Subgroup analysis for heterogeneous Cox model and statistical inference for panel count data with terminal event

Abstract

Survival analysis is an essential branch of statistics, which focuses on analyzing the duration time until one or more events happen. It is applied widely to medicine, economics, engineering, and sociology. Despite the rapid development during the past several decades, there are still many interesting researches to study in this area. This thesis considers two topics: subgroup analysis for the heterogeneous Cox model and statistical inference for panel count data with an informative terminal event. In survival analysis, Cox model is commonly used to study the covariate effects. Nevertheless, the homogeneous effect assumption in the classical Cox model is usually not satisfied in many applications due to the differences among underlying groups of individuals. Then the homogeneous model will lead to inaccurate estimation results. To remove the bias, we conduct the subgroup analysis and build the Cox model with individual-specific coefficients. We introduce the pairwise fusion penalty function and minimize the penalized criterion function by the majorized alternating direction method of multipliers (ADMM) algorithm. Then our estimation procedure automatically clusters individuals having similar treatment effects into the same subgroup and estimates the treatment effects simultaneously. For the asymptotic theory, we first verify that the oracle estimator, the estimator with prior information about the subgroup structure, is asymptotically consistent and has the asymptotic normal distribution. Then we prove that under some mild conditions, the oracle estimator is a local minimizer of our criterion function with high probability. This implies the asymptotic consistency and normality of our estimator. We show the finite sample estimation results by the simulation studies. Furthermore, we use our method to analyze the breast cancer data collected by the Netherlands Cancer Institute (NKI). In the long-term follow-up study of recurrent events, panel count data occurs when the observations of individuals are some discrete time points such that only the occurrence numbers of recurrent events between the adjacent time points are available. In general, the follow-up study often ends with some events having intricate interactions with the recurrent events, which motivates us to study the statistical inference approaches for panel count data with an informative terminal event. This thesis builds the nonparametric and semiparametric models for this problem with the least squares-based loss functions. Treating the distribution of terminal event time as a nuisance functional parameter, we consider the two-stage estimation procedures. We approximate the nonparametric function by the monotone I-spline function because the spline estimation converges faster than the estimation approximated by the step function. Using the empirical process theories, we verify the asymptotic properties for the proposed estimators. We also conduct the two-sample hypothesis test for the mean function in the nonparametric model. Our simulation studies demonstrate that the proposed estimations perform well. Finally, we use our methods to analyze the dataset of the Chinese Longitudinal Healthy Longevity Survey (CLHLS).