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Failure time data subject to various types of censoring commonly arise in epidemiological and biomedical studies. Motivated by an AIDS clinical trial, we consider regression analysis of failure time data that include exact and left‐, interval‐, and/or right‐censored observations, which are often referred to as partly interval‐censored failure time data. We study the effects of potentially time‐dependent covariates on partly interval‐censored failure time via a class of semiparametric transformation models that includes the widely used proportional hazards model and the proportional odds model as special cases. We propose an EM algorithm for the nonparametric maximum likelihood estimation and show that it unifies some existing approaches developed for traditional right‐censored data or purely interval‐censored data. In particular, the proposed method reduces to the partial likelihood approach in the case of right‐censored data under the proportional hazards model. We establish that the resulting estimator is consistent and asymptotically normal. In addition, we investigate the proposed method via simulation studies and apply it to the motivating AIDS clinical trial.

Interval‐censored failure time data commonly arise in epidemiological and biomedical studies where the occurrence of an event or a disease is determined via periodic examinations. Subject to interval‐censoring, available information on the failure time can be quite limited. Cost‐effective sampling designs are desirable to enhance the study power, especially when the disease rate is low and the covariates are expensive to obtain. In this work, we formulate the case‐cohort design with multiple interval‐censored disease outcomes and also generalize it to nonrare diseases where only a portion of diseased subjects are sampled. We develop a marginal sieve weighted likelihood approach, which assumes that the failure times marginally follow the proportional hazards model. We consider two types of weights to account for the sampling bias, and adopt a sieve method with Bernstein polynomials to handle the unknown baseline functions. We employ a weighted bootstrap procedure to obtain a variance estimate that is robust to the dependence structure between failure times. The proposed method is examined via simulation studies and illustrated with a dataset on incident diabetes and hypertension from the Atherosclerosis Risk in Communities study.