A semiparametric additive rates model for the weighted composite endpoint of recurrent and terminal events

Abstract

Recurrent event data with a terminal event commonly arise in longitudinal follow-up studies. We use a weighted composite endpoint of all recurrent and terminal events to assess the overall effects of covariates on the two types of events. A semiparametric additive rates model is proposed to analyze the weighted composite event process and the dependence structure among recurrent and terminal events is left unspecified. An estimating equation approach is developed for inference, and the asymptotic properties of the resulting estimators are established. The finite-sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is illustrated.

Appendix: Proofs of theorems

In order to study the asymptotic properties of the proposed estimators, we impose the following regularity conditions.

• (C1) $\{N_i(\cdot ),\ T_i,\ \delta _i,\ Z_i(\cdot )\}$ are independent and identically distributed.

• (C2) $P\{T_i \ge \tau \}>0$, and $E\{N_i(\tau )\} < \infty .$

• (C3) $Z_i(t)$ is almost surely of bounded variation on $[0, \tau ]$.

• (C4) The matrix A is nonsingular, where

  $$\begin{aligned} A=E\left[ \int _0^\tau I(C_i \ge t)\left\{ Z_i(t)-{\bar{z}}(t)\right\} ^{\otimes 2}dt \right] , \end{aligned}$$

  ${\bar{z}}(t)$ is the limit of ${\bar{Z}}(t)$, and

  $$\begin{aligned} {\bar{Z}}(t)= \frac{\sum _{i=1}^{n}I(C_i\ge t)Z_i(t)}{\sum _{i=1}^{n}I(C_i\ge t)}. \end{aligned}$$

Proof of Theorem 1

Using the uniform strong law of large numbers (Pollard 1990) and the uniform consistency of ${{\widehat{\gamma }}}$ and ${{\widehat{\Lambda }}}_0^C(\cdot )$ under model (6) (Andersen and Gill 1982), it follows from that ${\bar{Z}}^C(t)$ converges almost surely to ${\bar{z}}(t)$ uniformly in $t\in [0,\tau ]$, and almost surely,

$$\begin{aligned} {\widehat{A}} = \frac{1}{n}\sum _{i=1}^n \int _0^\tau {\widehat{W}}_i(t)\{Z_i(t)-{\bar{Z}}^C(t)\}^{\otimes 2}dt \longrightarrow A. \end{aligned}$$

(A.1)

Note that $M_i(t;\, \theta _0) = N_i(t)- \int _0^t I(C_i\ge u)\left\{ d\mu _0(u)+\theta _0'Z_i(u)du\right\}$ is a zero-mean stochastic process under model (2). In a similar manner, we have that almost surely,

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n \int _0^\tau \left\{ Z_i(t)-{\bar{Z}}^C(t)\right\} \left[ dN_i(t)-{\widehat{W}}_i(t)\left\{ d\mu _0(t)+\theta _0'Z_i(t)dt\right\} \right] \longrightarrow 0.\nonumber \\ \end{aligned}$$

(A.2)

Thus, it follows from (A.1) and (A.2) that almost surely,

$$\begin{aligned} {\widehat{\theta }}-\theta _0&= {\widehat{A}}^{-1} \left( \frac{1}{n}\sum _{i=1}^n \int _0^\tau \left\{ Z_i(t)-{\bar{Z}}^C(t)\right\} \left[ dN_i(t)-{\widehat{W}}_i(t)\left\{ d\mu _0(t)+\theta _0'Z_i(t)dt\right\} \right] \right) \\&\longrightarrow 0. \end{aligned}$$

That is, ${{\widehat{\theta }}}$ is strongly consistent.

To prove the asymptotic normality of ${{\widehat{\theta }}}$, define

$$\begin{aligned} M_i(t) = N_i(t)- \int _0^t {W}_i(u)\left\{ d\mu _0(u)+\theta _0'Z_i(u)du\right\} , \end{aligned}$$

where

$$\begin{aligned} W_i(t)=\frac{I(C_i \ge D_i \wedge t)\,S^{C}(t|Z_i)}{S^{C}(T_i\wedge t|Z_i)}. \end{aligned}$$

Write

$$\begin{aligned} n^{1/2}({\widehat{\theta }}-\theta _0) =&A^{-1}n^{-1/2} \bigg [ \sum _{i=1}^n \int _0^\tau \left\{ Z_i(t)-{\bar{Z}}^C(t)\right\} dM_i(t)\nonumber \\&+ \sum _{i=1}^n \int _0^\tau \left\{ Z_i(t)-{\bar{Z}}^C(t)\right\} I(C_i \ge D_i\wedge t) \left\{ \frac{S^C(t|Z_i)}{S^C(T_i\wedge t|Z_i)} -\frac{{\widehat{S}}^C(t|Z_i)}{{\widehat{S}}^C(T_i\wedge t|Z_i)}\right\} \nonumber \\&\times \left\{ d\mu _0(t)+\theta _0'Z_i(t)dt\right\} \bigg ]+o_p(1). \end{aligned}$$

