Robust estimation for panel count data with informative observation times and censoring times

Abstract

We consider the semiparametric regression of panel count data occurring in longitudinal follow-up studies that concern occurrence rate of certain recurrent events. The analysis of panel count data involves two processes, i.e, a recurrent event process of interest and an observation process controlling observation times. However, the model assumptions of existing methods, such as independent censoring time and Poisson assumption, are restrictive and questionable. In this paper, we propose new joint models for panel count data by considering both informative observation times and censoring times. The asymptotic normality of the proposed estimators are established. Numerical results from simulation studies and a real data example show the advantage of the proposed method.

Appendix: Proofs of asymptotics

To establish the consistency and asymptotic normality of $\hat{\eta }$ and $\hat{\beta }$, we assume that $\mathbf X _i(t)'s$ are of bounded variations. Furthermore, we assume that, as $n\rightarrow +\infty$,

$$\begin{aligned} S_{1}(t;\eta _0)= & {} \frac{1}{n}\sum _{i=1}^n \exp \{\eta _0'{} \mathbf X _i^*(t)\} \rightarrow s_{1}(t;\eta _0)\\ S_{2}(t;\eta _0)= & {} \frac{1}{n}\sum _{i=1}^n \exp \{\eta _0'{} \mathbf X _i^*(t)\}{} \mathbf X _i^*(t)(\mathbf X _i^*(t))' \rightarrow s_{2}(t;\eta _0)\\ S_{3}(t;\eta _0)= & {} \frac{1}{n}\sum _{i=1}^n \exp \{\eta _0'{} \mathbf X _i^*(t)\}{} \mathbf X _i^*(t) \rightarrow s_{3}(t;\eta _0) \\ S_4(t;\eta _0)= & {} \frac{1}{n}\sum _{i=1}^n \exp \{\eta _0'{} \mathbf X _i^*(t)\}{} \mathbf X _i \rightarrow s_{4}(t;\eta _0)\\ S_{5}(t;\eta _0)= & {} \frac{1}{n}\sum _{i=1}^n \exp \{\eta _0'{} \mathbf X _i^*(t)\}{} \mathbf X _i(\mathbf X _i^*(t))' \rightarrow s_{5}(t;\eta _0) \end{aligned}$$

Define

$$\begin{aligned} dM_i(t)=\varDelta _i(t)dO_i(t)-\exp \{\eta _0'{} \mathbf X _i^*(t)\}d\mathcal{B}(t) \end{aligned}$$

and

$$\begin{aligned} dR_i(t)=N_i(t)\exp \left( -\beta _0'{} \mathbf X _i\right) \varDelta _i(t)dO_i(t)-\exp \left\{ \eta _0'{} \mathbf X _i^*(t)\right\} d\mathcal{A}(t). \end{aligned}$$

It is easy to show that $dM_i(t)$ and $dR_i(t)$ are mean 0 stochastic process. Also, define $\bar{\mathbf{x }}(t)=\frac{s_4(t;\eta _0)}{s_1(t;\eta _0)}$, $\bar{\mathbf{x }}^*(t)=\frac{s_3(t;\eta _0)}{s_1(t;\eta _0)}$.

1.1 A.1 Proof of consistency and asymptotic normality of $\hat{\eta } =(\hat{\gamma }, \hat{\xi })$

Recall that $\hat{\eta }$ is the solution to $V(\eta )=0$ where

$$\begin{aligned} V(\eta )=\sum _{i=1}^n \int _{0}^{\tau } R(t)\left\{ \mathbf{X }_i^*(t)-\bar{\mathbf{X }}^*(t;\eta )\right\} \varDelta _i(t)dO_i(t). \end{aligned}$$

