Generalized mean residual life models for case-cohort and nested case-control studies

Abstract

Mean residual life (MRL) is the remaining life expectancy of a subject who has survived to a certain time point and can be used as an alternative to hazard function for characterizing the distribution of a time-to-event variable. Inference and application of MRL models have primarily focused on full-cohort studies. In practice, case-cohort and nested case-control designs have been commonly used within large cohorts that have long follow-up and study rare diseases, particularly when studying costly molecular biomarkers. They enable prospective inference as the full-cohort design with significant cost-saving benefits. In this paper, we study the modeling and inference of a family of generalized MRL models under case-cohort and nested case-control designs. Built upon the idea of inverse selection probability, the weighted estimating equations are constructed to estimate regression parameters and baseline MRL function. Asymptotic properties of the proposed estimators are established and finite-sample performance is evaluated by extensive numerical simulations. An application to the New York University Women’s Health Study is presented to illustrate the proposed models and demonstrate a model diagnostic method to guide practical implementation.

Appendix

1.1 Simulation study

We conducted numerical simulations to compare the IPCW method and the QPS estimator under the full cohort of 1000 subjects when censoring rates were approximately 10%, 30% and 80%. A total of 500 simulations were conducted for both the proportional and additive MRL models. We reported the bias, the standard deviation (SD) of the estimates, the average of estimated standard error (SE) and the coverage rate (CP) of 95% Wald-type confidence intervals (see results in Table 6). The SEs of the estimates were calculated through standard bootstrap method. Based on the simulation results, the two estimators had similar performance when censoring probability was low. The biases were all small and the means of estimated SEs were close to the empirical SDs of the parameter estimators. The 95% Wald-type confidence intervals had proper coverage rate. However, when the censoring rate was 80%, the IPCW performance dropped and underestimated SEs, which led to low coverage probabilities.

Table 6 Comparison between IPCW estimator and QPS estimator

1.2 Regularity conditions

Let $u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }), u_{\breve{\varvec{Z}}}(t;\varvec{\beta })$ and $u_{\bar{\varvec{Z}}}(t)$ be the limits of $\tilde{\varvec{Z}}(t;\varvec{\beta }), \breve{\varvec{Z}}(t;\varvec{\beta })$ and $\bar{\varvec{Z}}(t;\varvec{\beta })$, respectively. We assume the following regularity conditions:

• (C1) sup supp(F) $\le$ sup supp(G), where $F(\cdot )$ and $G(\cdot )$ are the distribution functions of T and C, respectively;

• (C2) $\varvec{Z}$ is bounded;

• (C3) $m_*(t)$ is continuously differentiable on [0,$\tau$];

• (C4) $A=\int _0^\tau E[(\varvec{Z}-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)) (\varvec{Z}-u_{\breve{\varvec{Z}}}(t;\varvec{\beta }_*))' (\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}\}dN(t) -Y(t)d\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}\})]$ is nonsingular;

Proof of Theorem 1(i)

First, we want to establish the consistency of the estimators $\hat{m}_0(t)$ and $\hat{\varvec{\beta }}$. Condition (C3) implies that $m_0(t)$ is of bounded variation on $[0,\tau ]$. Define $\mathcal {B}=\{\varvec{\beta }:||\varvec{\beta }-\varvec{\beta }_*||\le \epsilon \}$ for any $\epsilon >0$ and we have $E(w_i|\mathcal {F})=1$. By the strong law of large numbers and the fact that $E\{w_idM_i(t)\}=0$, for large n, $t\in [0,\tau ], \varvec{\beta }\in \mathcal {B}$, and sufficiently large $\theta$,

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^nw_i \left[ dN_i(t)-Y_i(t)\frac{dg\{m_0(t)+\theta +\varvec{\beta }'\varvec{Z}_i\}+dt}{g\{m_0(t)+\theta +\varvec{\beta }'\varvec{Z}_i\}}\right] <0, \end{aligned}$$

(11)

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^nw_i \left[ dN_i(t)-Y_i(t)\frac{dg\{m_0(t)-\theta +\varvec{\beta }'\varvec{Z}_i\}+dt}{g\{m_0(t)-\theta +\varvec{\beta }'\varvec{Z}_i\}}\right] >0. \end{aligned}$$