(A.3)

Note that ${\widehat{\Lambda }}^C_{0}(\cdot )$ is $n^{1/2}$-consistent. Applying the functional delta method (van der Vaart and Wellner 1996, Theorem 3.9.4, p. 374) and the martingale central limit theorem (Fleming and Harrington 1991, p. 227), we obtain

$$\begin{aligned}&n^{1/2} \left\{ \frac{S^C(t|Z_i)}{S^C(T_i\wedge t|Z_i)} - \frac{{\widehat{S}}^C(t|Z_i)}{{\widehat{S}}^C(T_i\wedge t|Z_i)} \right\} \nonumber \\&\quad = \frac{I(T_i < t)S^C(t|Z_i)}{S^C(T_i\wedge t|Z_i)} \bigg [ n^{-1/2}\sum _{j=1}^n \int _{T_i}^t \frac{\exp \{\gamma _0'Z_i(u)\}dM_j^C(u)}{s^{(0)}(u;\gamma _0)} \nonumber \\&\qquad +H(t;T_i, Z_i)'\,\Omega ^{-1}\, n^{-1/2}\sum _{j=1}^n \int _0^\tau \bigg \{Z_j(u) -\frac{s^{(1)}(u;\gamma _0)}{s^{(0)}(u;\gamma _0)}\bigg \} dM_j^C(u) \bigg ] +o_p(1), \end{aligned}$$

(A.4)

where

$$\begin{aligned} H(t;T_i, Z_i) = \int _{T_i}^t \exp \{\gamma _0'Z_i(u)\} \left\{ Z_i(u)-\frac{s^{(1)}(u;\gamma _0)}{s^{(0)}(u;\gamma _0)}\right\} d\Lambda ^C_0(u), \end{aligned}$$

and $s^{(d)}(t;\gamma )$ and $\Omega$ are the limits of $S^{(d)}(t;\gamma )$ and ${{\widehat{\Omega }}}$ is defined in Sect. 3, respectively.

Plugging (A.4) into (A.3) and interchanging integrals, we get

$$\begin{aligned} n^{1/2}\{{{\widehat{\theta }}}-\theta _0\} =&A^{-1}\,n^{-1/2}\sum _{i=1}^n \bigg [ \int _0^\tau \left\{ Z_i(t)-{\bar{Z}}^C(t)\right\} dM_i(t) \nonumber \\&+ \int _0^\tau \left\{ {\widetilde{B}} \bigg (Z_i(t)-\frac{s^{(1)}(t;\gamma _0)}{s^{(0)}(t;\gamma _0)}\bigg ) + \frac{{{\widetilde{F}}}(t)}{s^{(0)}(t;\gamma _0)}\right\} dM_i^C(t) \bigg ] +o_p(1), \end{aligned}$$

(A.5)

where

$$\begin{aligned} {\widetilde{B}} = \frac{1}{n}\sum _{i=1}^n \int _0^\tau \left\{ Z_i(t)-{\bar{Z}}^C(t)\right\} H(t;T_i, Z_i)'\,\Omega ^{-1}I(T_i < t)W_i(t) \left\{ d\mu _0(t)+\theta _0'Z_i(t)dt\right\} , \end{aligned}$$

and

$$\begin{aligned} {\widetilde{F}}(t)= & {} \frac{1}{n}\sum _{i=1}^n \int _0^\tau \left\{ Z_i(u)-{\bar{Z}}^C(u)\right\} \\&\quad \exp \left\{ \gamma _0'Z_i(t)\right\} I(T_i < t \le u)W_i(u) \left\{ d\mu _0(u)+\theta _0'Z_i(u)du\right\} . \end{aligned}$$

Let B and F(t) to be the limits of ${\widetilde{B}}$ and ${\widetilde{F}}(t)$, respectively. In view of (A.5), by the uniform strong law of large numbers, we have

$$\begin{aligned} n^{1/2}\{{{\widehat{\theta }}}-\theta _0\} =A^{-1}n^{-1/2}\sum _{i=1}^n \{\xi _i+\eta _i\}+o_p(1), \end{aligned}$$

where

$$\begin{aligned} \xi _i = \int _0^\tau \left\{ Z_i-{{\bar{z}}}(t)\right\} dM_i(t), \end{aligned}$$

and

$$\begin{aligned} \eta _i =\int _0^\tau \left[ B\left\{ Z_i(t)-\frac{s^{(1)}(t;\gamma _0)}{s^{(0)}(t;\gamma _0)}\right\} + \frac{F(t)}{s^{(0)}(t;\gamma _0)} \right] dM_i^C(t). \end{aligned}$$

It follows from the multivariate central limit theorem that $n^{1/2}({{\widehat{\theta }}}-\theta _0)$ converges in distribution to a zero-mean normal random vector with a covariance matrix $A^{-1}\Sigma A^{-1}$, where $\Sigma =E\{\xi _1+\eta _1\}^{\otimes 2}$. By replacing all the unknown quantities in A and $\Sigma$ with their empirical counterparts, the variance matrix can be consistently estimated by ${{\widehat{A}}}^{-1}{\widehat{\Sigma }}\widehat{A}^{-1}$ defined in Theorem 1. $\square$

