Nonparametric maximum likelihood estimation for competing risks frailty models

In clustered competing risks data, both event times within the same cluster and competing events are often correlated. This dependency can be modeled using cluster-specific frailties. This paper considers both hierarchical likelihood and nonparametric likelihood methods for fitting the so-called shared bivariate frailty model, a special case of the cause-specific competing risks frailty model with two causes of events, to clustered competing risks data. For this purpose, it first theoretically investigates the robustness of the maximum hierarchical likelihood estimators (MHLEs) under the common normal log-frailty assumption, while the true frailty distribution deviates from log-normality. It further proposes the nonparametric maximum likelihood method, which does not assume a specific parametric form for the frailty distribution, along with a hybrid algorithm for computing the nonparametric maximum likelihood estimators (NPMLEs). Simulation studies under various frailty distributions indicate that the proposed NPMLEs demonstrates good overall performance and that the proposed hybrid algorithm typically has a fast speed of convergence. Furthermore, the MHLEs of the regression coefficients yield comparable performance in terms of average relative bias and root mean squared error, which aligns with our theoretical findings. Both methods are applied to real-world datasets from multi-center studies on bladder cancer and bone marrow transplantation to illustrate their practical utility.