A likelihood-based approach for cure regression models

Abstract

We propose a new likelihood-based approach for estimation, inference and variable selection for parametric cure regression models in time-to-event analysis under random right-censoring. In this context, it often happens that some subjects are “cured”, i.e., they will never experience the event of interest. Then, the sample of censored observations is an unlabeled mixture of cured and “susceptible” subjects. Using inverse probability censoring weighting (IPCW), we propose a likelihood-based estimation procedure for the cure regression model without making assumptions about the distribution of survival times for the susceptible subjects. The IPCW approach does require a preliminary estimate of the censoring distribution, for which general parametric, semi- or nonparametric approaches can be used. The incorporation of a penalty term in our estimation procedure is straightforward; in particular, we propose $\ell _1$-type penalties for variable selection. Our theoretical results are derived under mild assumptions. Simulation experiments and real data analysis illustrate the effectiveness of the new approach.

Appendix

Lemma 2

Let r(Y, X) be an integrable real-valued function. Under conditions (1) and (2),

$$\begin{aligned} \mathbb {E}\left( \left. \frac{\Delta r(Y,X)}{S_C(Y-\mid X)} ~\right| X\right) ~=~ \mathbb {E}\{r(T_0,X)\mid X\} \{1-\pi (X)\}. \end{aligned}$$

Proof of Lemma  2

First, we have that

$$\begin{aligned} \mathbb {E}(\Delta \mid T_0, X )= & {} \mathbb {E}\{\mathbb {1}(T_0 \le C) (1-B) \mid T_0, X \} \\= & {} \mathbb {E}\{\mathbb {1}(T_0 \le C)\mathbb {E}(1-B \mid C, T_0, X ) \mid T_0, X \}\\= & {} S_C(T_0-\mid X) \,\{1-\pi (X)\} , \end{aligned}$$

where $\mathbb {E}(1-B \mid C, T_0, X ) = \mathbb {E}(1-B \mid X) = 1-\pi (X)$ follows from (2), and $\mathbb {E}[\mathbb {1}(T_0 \le C) \mid T_0, X ] = S_C(T_0-\mid X)$ follows from (1). Thus,

$$\begin{aligned}&\mathbb {E}\left( \left. \frac{\Delta r(Y,X)}{S_C(Y-\mid X)} ~\right| X\right) \\&= \mathbb {E}\left( \left. \frac{\Delta r(T_0,X)}{S_C(T_0-\mid X)} ~\right| X\right) \\&= \mathbb {E}\left\{ \left. \frac{ r(T_0,X)}{S_C(T_0-\mid X)} \mathbb {E}\left( \left. \Delta ~\right| T_0, X\right) ~\right| X\right\} \\&=\mathbb {E}\{r(T_0,X)\mid X\} \{1-\pi (X)\}, \end{aligned}$$

as required. (The first equality holds since only the $\Delta =1$ case contributes where $Y=T_0$.) $\square$