Weighted NPMLE for the Subdistribution of a Competing Risk

ABSTRACT

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.

Keywords:

• Cumulative incidence function
• Fine–Gray model
• Nonparametric maximum likelihood estimation
• Semiparametric transformation models
• Time-varying covariates

Acknowledgment

The authors thank Per Kragh Andersen’s permission to use the bmt dataset collected by CIBMTR with Public Health Service Grant/Cooperative Agreement no. U24-CA76518 from the US National Cancer Institute (NCI), the U.S. National Heart, Lung and Blood Institute (NHLBI), and the US National Institute of Allergy and Infectious Diseases (NIAID). The authors would like to thank Drs. Donglin Zeng, Hein Putter, Erik Parner and the two referees for their helpful comments and advice.

Additional information

Funding

A. Bellach was funded by the European Community's programme FP7/2011[290025]: Marie Curie initial training network MEDIASRES (www.mediasres-itn.eu). National Cancer Institute [P01 CA142538].