A case-base sampling method for estimating recurrent event intensities

Abstract

Case-base sampling provides an alternative to risk set sampling based methods to estimate hazard regression models, in particular when absolute hazards are also of interest in addition to hazard ratios. The case-base sampling approach results in a likelihood expression of the logistic regression form, but instead of categorized time, such an expression is obtained through sampling of a discrete set of person-time coordinates from all follow-up data. In this paper, in the context of a time-dependent exposure such as vaccination, and a potentially recurrent adverse event outcome, we show that the resulting partial likelihood for the outcome event intensity has the asymptotic properties of a likelihood. We contrast this approach to self-matched case-base sampling, which involves only within-individual comparisons. The efficiency of the case-base methods is compared to that of standard methods through simulations, suggesting that the information loss due to sampling is minimal.

Appendix

Because $M_i(t)$ and $M_i^*(t)$ are orthogonal, the predictable variation process of the score process can be expressed as

$$\begin{aligned}&\langle U \rangle (t; \theta ) \\&\quad = \sum _{i=1}^n \int _0^{t} \left( \frac{\partial }{\partial \theta } \left\{ \log \lambda _i(u; \theta ) - \log \left[ \lambda _i(u; \theta ) + \rho _i(u)\right] \right\} \right) ^{\otimes 2}Y_i(u)\lambda _i(u; \theta )\,{\text {d}} u \\&\qquad + \sum _{i=1}^n \int _0^{t} \left( \frac{\partial }{\partial \theta } \log \left[ \lambda _i(u; \theta ) + \rho _i(u)\right] \right) ^{\otimes 2} Y_i(u)\rho _i(u) \,{\text {d}} u \\&\quad = \sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i'(u; \theta )}{\lambda _i(u; \theta )} - \frac{\lambda _i'(u; \theta )}{\lambda _i(u; \theta ) + \rho _i(u)}\right) ^{\otimes 2}Y_i(u)\lambda _i(u; \theta )\,{\text {d}} u\\&\qquad + \sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i'(u; \theta )}{\lambda _i(u; \theta ) + \rho _i(u)}\right) ^{\otimes 2} Y_i(u)\rho _i(u) \,{\text {d}} u \\&\quad = \sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i'(u; \theta )^{\otimes 2}}{\lambda _i(u; \theta )^2}\right. - \frac{2 \lambda _i'(u; \theta )^{\otimes 2}}{\lambda _i(u; \theta )\left[ \lambda _i(u; \theta ) + \rho _i(u)\right] } +\left. \frac{\lambda _i'(u; \theta )^{\otimes 2}}{[\lambda _i(u; \theta ) + \rho _i(u)]^2} \right) \\&\qquad \times Y_i(u)\lambda _i(u; \theta )\,{\text {d}} u + \sum _{i=1}^n \int _0^{t} \frac{\lambda _i'(u; \theta )^{\otimes 2}}{[\lambda _i(u; \theta ) + \rho _i(u)]^2} Y_i(u)\rho _i(u)\,{\text {d}} u \\&\quad = \sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i'(u; \theta )^{\otimes 2}}{\lambda _i(u; \theta )} - \frac{\lambda _i'(u; \theta )^{\otimes 2}}{\lambda _i(u; \theta ) + \rho _i(u)}\right) Y_i(u)\,{\text {d}} u. \end{aligned}$$

The observed information process is given by

$$\begin{aligned} J(t; \theta )&= -\sum _{i=1}^n \int _0^{t} \frac{\partial ^2}{\partial \theta \partial \theta ^\top }\log \lambda _i(u; \theta ) {\text {d}}N_i(u) \\&\quad + \sum _{i=1}^n \int _0^{t} \frac{\partial ^2}{\partial \theta \partial \theta ^\top } \log \left[ \lambda _i(u; \theta ) + \rho _i(u)\right] {\text {d}}Q_i(u) \\&= -\sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i''(u; \theta )\lambda _i(u; \theta ) - \lambda _i'(u; \theta )^{\otimes 2}}{\lambda _i(u; \theta )^2}\right) {\text {d}}N_i(u) \\&\quad +\sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i''(u; \theta )\left[ \lambda _i(u; \theta ) + \rho _i(u)\right] - \lambda _i'(u; \theta )^{\otimes 2}}{\left[ \lambda _i(u; \theta ) + \rho _i(u)\right] ^2}\right) {\text {d}}Q_i(u), \end{aligned}$$

where we denoted $\lambda _i''(u; \theta ) \equiv \frac{\partial ^2}{\partial \theta \partial \theta ^\top } \lambda _i(u; \theta )$.