(12)

By (11), (12), and the monotonicity and continuity of g function, for any $t\in [0,\tau ]$ and $\varvec{\beta }\in \mathcal {B}$, there exists a unique $\hat{m}_0(t;\varvec{\beta })$ that satisfies

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^nw_i\left[ dN_i(t)-Y_i(t)\frac{dg\{\hat{m}_0(t;\varvec{\beta })+\varvec{\beta }'\varvec{Z}_i\}+dt}{g\{\hat{m}_0(t;\varvec{\beta })+\varvec{\beta }'\varvec{Z}_i\}}\right] =0. \end{aligned}$$

(13)

Note that (11) and (12) hold for any $\theta >0$ when and only when $\varvec{\beta }=\varvec{\beta }_*$. Then we have that $\hat{m}_0(t;\varvec{\beta })$ converges to $m_0(t;\varvec{\beta })$ uniformly in $t\in [0,\tau ]$ and $\varvec{\beta }$ in a compact set which contains the true parameter $\varvec{\beta }_*$, and $m_0(t;\varvec{\beta }_*) = m_*(t)$. Thus, to prove the existence and uniqueness of $\hat{\varvec{\beta }}$ and $\hat{m}_0(t)$, it suffices to show that there exists a unique solution to $U(\varvec{\beta })=0$. Take derivative of (13) with respect to $\varvec{\beta }$, we have

$$\begin{aligned}&\frac{d\hat{m}_0(t)}{d\varvec{\beta }}\frac{\sum _{i=1}^nw_i[\dot{g}\{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}dN_i(t)-Y_i(t)d\dot{g}\{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}]}{\sum _{i=1}^nw_iY_i(t)\dot{g}\{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}} - d(\frac{d\hat{m}_0(t)}{d\varvec{\beta }}) \\&\quad = - \frac{\sum _{i=1}^nw_i\varvec{Z}_i[\dot{g}\{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}dN_i(t)-Y_i(t)d\dot{g}\{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}]}{\sum _{i=1}^nw_iY_i(t)\dot{g}\{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}}, \end{aligned}$$

which is a first-order linear ordinary differential equation about $d\hat{m}_0(t)/d\varvec{\beta }$. The solution is

$$\begin{aligned} \frac{d\hat{m}_0(t)}{d\varvec{\beta }}= \frac{d\hat{m}_0(t;\varvec{\beta })}{d\varvec{\beta }}=-\frac{1}{K(t;\varvec{\beta })}\int _t^\tau K(u;\varvec{\beta })Q(u;\varvec{\beta }) \equiv -\breve{\varvec{Z}}(t;\varvec{\beta }), \end{aligned}$$

where

$$\begin{aligned} K(t;\varvec{\beta })= & {} \exp \left\{ -\int _0^t\frac{\sum _{i=1}^nw_i[\dot{g} \{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}dN_i(t)-Y_i(t)d\dot{g} \{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}]}{\sum _{i=1}^nw_iY_i(t) \dot{g}\{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}}\right\} ,\\ Q(t;\varvec{\beta })= & {} \frac{\sum _{i=1}^nw_i\varvec{Z}_i[\dot{g}\{\hat{m}_0(t)+ \varvec{\beta }'\varvec{Z}_i\}dN_i(t)-Y_i(t)d\dot{g} \{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}]}{\sum _{i=1}^nw_iY_i(t)\dot{g} \{\hat{m}_0(t)+\varvec{\beta }'\varvec{Z}_i\}}. \end{aligned}$$

