Case-cohort analysis of clusters of recurrent events

Abstract

The case-cohort sampling, first proposed in Prentice (Biometrika 73:1–11, 1986), is one of the most effective cohort designs for analysis of event occurrence, with the regression model being the typical Cox proportional hazards model. This paper extends to consider the case-cohort design for recurrent events with certain specific clustering feature, which is captured by a properly modified Cox-type self-exciting intensity model. We discuss the advantage of using this model and validate the pseudo-likelihood method. Simulation studies are presented in support of the theory. Application is illustrated with analysis of a bladder cancer data.

Appendix

1.1 Sketch proof of Proposition 1

Although due to the presence of the self-exciting component, our model is not a special case of the model considered by Self and Prentice (1988), the asymptotic properties of the pseudo likelihood based estimator can be proved along the same lines of Self and Prentice (1988) for the Cox proportional intensity model. Here we present a sketch of the technical arguments using the empirical approximation method. The conditions C3–C5 ensure that $\varPsi (t, \theta )Y( t)$ is $P$-Glivenko–Cantelli over $[0, t_0] \times \varTheta$ for any fixed $t_0>0$. As a result,

$$\begin{aligned} \tilde{R}^{(k)} (t, \theta ) \rightarrow r^{(k)} (t, \theta ) \end{aligned}$$

uniformly over $[0, \tau ] \times \varTheta$ with probability one. It follows that, uniformly over $\varTheta$,

$$\begin{aligned} {1 \over n}\left[ \log \{L(\theta )\} -\log \{ L(\theta _0)\} \right] \rightarrow l(\theta ) - l(\theta _0), \end{aligned}$$

where

$$\begin{aligned} l(\theta )&= \mathrm{E }\left( \int _0^C \left[ \varPsi (t, \theta )- \log \mathrm{E }\{ \exp \{ \varPsi (t, \theta ) \} \mu _0(t) Y(t)\}\right] \,\mathrm d N(t) \right) \\&= \int _0^\tau \mathrm{E }\left( \left[ \varPsi (t, \theta )- \log \mathrm{E }\{ \exp \{ \varPsi (t, \theta ) \} \mu _0(t) Y(t) \}\right] \exp \{\varPsi (t, \theta _0) \} Y(t)\mu _0(t)\right) \mathrm d t. \end{aligned}$$

Observe that, for any positive random variable $\zeta$ and nonnegative $\eta$ with positive mean, Jensen’s inequality implies $\mathrm{E }[\eta \log \zeta ]/\mathrm{E }[\eta ] \le \log \mathrm{E }[\zeta \eta ] - \log \mathrm{E }[\eta ].$ Set $\zeta = \exp \{\varPsi (t, \theta ) - \varPsi (t, \theta _0)\}$ and $\eta = \exp \{\varPsi (t, \theta _0)\} Y(t) \mu (t)$. It is seen that the integrand in the second expression of $l(\theta )$ achieves a unique maximum when $\theta =\theta _0$. By C4, $l(\theta )$ achieves maximum only at $\theta _0$. The uniform convergence over $\varTheta$ implies that $\hat{\theta }$ is strongly consistent.

The asymptotic normality is proved by the Taylor expansion in a small neighborhood of $\theta _0$. Similar to the definition of $r^{(k)}$ and $\tilde{R}^{(k)}$, let

$$\begin{aligned} R^{(k)}(t, \theta ) = {1 \over n} \sum \limits _{i=1}^n \psi _i(t, \theta ) Y_i(t) \exp \{ \varPsi _i(t, \theta ) \}. \end{aligned}$$

The principal derivation is

$$\begin{aligned}&\hat{\theta }- \theta _0 \\&= - \left[ \frac{\partial ^2}{\partial \theta \partial \theta ^{\top }}\log \{L(\theta _0)\} \right] ^{-1} \left[ \sum \limits _{i=1}^n \int _0^{C_i } \left\{ \psi _i(t, \theta _0) - { R^{(1)}(t, \theta _0) \over R^{(0)} (t, \theta _0)} \right\} \,\mathrm d N_i(t) \right. \\ \nonumber&\left. \qquad \qquad \qquad \qquad \qquad +\sum \limits _{i=1}^n \int _0^{C_i} \left\{ { R^{(1)}(t, \theta _0) \over R^{(0)} (t, \theta _0)} - { \tilde{R}^{(1)}(t, \theta _0) \over \tilde{R}^{(0)} (t, \theta _0)} \right\} \,\mathrm d N_i(t) \right] +o_P(n^{-1/2})\\ \!&= \! \!-\! \left[ \frac{\partial ^2}{\partial \theta \partial \theta ^{\top }}\log \{L(\theta _0)\} \right] ^{-1} \left[ \sum \limits _{i=1}^n \int _0^{C_i} \left\{ \psi _i(t, \theta _0) - { R^{(1)}(t, \theta _0) \over R^{(0)} (t, \theta _0)} \right\} \,\mathrm d N_i(t)\right. \\&\qquad \qquad \qquad \qquad \qquad \left. + \sum \limits _{i=1}^n \left( {\epsilon _i \over n_*/n } - 1\right) \xi _i \right] + o_P(n^{-1/2}).\\ \end{aligned}$$

Observe that $\epsilon _j$ are membership indicators for simple random sampling without replacement and the asymptotic normality of $\hat{\theta }$ follows.