Proportional hazards regression in the presence of missing study eligibility information

We consider the study of censored survival times in the situation where the available data consist of both eligible and ineligible subjects, and information distinguishing the two groups is sometimes missing. A complete-case analysis in this context would use only subjects known to be eligible, resulting in inefficient and potentially biased estimators. We propose a two-step procedure which resembles the EM algorithm but is computationally much faster. In the first step, one estimates the conditional expectation of the missing eligibility indicators given the observed data using a logistic regression based on the complete cases (i.e., subjects with non-missing eligibility indicator). In the second step, maximum likelihood estimators are obtained from a weighted Cox proportional hazards model, with the weights being either observed eligibility indicators or estimated conditional expectations thereof. Under ignorable missingness, the estimators from the second step are proven to be consistent and asymptotically normal, with explicit variance estimators. We demonstrate through simulation that the proposed methods perform well for moderate sized samples and are robust in the presence of eligibility indicators that are missing not at random. The proposed procedure is more efficient and more robust than the complete case analysis and, unlike the EM algorithm, does not require time-consuming iteration. Although the proposed methods are applicable generally, they would be most useful for large data sets (e.g., administrative data), for which the computational savings outweigh the price one has to pay for making various approximations in avoiding iteration. We apply the proposed methods to national kidney transplant registry data.