such that $P(C_i\ge \tau )>0$.

(C3):

Both $\widetilde{N}_i(t)$ and $Y_i(t)$ ($0 \le t \le \tau , i=1,\ldots ,n$) are bounded.

(C4):

W(t) and $\mathbf{Z}_i,\, i=1,\dots ,n$, have bounded variations and W(t) converges almost surely to a deterministic function w(t) uniformly in $t\in [0,\tau ]$.

(C5):

$A_{\beta }=E\{\int _0^{\tau } w(t) e^{\beta '_0 \mathbf{Z}_i+\eta _0' \mathbf{X}_i(t)} [\mathbf{Z}_i-e_z(t)]^{\otimes 2}d{\varLambda }^*_2(t)\}$ and ${\varOmega }_\eta =E\Big [\int _0^{\tau }\big \{\mathbf{X}_i(t)-\bar{x}(t)\big \}^{\otimes 2} e^{\eta _0' \mathbf{X}_i(t)}d{\varLambda }^*_1(t)\Big ]$ are both positive definite.

Under condition (C2), we define

$$\begin{aligned} U_1(\beta ;\hat{\eta })=\sum _{i=1}^n \int _0^{\tau } W(t)\mathbf{Z}_i\bigg [ Y_i(t)d\widetilde{N}_i(t)-e^{\beta ' \mathbf{Z}_i + \hat{\eta }^{\prime }{} \mathbf{X}_i(t)}d\widehat{{\varLambda }}^*_2(t)\bigg ], \end{aligned}$$

which is integrable under conditions (C3) and (C4). Also note that $d\widehat{{\varLambda }}^*_2(t)$ satisfies

$$\begin{aligned} \sum _{i=1}^n \bigg [ Y_i(t)d\widetilde{N}_i(t)-e^{\beta ' \mathbf{Z}_i + \hat{\eta }^{\prime }{} \mathbf{X}_i(t)}d\widehat{{\varLambda }}^*_2(t)\bigg ]=0,\;\,0\le t \le \tau . \end{aligned}$$

(16)

Let

$$\begin{aligned} \widehat{A}_\beta (\beta )=-n^{-1}\partial U_1(\beta ,\hat{\eta })/\partial \beta ', \widehat{A}_{\eta }(\eta )=-n^{-1}\partial U_1(\beta _0,\eta )/\partial {\eta }', \end{aligned}$$

and under (C1), let

$$\begin{aligned} A_\beta =\lim _{n\rightarrow \infty }\widehat{A}_\beta (\beta _0),\; A_\eta =\lim _{n\rightarrow \infty }\widehat{A}_\eta (\eta _0). \end{aligned}$$

The consistency of $\hat{\beta }$ and $\hat{\eta }$ follows from the facts that $U_1(\beta _0;\hat{\eta })$ and $U_{\eta }(\eta _0)$ both tend to 0 in probability as $n\rightarrow \infty$, and that under condition (C5), $\widehat{A}_\beta (\beta )$ and $-n^{-1} \partial U_{\eta }(\eta ) / \partial \eta '$ both converge uniformly to the positive definite matrices $A_{\beta }$ and ${\varOmega }_{\eta }$ over $\beta$ and $\eta$, respectively, in neighborhoods around the true values $\beta _0$ and $\eta _0$. Then the Taylor series expansions of $U_1(\hat{\beta };\hat{\eta })$ at $(\beta _0;\hat{\eta })$ and $(\beta _0,\eta _0)$ yield $n^{1/2}(\hat{\beta }-\beta _0)=A_\beta ^{-1}n^{-1/2}U_1(\beta _0;\hat{\eta })+o_p(1)= A_\beta ^{-1}\Big \{n^{-1/2}U_1(\beta _0;\eta _0)-A_\eta n^{1/2} (\hat{\eta }-\eta _0)\Big \}+o_p(1)$. The proof of Theorem 1 is sketched as follows:

• (1) First, using some derivation operation to $U_1(\beta ;\hat{\eta })$ and (16), we can get

  $$\begin{aligned} \widehat{A}_\beta (\beta )=n^{-1}\sum _{i=1}^n\int _0^{\tau }W(t)\big \{\mathbf{Z}_i-\widehat{E}_Z(t;\beta ,\hat{\eta })\big \}^{\otimes 2}e^{\beta '\mathbf{Z}_i+\hat{\eta }^{\prime }{}\mathbf{X}_i(t)}d\widehat{{\varLambda }}^*_2(t;\beta ,\hat{\eta }). \end{aligned}$$

• (2) Solving $d\widehat{{\varLambda }}^*_2(t;\beta _0,\eta _0)$ from (16) and applying to $U_1(\beta _0;\eta _0)$ yields

  $$\begin{aligned} U_1(\beta _0;\eta _0)=\sum _{i=1}^n\int _0^{\tau }w(t)\Big (\mathbf{Z}_i-e_z(t)\Big )dM_i(t)+o_p(n^{1/2}), \end{aligned}$$

  where $e_z(t)=lim_{n\rightarrow \infty }\widehat{E}_Z(t;\beta _0,\eta _0)$ as defined earlier in Sect. 3 and w(t) is a deterministic function defined under (C5).