By the Taylor series expansion, we have

$$\begin{aligned} V(\hat{\eta })-V(\eta _0)=\frac{\partial V(\eta )}{\partial \eta }|_{\eta =\eta _0}(\hat{\eta }-\eta _0)+\frac{1}{2}(\hat{\eta }-\eta _0)\frac{\partial ^2 V(\eta )}{\partial \eta \partial \eta '}|_{\eta =\eta ^*}(\hat{\eta }-\eta _0)', \end{aligned}$$

where $\eta ^* \in (\eta _0, \hat{\eta })$. Since,

$$\begin{aligned} \frac{\partial \bar{\mathbf{X }}^*(t;\eta )}{\partial \eta }|_{\eta =\eta _0}= & {} \frac{\sum _{i=1}^n\sum _{j=1}^n \exp \{\eta _0'{} \mathbf X _i^*(t)\}\exp \{\eta _0'{} \mathbf X _j^*(t)\} (\mathbf X _j^*(t)-\mathbf X _i^*(t)) (\mathbf X _j^*(t))'}{ \left( \sum _{i=1}^n\exp \{\eta _0'{} \mathbf X _i^*(t)\}\right) ^2}\\= & {} \frac{S_2(t;\eta _0)}{S_1(t;\eta _0)}- \left( \frac{S_3(t;\eta _0)}{S_1(t;\eta _0)}\right) ^{\otimes 2},\\ \end{aligned}$$

thus, we have,

$$\begin{aligned}&\frac{\partial V(\eta )}{\partial \eta }|_{\eta =\eta _0}=-\sum _{i=1}^n\int _{0}^{\tau }R(t)\frac{\partial \bar{\mathbf{X }}^*(t;\eta )}{\partial \eta }|_{\eta =\eta _0}\varDelta _i(t)dO_i(t)\\&\quad =-\sum _{i=1}^n\int _{0}^{\tau }R(t)\left\{ \frac{S_2(t;\eta _0)}{S_1(t;\eta _0)}-\left( \frac{S_3(t;\eta _0)}{S_1(t;\eta _0)}\right) ^{\otimes 2}\right\} [dM_i(t)+\exp \{\eta _0'{} \mathbf X _i^*(t)\}d\mathcal{B}(t)]\\&\qquad -\sum _{i=1}^n\int _{0}^{\tau }R(t) \left\{ \frac{s_2(t;\eta _0)}{s_1(t;\eta _0)} -\left( \frac{s_3(t;\eta _0)}{s_1(t;\eta _0)}\right) ^{\otimes 2}\right\} dM_i(t)\\&\qquad -n\int _{0}^{\tau }R(t) \left\{ s_2(t;\eta _0)-\frac{s_3(t;\eta _0)^{\otimes 2}}{s_1(t;\eta _0) }\right\} d\mathcal{B}(t)+o_p(1). \end{aligned}$$

Define $P_n=-\frac{1}{n}\frac{\partial V(\eta )}{\partial \eta }|_{\eta =\eta _0}$. Then we have,

$$\begin{aligned} \sqrt{n}(\hat{\eta }-\eta _0)= P_n^{-1}\frac{1}{\sqrt{n}}V(\eta _0)+o_p(1). \end{aligned}$$

Since $\frac{\partial ^2 V(\eta )}{\partial \eta \partial \eta '}|_{\eta =\eta ^*}$ is bounded in probability. Furthermore, following the arguments similar to those given in Appendix 2 of Lin and Ying (2001), we have,

$$\begin{aligned}&\frac{1}{\sqrt{n}}V(\eta _0)=\frac{1}{\sqrt{n}}\sum _{i=1}^n \int _{0}^{\tau } R(t)\left\{ \mathbf{X }_i^*(t)-\bar{\mathbf{X }}^*(t;\eta _0)\right\} \varDelta _i(t)dO_i(t)\\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n \int _{0}^{\tau } R(t)\left\{ \mathbf{X }_i^*(t)-\bar{\mathbf{X }}^*(t;\eta _0)\right\} \left\{ \varDelta _i(t)dO_i(t)-\exp \left\{ \eta _0'{} \mathbf X _i^*(t)\right\} d\mathcal{B}(t)\right\} \\&\qquad +\frac{1}{\sqrt{n}}\sum _{i=1}^n \int _{0}^{\tau } R(t)\left\{ \mathbf{X }_i^*(t)-\bar{\mathbf{X }}^*(t;\eta _0)\right\} \exp \left\{ \eta _0'{} \mathbf X _i^*(t)\right\} d\mathcal{B}(t)\\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n \int _{0}^{\tau } R(t)\left\{ \mathbf{X }_i^*(t)-\bar{\mathbf{X }}^*(t;\eta _0)\right\} dM_i(t)\\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n \int _{0}^{\tau } R(t)\left\{ \mathbf{X }_i^*(t)-\bar{\mathbf{x }}^*(t)\right\} dM_i(t)+o_p(1)\\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n V_i+o_p(1), \end{aligned}$$

