Survival curve estimation with partial non-random exposure information

The objective of this paper is to estimate survival curves for two different exposure groups when the exposure group is not known for all observations, and the data is subject to left truncation and right censoring. The situation we consider is when the probability that the exposure group is missing may depend on whether the observation is censored or uncensored, in which case the exposure is not missing at random. The problem was motivated by a study of Alzheimer's disease to estimate the distribution of ages at diagnosis for individuals with and without an apolipoprotein E4 allele (the exposure group). Genotyping for this risk factor was incomplete and performed more frequently on the cases of Alzheimer's disease (the uncensored observations) than the censored observations. The survival curves are estimated in discrete time using an EM algorithm. A bootstrapping procedure is proposed that guarantees each bootstrap sample has the same proportion of observations with missing exposure. A simulation is performed to evaluate the bias of the estimators and to investigate design and efficiency issues. The methods are applied to the Alzheimer's disease study.