Weighted NPMLE for the Subdistribution of a Competing Risk

Direct regression modeling of the subdistribution has become popular for analyzing data with multiple, competing event types. All general approaches so far are based on nonlikelihood-based procedures and target covariate effects on the subdistribution. We introduce a novel weighted likelihood function that allows for a direct extension of the Fine–Gray model to a broad class of semiparametric regression models. The model accommodates time-dependent covariate effects on the subdistribution hazard. To motivate the proposed likelihood method, we derive standard nonparametric estimators and discuss a new interpretation based on pseudo risk sets. We establish consistency and asymptotic normality of the estimators and propose a sandwich estimator of the variance. In comprehensive simulation studies, we demonstrate the solid performance of the weighted nonparametric maximum likelihood estimation in the presence of independent right censoring. We provide an application to a very large bone marrow transplant dataset, thereby illustrating its practical utility. Supplementary materials for this article are available online.