a sum of n independent mean 0 random vectors plus an asymptotically negligible term. Thus, by the multivariate CLT, $\frac{1}{\sqrt{n}}V(\eta _0)\rightarrow N(0,\varSigma _{\eta })$, where $\varSigma _{\eta }=E[\int _{0}^{\tau } R(t)\{\mathbf{X }_1^*(t)-\bar{\mathbf{x }}^*(t)\}dM_1(t)]^{\otimes 2}$.

Finally, we have $\sqrt{n}(\hat{\eta }-\eta _0)\rightarrow P^{-1}N(0,\varSigma _{\eta })$, as $n\rightarrow +\infty$, where

$$\begin{aligned} P=\lim _{n\rightarrow +\infty }P_n=\int _{0}^{\tau }R(t)\left\{ s_2(t;\eta _0)-\frac{s_3(t;\eta _0)^{\otimes 2}}{s_1(t;\eta _0)}\right\} d\mathcal{B}(t). \end{aligned}$$

It is easy to know that ${\hat{P}}$ and ${\hat{\varSigma }}_{\eta }$ is the consistent estimator of P and $\varSigma _{\eta }$, respectively.

1.2 A.2 Proof of consistency and asymptotic normality of $\hat{\beta }$

Recall that $\hat{\beta }$ is the solution to $U(\beta ;\hat{\eta })=0$ where

$$\begin{aligned} U(\beta ; \eta )= & {} \sum ^n_{i=1}\int ^\tau _0 W(t)\{\mathbf{X }_i-\bar{\mathbf{X }}(t; \eta )\}N_i(t)\exp (-\beta '{} \mathbf X _i)\varDelta _i(t)dO_i(t). \end{aligned}$$

By the Taylor series expansion, we have

$$\begin{aligned} U(\hat{\beta }; \hat{\eta })-U(\beta _0; \hat{\eta })=\frac{\partial U(\beta ;\hat{\eta })}{\partial \beta }|_{\beta =\beta _0}(\hat{\beta }-\beta _0)+\frac{1}{2}(\hat{\beta }-\beta _0)\frac{\partial ^2 U(\beta ,\hat{\eta })}{\partial \beta \partial \beta '}|_{\beta =\beta ^*}(\hat{\beta }-\beta _0)', \end{aligned}$$

where $\frac{\partial U(\beta ;\hat{\eta })}{\partial \beta }|_{\beta =\beta _0}=-\sum _{i=1}^n\int _{0}^{\tau }W(t)\{\mathbf{X }_i-\bar{\mathbf{X }}(t;\hat{\eta })\}N_i(t)\exp (-\beta _0'{} \mathbf X _i){} \mathbf X _i'\varDelta _i(t)dO_i(t)$.

Define $Q_n=-\frac{1}{n}\frac{\partial U(\beta ;\hat{\eta })}{\partial \beta }|_{\beta =\beta _0}$. Then we have $\sqrt{n}(\hat{\beta }-\beta _0)=Q_n^{-1}\frac{1}{\sqrt{n}}U(\beta _0;\hat{\eta })+o_p(1)$, since $\frac{\partial ^2 U(\beta ,\hat{\eta })}{\partial \beta \partial \beta '}|_{\beta =\beta ^*}$ is bounded. Therefore, it is sufficient to find the limit distribution of $\frac{1}{\sqrt{n}}U(\beta _0;\hat{\eta })$. Again, by the Taylor series expansion, we have

