The cumulative incidence is the probability of failure from the cause of interest over a certain time period in the presence of other risks. A semiparametric regression model proposed by Fine and Gray (1999) has become the method of choice for formulating the effects of covariates on the cumulative incidence. Its estimation, however, requires modeling of the censoring distribution and is not statistically efficient. In this paper, we present a broad class of semiparametric transformation models which extends the Fine and Gray model, and we allow for unknown causes of failure. We derive the nonparametric maximum likelihood estimators (NPMLEs) and develop simple and fast numerical algorithms using the profile likelihood. We establish the consistency, asymptotic normality, and semiparametric efficiency of the NPMLEs. In addition, we construct graphical and numerical procedures to evaluate and select models. Finally, we demonstrate the advantages of the proposed methods over the existing ones through extensive simulation studies and an application to a major study on bone marrow transplantation.
Keywords: Censoring; Nonparametric maximum likelihood estimation; Profile likelihood; Proportional hazards; Semiparametric efficiency; Survival analysis.