Using the decompositions (4) and (5), the observed information process can be further written as

$$\begin{aligned}&J(t; \theta ) \\&\quad = -\sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i''(u; \theta )\lambda _i(u; \theta ) - \lambda _i'(u; \theta )^{\otimes 2}}{\lambda _i(u; \theta )^2}\right) Y_i(u)\lambda _i(u; \theta )\,{\text {d}} u\\&\qquad +\,\sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i''(u; \theta )[\lambda _i(u; \theta ) + \rho _i(u)] {-} \lambda _i'(u; \theta )^{\otimes 2}}{[\lambda _i(u; \theta ) + \rho _i(u)]^2}\right) Y_i(u)[\lambda _i(u; \theta ) + \rho _i(u)]\,{\text {d}} u \\&\qquad +\,{\mathcal {E}}(t) \\&\qquad = \sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i'(u; \theta )^{\otimes 2}}{\lambda _i(u; \theta )} - \frac{\lambda _i'(u; \theta )^{\otimes 2}}{\lambda _i(u; \theta ) + \rho _i(u)}\right) Y_i(u) \,{\text {d}} u + {\mathcal {E}}(t), \end{aligned}$$

where we denoted

$$\begin{aligned} {\mathcal {E}}(t)\equiv & {} -\sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i''(u; \theta )\lambda _i(u; \theta ) - \lambda _i'(u; \theta )^{\otimes 2}}{\lambda _i(u; \theta )^2}\right) {\text {d}} M_i(u)\\&\quad +\,\sum _{i=1}^n \int _0^{t} \left( \frac{\lambda _i''(u; \theta )\left[ \lambda _i(u; \theta ) + \rho _i(u)\right] {-} \lambda _i'(u; \theta )^{\otimes 2}}{\left[ \lambda _i(u; \theta ) + \rho _i(u)\right] ^2}\right) \left[ {\text {d}} M_i(u) + {\text {d}} M_i^*(u)\right] . \end{aligned}$$

Therefore, $E[\langle U \rangle (t; \theta _0)] = E[J(t; \theta _0)]$. With these results, motivating the asymptotic normality of the maximum partial likelihood estimator $\hat{\theta }$ can proceed similarly as for parametric survival models (e.g. Kalbfleisch and Prentice 2002, p. 180). Briefly, assume a scalar $\theta$ for notational simplicity, and denote $U(\theta ) \equiv U(\tau ; \theta )$ and $J(\theta ) \equiv J(\tau ; \theta )$. From the martingale central limit theorem, it follows under the standard regularity conditions that

$$\begin{aligned} \frac{\sqrt{n}}{n} U(\theta _0) \mathop {\rightarrow }\limits ^{d} N\left( 0, \varSigma (\theta _0)\right) , \end{aligned}$$

where the matrix $\varSigma (\theta _0)$ is such that $\frac{1}{n} \langle U \rangle (\tau ; \theta _0) \mathop {\rightarrow }\limits ^{p} \varSigma (\theta _0)$. The Taylor expansion

$$\begin{aligned} U(\hat{\theta }) = U(\theta _0) - J(\theta _0)\left( \hat{\theta }- \theta _0\right) + \frac{1}{2} \frac{\partial ^3 l(\theta ^*)}{\partial \theta ^3} \left( \hat{\theta }- \theta _0\right) ^2 \end{aligned}$$

can be used to motivate both the consistency and asymptotic normality of $\hat{\theta }$ by assuming that the third term on the right hand side is bounded in probability. In particular, we get

$$\begin{aligned} \sqrt{n} (\hat{\theta }- \theta _0) \mathop {\rightarrow }\limits ^{d} N\left( 0, \varSigma (\theta _0)^{-1}\right) , \end{aligned}$$

where $\varSigma (\theta _0)$ is in practice estimated by the average observed information $\frac{1}{n} J(\hat{\theta })$ at the maximum likelihood point.