Fitting the Cox proportional hazards model to interval-censored data

Interval-censored data arise frequently in clinical, epidemiological, financial, and sociological studies, where the event or failure of interest is not observed at an exact time point but is rather known to occur within a time interval induced by periodic examinations. We formulate the effects of potentially time-dependent covariates on the failure time through the familiar Cox proportional hazards model, under which the failure time distribution is completely arbitrary. We consider nonparametric maximum-likelihood estimation with an arbitrary number of examination times for each study subject. We present an EM algorithm that involves very simple calculations and converges stably for any dataset, even in the presence of time-dependent covariates. The resulting estimators for the regression parameters are consistent, asymptotically normal, and asymptotically efficient with an easily estimated covariance matrix. In addition, we extend the EM algorithm and the theoretical results to multivariate failure time data, in which there are multiple events per subjects or clustering of study subjects. Finally, we provide illustrations with real medical studies.