Let $\hat{A}(\varvec{\beta }_*) \doteq dU(\varvec{\beta })/d\varvec{\beta }|_{\varvec{\beta }=\varvec{\beta }_*}$. We have

$$\begin{aligned} \hat{A}(\varvec{\beta }_*)\doteq & {} \frac{1}{n}\sum _{i=1}^n\int _0^\tau w_i[\varvec{Z}_i-\varvec{\bar{Z}}(t;\varvec{\beta }_*)][\varvec{Z}_i-\varvec{\breve{Z}}(t;\varvec{\beta }_*)]'[\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dN_i(t)\\&\quad -Y_i(t)d\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}]\\= & {} \frac{1}{n}\sum _{i=1}^n\int _0^\tau w_i[\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)][\varvec{Z}_i-u_{\breve{\varvec{Z}}}(t;\varvec{\beta }_*)]'[\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dN_i(t)\\&\quad -Y_i(t)d\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}] +o_p(1)\\= & {} A+o_p(1) \end{aligned}$$

Thus, $\hat{A}(\varvec{\beta }_*)$ converges in probability to a nonrandom A. It is easy to check that $U(\varvec{\beta }_*)\rightarrow 0$ almost surely, and A is nonsingular by (C4). The convergence of $\hat{A}(\varvec{\beta }_*)$ and the continuity of ${A}(\varvec{\beta })$ imply that we can find a small neighborhood of $\varvec{\beta }_*$ in which $\hat{A}(\varvec{\beta }_*)$ is nonsingular when n is large enough. Therefore, it follows from the inverse function theorem that within a small neighborhood of $\varvec{\beta }_*$, there exists a unique solution $\hat{\varvec{\beta }}$ to $U(\varvec{\beta })=0$ for sufficiently large n. Thus, there exists unique estimators $\hat{\varvec{\beta }}$ and $\hat{m}(t)$. Since $\hat{\varvec{\beta }}$ is strongly consistent to $\varvec{\beta }_*$, then it follows the uniform convergence of $\hat{m}_0(t;\varvec{\beta })$ to $m_0(t;\varvec{\beta })$ that $\hat{m}_0(t)\doteq \hat{m}_0(t;\hat{\varvec{\beta }}) \rightarrow m_0(t;\varvec{\beta }_*) = m_*(t)$ almost surely in $[0,\tau ]$.

Proof of Theorem 1(ii)

In this section, we first prove the Theorem 1(ii) under the CC design, then prove the theorem under the NCC design following the proof from Lu and Liu (2012). We know from Eq. (8) that

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^n w_i\left[ g\{m_0(t)+\varvec{\beta }^{'}\varvec{Z}_i\}dN_i(t)-Y_i(t)d[g\{m_0(t)+\varvec{\beta }^{'}\varvec{Z}_i\}+t]\right] \\&\quad = \frac{1}{n}\sum _{i=1}^nw_ig\{m_0(t)+\varvec{\beta }'\varvec{Z}_idM_i(t)\} \\&\qquad \frac{1}{n}\sum _{i=1}^n w_i\left[ g\{\hat{m}_0(t)+\varvec{\beta }^{'}\varvec{Z}_i\}dN_i(t)-Y_i(t)d[g\{\hat{m}_0(t)+\varvec{\beta }^{'}\varvec{Z}_i\}+t]\right] =0 \end{aligned}$$

Subtract the above two equations and use Taylor expansion, we have,

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^nw_i\dot{g}\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}[\hat{m}_0(t)-m_0(t)]dN_i(t) \\&\qquad -\frac{1}{n}\sum _{i=1}^nw_iY_i(t)d\dot{g}\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}[\hat{m}_0(t)-m_0(t)] \\&\qquad -\frac{1}{n}\sum _{i=1}^nw_iY_i(t)d\dot{g}\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}[d\hat{m}_0(t)-dm_0(t)] \\&\quad =-\frac{1}{n}\sum _{i=1}^nw_ig\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}dM_i(t) \end{aligned}$$

Hence, following the first-order ordinary differential equation,

$$\begin{aligned}&[\hat{m}_0(t)-m_0(t)]\frac{\sum _{i=1}^nw_i[\dot{g}\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}dN_i(t)-Y_i(t)d\dot{g}\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}]}{\sum _{i=1}^nw_iY_i(t)\dot{g}\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}} \\&\qquad -[d\hat{m}_0(t)-dm_0(t)] =-\frac{\sum _{i=1}^nw_ig\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}dM_i(t)}{\sum _{i=1}^nw_iY_i(t)\dot{g}\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}}, \\&\hat{m}_0(t)-m_0(t) = -\frac{1}{K(t;\varvec{\beta })}\int _t^\tau K(u;\varvec{\beta })\frac{\sum _{i=1}^nw_ig\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}dM_i(t)}{\sum _{i=1}^nw_iY_i(t)\dot{g}\{m_0(t)+\varvec{\beta }'\varvec{Z}_i \}}. \end{aligned}$$

