Efficient Shrinkage for Generalized Linear Mixed Models Under Linear Restrictions

An efficient estimation approach to joint modeling of longitudinal and survival data

The joint models for longitudinal and survival data have recently received significant attention in medical and epidemiological studies. Joint models typically combine linear mixed effects models for repeated measurement data and Cox models for survival time. When we are jointly modeling the longitudinal and survival data, variable selection and efficient estimation of parameters are especially important for performing reliable statistical analyzes, both of which are currently lacking in the literature. In this paper we discuss the pretest and shrinkage estimation methods for jointly modeling longitudinal data and survival time data when some of the covariates in both longitudinal and survival components may not be relevant for predicting survival times. In this situation, we fit two models: the full model that contains all the covariates and the subset model that contains a reduced number of covariates. We combine the full model estimators and the estimators that are restricted by a linear hypothesis to define pretest and shrinkage estimators. We provide their numerical mean squared errors (MSE) and relative MSE. We show that if the shrinkage dimension exceeds two, the risk of the shrinkage estimators is strictly less than that of the full model estimators. Our proposed methods are illustrated by extensive simulation studies and a real-data example.