$$\begin{aligned} U(\beta _0;\hat{\eta }){\,=\,}U(\beta _0;\eta _0)+\frac{\partial U(\beta _0;\eta )}{\partial \eta }|_{\eta =\eta _0}(\hat{\eta }-\eta _0)+\frac{1}{2}(\hat{\eta }-\eta _0)\frac{\partial ^2 U(\beta _0;\eta )}{\partial \eta \partial \eta '}|_{\eta =\eta ^*}(\hat{\eta }-\eta _0)', \end{aligned}$$

where $\eta ^* \in (\eta _0,\hat{\eta })$.

As $\hat{\eta }$ is a consistent estimator of $\eta$ as showed in previous section and $\frac{\partial ^2 U(\beta _0;\eta )}{\partial \eta \partial \eta '}|_{\eta =\eta ^*}$ is bounded, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}U(\beta _0;\hat{\eta })=\frac{1}{\sqrt{n}}U(\beta _0;\eta _0)+ \frac{1}{n}\frac{\partial U(\beta _0;\eta )}{\partial \eta }|_{\eta =\eta _0}\sqrt{n}(\hat{\eta }-\eta _0)+o_p(1). \end{aligned}$$

Similar to the arguments of $V(\eta _0)$, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}U(\beta _0; \eta _0)= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\int ^\tau _0 W(t)\{\mathbf{X }_i-\bar{\mathbf{X }}(t; \eta _0)\}N_i(t)\exp (-\beta _0'{} \mathbf X _i)\varDelta _i(t)dO_i(t)\\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\int ^\tau _0 W(t)\{\mathbf{X }_i-\bar{\mathbf{X }}(t; \eta _0)\}[dR_i(t)+\exp \{\eta _0'{} \mathbf X _i^*(t)d\mathcal{A}(t)\}] \\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\int ^\tau _0 W(t)\{\mathbf{X }_i-\bar{\mathbf{X }}(t; \eta _0)\}dR_i(t)\\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\int ^\tau _0 W(t)\{\mathbf{X }_i-\bar{\mathbf{x }}(t)\}dR_i(t)+o_p(1)\\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n U_i+o_p(1),\\ \end{aligned}$$

a sum of n independent mean 0 random vectors plus an asymptotically negligible term.

Now, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}U(\beta _0;\hat{\eta })= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n U_i+ T_nP_n^{-1}\frac{1}{\sqrt{n}}\sum _{i=1}^n V_i+o_p(1)\\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n W_i(M_n)+o_p(1), \end{aligned}$$

where $T_n=\frac{1}{n}\frac{\partial U(\beta _0;\eta )}{\partial \eta }|_{\eta =\eta _0}, M_n=T_nP_n^{-1}, W_i(M_n)=U_i+M_nV_i$. Thus, by the multivariate CLT, we have $\sqrt{n}(\hat{\beta }-\beta _0)\rightarrow Q^{-1}N(0,\varSigma _{\beta })$, where $\varSigma _{\beta }=EW_1(M)^{\otimes 2}$, $M=\lim _{n\rightarrow +\infty }M_n$, and

$$\begin{aligned} Q= & {} \lim _{n\rightarrow +\infty }\frac{1}{n}\sum _{i=1}^n\int _{0}^{\tau }W(t)\{\mathbf{X }_i-\bar{\mathbf{X }}(t;\hat{\eta })\}N_i(t)\exp (-\beta _0'{} \mathbf X _i)\mathbf X _i'\varDelta _i(t)dO_i(t)\\= & {} \lim _{n\rightarrow +\infty }\frac{1}{n}\sum _{i=1}^n\int _{0}^{\tau }W(t)\{\mathbf{X }_i-\bar{\mathbf{X }}(t;\hat{\eta })\}N_i(t)\exp (-\beta _0'{} \mathbf X _i)\mathbf X _i'[dM_i(t)\\&+\exp \{\eta _0'{} \mathbf X _i^*(t)\}d\mathcal{B}(t)]\\= & {} E\int _{0}^{\tau }W(t)[\mathbf X _1-\bar{\mathbf{x }}(t)]N_1(t)\exp (-\beta _0'{} \mathbf X _1)\mathbf X _1'\exp \{\eta _0'{} \mathbf X _1^*(t)\}d\mathcal{B}(t). \end{aligned}$$

