Bayesian influence analysis of generalized partial linear mixed models for longitudinal data

Abstract

This paper develops a Bayesian local influence approach to assess the effects of minor perturbations to the prior, sampling distribution and individual observations on the statistical inference in generalized partial linear mixed models (GPLMMs) with the distribution of random effects specified by a truncated and centered Dirichlet process (TCDP) prior. A perturbation manifold is defined. The metric tensor is employed to select an appropriate perturbation vector. Several Bayesian local influence measures are proposed to quantify the degree of various perturbations to statistical models based on the first and second-order approximations to the objective functions including the ϕ-divergence, the posterior mean distance and Bayes factor. We develop two Bayesian case influence measures to detect the influential observations in GPLMMs based on the ϕ-divergence and Cook’s posterior mean distance. The computationally feasible formulae for Bayesian influence analysis are given. Several simulation studies and a real example are presented to illustrate the proposed methodologies.