Proof of Theorem 2

For the consistency of ${{\widehat{\mu }}}_0(t)$, we have

$$\begin{aligned} {{\widehat{\mu }}}_0(t)-\mu _0(t) = \left\{ {\widehat{\mu }}_0(t)-{\widehat{\mu }}_0(t;\theta _0)\right\} + \left\{ {\widehat{\mu }}_0(t;\theta _0)-\mu _0(t)\right\} , \end{aligned}$$

(A.6)

where

$$\begin{aligned} {\widehat{\mu }}_0(t; \theta _0) =\int _0^t \frac{\sum _{i=1}^n \left\{ dN_i(u)-{\widehat{W}}_i(u)\,\theta _0'Z_i(u)du\right\} }{\sum _{i=1}^n {\widehat{W}}_i(u)}. \end{aligned}$$

Following the similar techniques in the proof of Theorem 1 and some algebraic manipulations, we obtain

$$\begin{aligned} {\widehat{\mu }}_0(t;{\widehat{\theta }})-{\widehat{\mu }}_0(t;\theta _0) = -\left[ \int _0^t {{\bar{z}}}(u)'du \right] \, A^{-1}\, \left[ \frac{1}{n}\sum _{i=1}^n \{\xi _i+\eta _i\} \right] +o_p(1),\nonumber \\ \end{aligned}$$

(A.7)

and

$$\begin{aligned} {\widehat{\mu }}_0(t;\theta _0)-\mu _0(t) =&\frac{1}{n}\sum _{i=1}^n \bigg [ \int _0^t \frac{dM_i(u)}{n^{-1}\sum _{j=1}^n{\widehat{W}}_i(u)} \nonumber \\&+ \int _0^t \frac{I(C_i \ge D_i\wedge u)}{n^{-1}\sum _{j=1}^n{\widehat{W}}_i(u)} \bigg \{\frac{S^C(u|Z_i)}{S^C(T_i\wedge u|Z_i)} \bigg . \bigg . \nonumber \\&\quad - \bigg . \bigg . \frac{{\widehat{S}}^C(u|Z_i)}{{\widehat{S}}^C(T_i\wedge u|Z_i)}\bigg \} \left\{ d\mu _0(u)+\theta _0'Z_i(u)du\right\} \bigg ]. \end{aligned}$$

(A.8)

Due to (A.6), (A.7) and (A.8), applying the uniform strong law of large numbers and the strong consistency of ${{\widehat{\theta }}}$, we can show that ${{\widehat{\mu }}}_0(t)$ converges almost surely to $\mu _0(t)$ uniformly in $t \in [0, \tau ]$.

Furthermore, using the functional central limit theorem (Pollard 1990) and the fact that $M_i^C(t)$ ($i=1 ,\ldots , n$) are martingales, we have

$$\begin{aligned} n^{1/2}\left\{ {{\widehat{\mu }}}_0(t)-\mu _0(t)\right\} =n^{-1/2}\sum _{i=1}^n \phi _i(t)+o_p(1), \end{aligned}$$

where

$$\begin{aligned} \phi _i(t) =&\int _0^t \frac{dM_i(u)}{E\{I(C_i \ge u)\}}+\int _0^\tau \left[ P_1(t)\left\{ Z_i(u)-\frac{s^{(1)}(u;\gamma _0)}{s^{(0)}(u;\gamma _0)}\right\} +\frac{P_2(u,t)}{s^{(0)}(u;\gamma _0)} \right] dM_i^C(u)\\&-\int _0^t {\bar{z}}(u)'duA^{-1}\{\xi _i+\eta _i\}, \end{aligned}$$

with $P_1(t)$ and $P_2(u,t)$ being the limits of ${{\widehat{P}}}_1(t)$ and ${{\widehat{P}}}_2(u,t)$ given in Theorem 2, respectively. Because $\phi _i$ ($i=1 ,\ldots , n$) are independent zero-mean random variables for each t, the multivariate central limit theorem implies that $n^{1/2}\left\{ {{\widehat{\mu }}}_0(t)-\mu _0(t)\right\}$ converges in finite-dimensional distribution to a zero-mean Gaussian process. Since $\phi _i$ can be written as sums or products of monotone functions of t and are thus tight (van der Vaart and Wellner 1996). Therefore, $n^{1/2}\left\{ {{\widehat{\mu }}}_0(t)-\mu _0(t)\right\}$ is tight and converges weakly to a zero-mean Gaussian process. The covariance function for $n^{1/2}\left\{ {{\widehat{\mu }}}_0(t)-\mu _0(t)\right\}$ at (t, s) is given by $E\{\phi _1(t)\phi _1(s)\}$, which can be consistently estimated by ${\widehat{\Phi }}(t,s)$ defined in Theorem 2. $\square$