Next we compute M. Note that

$$\begin{aligned} T_n= & {} -\frac{1}{n}\sum _{i=1}^n \int _{0}^{\tau } W(t)\frac{\partial \bar{\mathbf{X }}(t;\eta )}{\partial \eta }|_{\eta =\eta _0}N_i(t)\exp (-\beta _0'{} \mathbf X _i)\varDelta _i(t)dO_i(t). \end{aligned}$$

Since

$$\begin{aligned} \frac{\partial \bar{\mathbf{X }}(t;\eta )}{\partial \eta }|_{\eta =\eta _0}= & {} \frac{\sum _{i=1}^n\sum _{j=1}^n \exp \{\eta _0'{} \mathbf X _i^*(t)\}\exp \{\eta _0'{} \mathbf X _j^*(t)\}(\mathbf X _j-\mathbf X _i)(\mathbf X _j^*(t))'}{(\sum _{i=1}^n\exp \{\eta _0'{} \mathbf X _i^*(t)\})^2}\\= & {} \frac{S_5(t;\eta _0)}{S_1(t;\eta _0)}-\frac{S_4(t;\eta _0)S_3'(t;\eta _0)}{S_1^2(t;\eta _0)},\\ \end{aligned}$$

we have

$$\begin{aligned} T_n= & {} -\frac{1}{n}\sum _{i=1}^n \int _{0}^{\tau } W(t)\left\{ \frac{S_5(t;\eta _0)}{S_1(t;\eta _0)} -\frac{S_4(t;\eta _0)S_3'(t;\eta _0)}{S_1^2(t;\eta _0)}\right\} N_i(t)\exp (-\beta _0'{} \mathbf X _i)\varDelta _i(t)dO_i(t)\\= & {} -\frac{1}{n}\sum _{i=1}^n \int _{0}^{\tau } W(t)\left\{ \frac{S_5(t;\eta _0)}{S_1(t;\eta _0)} -\frac{S_4(t;\eta _0)S_3'(t;\eta _0)}{S_1^2(t;\eta _0)}\right\} [dR_i(t)+\exp \{\eta _0'{} \mathbf X _i^*(t)\}d\mathcal{A}(t)]\\= & {} -\frac{1}{n}\sum _{i=1}^n \int _{0}^{\tau } W(t)\left\{ \frac{s_5(t;\eta _0)}{s_1(t;\eta _0)} -\frac{s_4(t;\eta _0)s_3'(t;\eta _0)}{s_1^2(t;\eta _0)}\right\} dR_i(t)\\&-\int _{0}^{\tau } W(t)\left\{ s_5(t;\eta _0)-\frac{s_4(t;\eta _0)s_3'(t;\eta _0)}{s_1(t;\eta _0)}\right\} d\mathcal{A}(t)+o_p(1)\\&\rightarrow _p-\int _{0}^{\tau } W(t)\left\{ s_5(t;\eta _0) -\frac{s_4(t;\eta _0)s_3'(t;\eta _0)}{s_1(t;\eta _0)}\right\} d\mathcal{A}(t). \end{aligned}$$

Thus, we have $M=TP^{-1}$, where

$$\begin{aligned} T=-\int _{0}^{\tau } W(t)\left\{ s_5(t;\eta _0)-\frac{s_4(t;\eta _0)s_3'(t;\eta _0)}{s_1(t;\eta _0)}\right\} d\mathcal{A}(t). \end{aligned}$$

It is easy to show that ${\hat{Q}}$, ${\hat{\varSigma }}_{\beta }$ and ${\hat{M}}$ are the consistent estimators of $Q, \varSigma _{\beta }$ and M, respectively.