Multiple event times in the presence of informative censoring: modeling and analysis by copulas

Abstract

Motivated by a breast cancer research program, this paper is concerned with the joint survivor function of multiple event times when their observations are subject to informative censoring caused by a terminating event. We formulate the correlation of the multiple event times together with the time to the terminating event by an Archimedean copula to account for the informative censoring. Adapting the widely used two-stage procedure under a copula model, we propose an easy-to-implement pseudo-likelihood based procedure for estimating the model parameters. The approach yields a new estimator for the marginal distribution of a single event time with semicompeting-risks data. We conduct both asymptotics and simulation studies to examine the proposed approach in consistency, efficiency, and robustness. Data from the breast cancer program are employed to illustrate this research.

Keywords: Efficiency and robustness; Marginal distribution; Pseudo-likelihood estimation; Variable correlation; Variance estimation.

Figure 1:

Estimates of marginal survivor function of time to RSC, S₁(·) with 95% CB, by the proposed approach and the Kaplan-Meier estimator using simulated data with different sample sizes under trivariate Clayton copula: τ = 0.6

Figure 2:

Estimates of marginal survivor functions S_j(·), j = 1,2,3,4 by the proposed approach and the Kaplan-Meier estimator using simulated data generated from 5-dimensional Clayton copula, sample size n = 1000.

Figure 3:

Estimated marginal survivor functions of times to RSC and CVD, S₁(·) and S₂(·), by the proposed approach with different copulas and the Kaplan-Meier estimator using the BRCA data. (a1)-(a2): early diagnosis stage; (b1)-(b2): late diagnosis stage.

Figure 4:

Estimated marginal survivor functions of times to RSC and CVD, S₁(·) and S₂(·) and times to disease free survival $S_1^{*}(\cdot )$ and $S_2^{*}(\cdot )$, using BRCA data (overall). (a): Comparison between S₁(·) and $S_1^{*}(\cdot )$, shaded area showing years lost due to death; (b):Comparison between S₂(·) and $S_2^{*}(\cdot )$, shaded area showing years lost due to death; (c):Comparison between S_D(·) and $S_1^{*}(\cdot )$, shaded area showing year lost due to RSC; (d):Comparison between S_D(·) and $S_2^{*}(\cdot )$, shaded area showing years lost due to CVD