Let $U(\varvec{\beta }_*) \doteq U(\varvec{\beta }_*,\hat{m}_0(t;\varvec{\beta }_*))$ and we have

$$\begin{aligned} U(\varvec{\beta }_*,m_*(t))= & {} \frac{1}{n}\sum _{i=1}^n\int _0^{\tau }w_i\varvec{Z}_i[g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dN_i(t) \\&-Y_i(t)dg\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}-Y_i(t)dt]. \end{aligned}$$

By using Taylor expansion again in $U(\varvec{\beta }_*,\hat{m}_0(t;\varvec{\beta }_*)) - U(\varvec{\beta }_*,m_*(\cdot ))$, we have

$$\begin{aligned}&U(\varvec{\beta }_*,\hat{m}_0(t;\varvec{\beta }_*)) - U(\varvec{\beta }_*,m_*(\cdot )) \\&\quad = \frac{1}{n}\sum _{i=1}^n\int _0^\tau w_i\varvec{Z}_i[\hat{m}_0(t)-m_0(t)]\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dN_i(t) \\&\qquad -w_i\varvec{Z}_iY_i(t)[\hat{m}_0(t)-m_0(t)]d\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\} \\&\qquad -w_i\varvec{Z}_iY_i(t)[d\hat{m}_0(t)-dm_0(t)]\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\} \\&\quad =\frac{1}{n}\sum _{i=1}^n\int _0^\tau w_i[\varvec{Z}_i-\bar{\varvec{Z}}(t;\varvec{\beta }_*)][\hat{m}_0(t)-m_0(t)][\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dN_i(t) \\&\qquad -Y_i(t)d\dot{g}\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}] - w_i\bar{\varvec{Z}}(t;\varvec{\beta }_*)g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t) \\&\quad =-\frac{1}{n}\sum _{i=1}^n\int _0^\tau w_i[\tilde{\varvec{Z}}(t;\varvec{\beta }_*)+\bar{\varvec{Z}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t). \end{aligned}$$

Thus,

$$\begin{aligned}&\sqrt{n}U(\varvec{\beta }_*) = \sqrt{n}U(\varvec{\beta }_*,\hat{m}_0(t;\varvec{\beta }_*)) \\&\quad = \sqrt{n}U(\varvec{\beta }_*,m_*)+\sqrt{n}[U(\varvec{\beta }_*,\hat{m}_0(t;\varvec{\beta }_*))-U(\varvec{\beta }_*,m_*)] \\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\tau w_i[\varvec{Z}_i-\bar{\varvec{Z}}(t;\varvec{\beta }_*)-\tilde{\varvec{Z}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*) \\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\tau w_i[\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*)\\&\qquad + o_p(1) \\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\tau [\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*)\\&\qquad +\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\tau (w_i-1)[\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)] \\&\qquad \quad g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*) + o_p(1)\\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\tau [\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*)\\&\qquad -\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\tau (1-\delta _i)(1-\gamma _i/p_{0i})[\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)] \\&\qquad \quad g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*) + o_p(1). \end{aligned}$$

As we defined in our manuscript, let $\mathcal {G}_i$ be the $\sigma$-field generated by $\{\tilde{T}_i,\delta _i,\varvec{Z}_i, i=1,\ldots ,n\}$ and $\mathcal {F}_i$ be the $\sigma$-field generated by $\{\tilde{T}_i,\delta _i, i=1,\ldots ,n\}$. We denote

$$\begin{aligned} \eta _i = (1-\delta _i)\int _0^\tau [\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)- u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*' \varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*). \end{aligned}$$