• (3) Differentiation of $U_1(\beta _0, \eta )$ and (16) with respect to $\eta$ yields

  $$\begin{aligned} \widehat{A}_\eta (\eta )=n^{-1}\sum _{i=1}^n\int _0^{\tau }W(t)\big [\mathbf{Z}_i-\widehat{E}_Z(t;\beta _0,\eta )\big ] e^{\beta _0'\mathbf{Z}_i+\eta '\mathbf{X}_i(t)}X'_i(t) d\widehat{{\varLambda }}^*_2(t;\beta _0,\eta ). \end{aligned}$$

• (4) According to equation (5) and by using the asymptotic results in Lin et al. (2000) (A.5), one can show that

  $$\begin{aligned} n^{1/2}\{\hat{\eta }\!-\!\eta _0\}\!=\!{\varOmega }_\eta ^{-1}n^{-1/2}\sum _{i=1}^n\bigg [ \int _0^{\tau }\Big (\mathbf{X}_i(t)-\frac{s^{(1)}(t)}{s^{(0)}(t)}\Big ) d M^*_i(t)\bigg ] +\, o_p(1),\quad \end{aligned}$$

  (17)

  where ${\varOmega }_\eta =E\Big [\int _0^{\tau }\big \{\mathbf{X}_i(t)-\bar{x}(t)\big \}^{\otimes 2} e^{\eta _0'\mathbf{X}_i(t)}d{\varLambda }^*_1(t)\Big ]$, which is invertible under (C5).

Combining the results in steps (1)–(4), we have

$$\begin{aligned} U_1(\beta _0;\hat{\eta })= & {} \sum _{i=1}^n\bigg [\int _0^{\tau }w(t)\big \{\mathbf{Z}_i- e_z(t)\big \} dM_i(t)\bigg ] \\&-A_{\eta } {\varOmega }_\eta ^{-1}\sum _{i=1}^n \bigg [\int _0^{\tau }\big \{\mathbf{X}_i(t)-\bar{x}(t)\big \}d M_i^*(t)\bigg ]+o_p(n^{1/2}). \end{aligned}$$

Since $A_{\beta }$ is also invertible under (C5), it then follows from the multivariate central limit theorem that the conclusions hold.

Appendix 2: Proof of the null distribution of ${\mathcal {F}}(t,z)$ in Section 3

Let $V(\hat{\beta },\hat{\eta })=\sum _{i=1}^n\int _0^t I(\mathbf{Z}_i\le z)d\widehat{M}_i(s; \hat{\beta },\hat{\eta })$, then by the Taylor expansion,

$$\begin{aligned} {\mathcal {F}}(t,z;\hat{\beta },\;\hat{\eta })= & {} n^{-1/2}V(\beta _0,\eta _0)+\frac{\partial V(\beta _0,\eta _0)}{n\partial \eta '}\sqrt{n} (\hat{\eta }-\eta _0) \\&+ \frac{\partial V(\beta _0,\hat{\eta })}{n\partial \beta '}\sqrt{n} (\hat{\beta }-\beta _0)+o_p(1). \end{aligned}$$

Using the arguments and algebra manipulation similar to those in Appendix 1, we have $V(\beta _0,\eta _0)=\sum _{i=1}^n u_{1i}(t,z)+o_p(n^{1/2})$, where $u_{1i}(t,z)=\int _0^t \{I(\mathbf{Z}_i\le z)-e_I(s,z)\}dM_i(s)$. Also, $\frac{\partial V(\beta _0,\eta _0)}{n\partial \eta '}$ and $\frac{\partial V(\beta _0,\hat{\eta })}{n\partial \beta '}$ can be estimated by $-\widehat{{\varPhi }}_\eta (t,z)$ and $-\widehat{{\varPhi }}_\beta (t,z)$, respectively.

In addition, it follows from (17) and Theorem 1 that

$$\begin{aligned} \sqrt{n}\{\hat{\eta }-\eta _0\}={\varOmega }_\eta ^{-1}n^{-1/2}\sum _{i=1}^n\bigg [ \int _0^{\tau }\Big (\mathbf{X}_i(t)-\frac{s^{(1)}(t)}{s^{(0)}(t)}\Big ) d M^*_i(t)\bigg ]+o_p(1), \end{aligned}$$

and

$$\begin{aligned} \sqrt{n}\{\hat{\beta }-\beta _0\}=A^{-1}_{\beta }n^{-1/2}\sum _{i=1}^n(v_{1i}-v_{2i})+o_p(1), \end{aligned}$$

where $v_{1i}=\int _0^{\tau }w(t)\big (\mathbf{Z}_i- e_z(t)\big )d M_i(t)$, and $v_{2i}=\int _0^{\tau } A_\eta {\varOmega }_\eta ^{-1}\big (\mathbf{X}_i(t)- \bar{x}(t)\big ) d M^*_i(t) \,$. Therefore, ${\mathcal {F}}(t,z;\hat{\beta },\hat{\eta })$ can be expressed as a sum of i.i.d. mean-zero terms for fixed t. By the multivariate central limit theorem, ${\mathcal {F}}(t,z)$ converges in finite-dimensional distribution to a mean-zero Gaussian process. Since ${\mathcal {F}}(t,z)$ is tight based on the empirical process theory, ${\mathcal {F}}(t,z)$ converges weakly to a mean-zero Gaussian process that can be approximated by $\widehat{\mathcal {F}}(t,z)$ given by Eq. (10).