Advances in Survival Analysis: Accurate Partial Likelihood Computation by Poisson-Binomial Distributions and Nonparametric Competing Risk Cox Model

Abstract

Two novel contributions to survival analysis are presented. The first project revisits the partial likelihood in the Cox model, which traditionally approximates conditional probabilities using risk score ratios under a continuous-time assumption. We propose a new accurate partial likelihood computation method based on the Poisson-binomial distribution. Although ties are common in real studies, existing Cox model theory largely overlooks tied data. In contrast, our approach accommodates both grouped data with ties and continuous data without ties, offering a unified theoretical framework for accurate partial likelihood computation regardless of data type. Simulations and real data analyses show that the method reduces bias and mean squared error while improving confidence interval coverage rates, particularly when ties are frequent or risk score variability is high. The second project develops a nonparametric regression model for competing risks survival data by combining the proportional cause-specific hazards framework with a smoothing spline ANOVA approach. We establish estimation procedures and theoretical convergence rates. Simulation studies demonstrate the method's effectiveness, and application to a multiple myeloma dataset reveals that for each gene expression covariate, at least one cause-specific effect is nonlinear and differs from the others. The proposed model fills a gap in the existing literature, where competing risks are often overlooked or covariate effects are assumed to follow parametric forms, by providing a flexible and practical framework for data analysis.