It is evident that $E(1-\gamma _i/p_{0i}|\mathcal {F}_i)=0$ and $E\{\eta _i(1-\gamma _i/p_{0i}|\mathcal {F}_i)\}=E\{\eta _i E (1-\gamma _i/p_{0i}|\mathcal {F}_i)\}$=0. Following the proof in Lu and Tsiatis (2006), $var\{\eta _i(1-\gamma _i/p_{0i})\}= E \{\eta _i^{\otimes 2}(1-p_{0i})/p_{0i}\}- E [\eta _i(1-p_{0i})/p_{0i}]^{\otimes 2} = \Sigma _2$. Condition on $\mathcal {F}_i, \{\eta _i(1-\gamma _i/p_{0i}),i=1,\ldots ,n\}$ and the first term of $\sqrt{n}U(\varvec{\beta }_*)$ are uncorrelated. Therefore, $\sqrt{n}U(\varvec{\beta }_*)$ is asymptotically normal with mean zero and variance-covariance matrix $\Sigma =\Sigma _1+\Sigma _2$. By Taylor expansion and consistency of $\hat{\varvec{\beta }}$, it follows

$$\begin{aligned}&\sqrt{n}(\hat{\varvec{\beta }}-\varvec{\beta }_*) \\&\quad = -A^{-1}\sqrt{n}U(\varvec{\beta }_*) +o_p(1) \\&\quad = -A^{-1}\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\tau w_i[\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]\\&\qquad g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*)+o_p(1), \end{aligned}$$

thus $\sqrt{n}(\hat{\varvec{\beta }}-\varvec{\beta }_*)\rightarrow N\{A^{-1}\Sigma (A^{-1})^{'}\}.$

Under the nested case-control design, the asymptotic distribution of $\hat{\varvec{\beta }}$ is more difficult to derive because the NCC sampling scheme is a dynamic process. The probability of being selected as a control is neither a constant not independent. Thus, we consider the idea of central limit theory for asymptotically negatively dependent random variables (Zhang 2000), which has been used in the proof of Lu and Liu (2012). Based on the following asymptotical representation, we have

$$\begin{aligned} U(\varvec{\beta }_*)= & {} \frac{1}{n}\sum _{i=1}^n\int _0^\tau w_i[\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)] \\&g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*) +o_p(1) \\\equiv & {} U_1(\varvec{\beta }_*) + U_2(\varvec{\beta }_*) +o_p(\frac{1}{\sqrt{n}}). \end{aligned}$$

By martingale central limit theorem, $\sqrt{n}U_1(\varvec{\beta }_*) = \frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^\tau w_i[\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t;\varvec{\beta }_*,m_*) \rightarrow N(0,\Sigma _1)$ as $n \rightarrow \infty$. Since $E(w_i-1|\mathcal {F}_i)$ = 0, it is evident that $U_1(\varvec{\beta }_*)$ and $U_2(\varvec{\beta }_*)$ are uncorrelated. However, because of the NCC sampling scheme, $w_i$ and $w_j$ ($i \ne j$) are correlated even after conditioning on $\mathcal {F}$. Since $(w_i-1)^2 = (1-\delta _i)(\gamma _i-p_{0i}^2)/p_{0i}^2$, then $E\{(w_i-1)^2|\mathcal {F}\} = (1-\delta _i)(1-p_{0i})/p_{0i}$. Thus, the conditional variance of $\sqrt{n}U_2(\varvec{\beta }_*)$ can be written as

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^n\frac{1-p_{0i}}{p_{0i}}(1-\delta _i) \left[ \int _0^\tau [\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t)\right] ^{\otimes 2} \\&\qquad +\frac{1}{n}\sum _{i\ne j}E\{(\frac{\gamma _i}{p_{0i}}-1)(\frac{\gamma _j}{p_{0j}}-1)|\mathcal {F}\} \\&\qquad *(1-\delta _i)(1-\delta _j)\int _0^\tau [\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t) \\&\qquad \quad \left[ \int _0^\tau [\varvec{Z}_j-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_j\}dM_j(t)\right] '. \end{aligned}$$

According to Samuelsen (1997), for $i\ne j, Cov(\gamma _i,\gamma _j|\mathcal {F})=\rho _{ij}(1-p_{0i})(1-p_{0j})$, where $\rho _{ij}=-\frac{m}{n}\int _0^{min(\tilde{T_i},\tilde{T_j})} \frac{\bar{g}_1(t)}{\bar{y}(t)}dm_*(t)+\frac{\bar{g}_2(t)}{\bar{y}(t)}dt+O_p(n^{-2})$. Thus, with some algebra, the Var$\{\sqrt{n}U_2(\varvec{\beta }_*)|\mathcal {F}\}$ can be written as

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^n(1-\delta _i)\frac{1-p_{0i}}{p_{0i}}\left[ \int _0^\tau [\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t)\right] ^{\otimes 2} \\&\qquad -m\int _0^\tau [\frac{1}{n}\sum _{i=1}^nY_i(t)\frac{1-p_{0i}}{p_{0i}}(1-\delta _i)\int _0^\tau [\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)] \\&\qquad \quad g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t)]^{\otimes 2}\left( \frac{\bar{g}_1(t)}{\bar{y}(t)}dm_*(t) +\frac{\bar{g}_2(t)}{\bar{y}(t)}dt\right) +o_p(1), \end{aligned}$$

where $\bar{g}_1(t) = \sum _{j=1}^n\frac{Y_j(t)\dot{g}\{\hat{m}_0(t)+\hat{\varvec{\beta }}'\varvec{Z}_j\}}{g\{\hat{m}_0(t)+\hat{\varvec{\beta }}'\varvec{Z}_j\}}, \bar{g}_2(t) = \sum _{j=1}^n\frac{Y_j(t)}{g\{\hat{m}_0(t)+\hat{\varvec{\beta }}'\varvec{Z}_j\}}, \bar{y}(t)=\sum _{j=1}^nY_j(t)$. According to strong law of large numbers, we have

$$\begin{aligned} \Sigma _2\equiv & {} lim_{n\rightarrow \infty } Var\{\sqrt{n}U_2(\varvec{\beta }_*)|\mathcal {F}\} \\= & {} E\left[ \frac{1-s_0}{s_0}(1-\delta )[[\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM_i(t)]^{\otimes 2}\right] \\&-m\int _0^\tau (E[Y(t)(1-\delta )\frac{1-s_0}{s_0} \\&\quad \int _0^\tau [\varvec{Z}_i-u_{\bar{\varvec{Z}}}(t;\varvec{\beta }_*)-u_{\tilde{\varvec{Z}}}(t;\varvec{\beta }_*)]g\{m_*(t)+\varvec{\beta }_*'\varvec{Z}_i\}dM(t)]) ^{\otimes 2} \\&\qquad \left( \frac{\bar{g}_1(t)}{\bar{y}(t)}dm_*(t)+\frac{\bar{g}_2(t)}{\bar{y}(t)}dt\right) , \end{aligned}$$

where

$$\begin{aligned} s_{0i} = lim_{n\rightarrow \infty } p_{0i} = 1-\exp \left\{ -m\int _0^{\tilde{T}_i} \frac{\bar{g}_1(t)}{\bar{y}(t)}dm_0(t)+\frac{\bar{g}_2(t)}{\bar{y}(t)}dt \right\} . \end{aligned}$$

Thus, by the central limit theory for asymptotically negatively dependent random variables (Zhang 2000), we have $\sqrt{n}U_2(\varvec{\beta }_*) \rightarrow N(0,\Sigma _2)$ as $n\rightarrow \infty$, and

$$\begin{aligned} \sqrt{n}U(\varvec{\beta }_*) \rightarrow N(0,\Sigma _1+\Sigma _2), \end{aligned}$$

in distribution as $n \rightarrow \infty$. It is easy to see that $\Sigma _1+\Sigma _2 = \Sigma$. Follow by Taylor expansion and consistency of $\hat{\varvec{\beta }}$, we have $\sqrt{n}(\hat{\varvec{\beta }}-\varvec{\beta }_*)\rightarrow N\{A^{-1}\Sigma (A^{-1})^{'}\}$.