We consider Bayesian inference in semiparametric mixed models (SPMMs) for longitudinal data. SPMMs are a class of models that use a nonparametric function to model a covariate effect, e.g., time effect, a parametric function to model other covariate effects, and parametric or nonparametric random effects to account for the within-subject correlation. We model the nonparametric function using the Bayesian formulation of a cubic smoothing spline, and the random effect distribution using a normal distribution and alternatively with a nonparametric Dirichlet process (DP) prior. When the random effect distribution is assumed to be normal, we propose a uniform shrinkage prior (USP) for the variance components and the smoothing parameter. When the random effect distribution is modeled nonparametrically, we use a DP prior with a normal base measure and propose a USP for the hyperparameters of the DP base measure. We argue that the commonly assumed DP prior implies a non-zero mean of the random effect distribution, even when a base measure with mean zero is specified. This leads to biases in Bayesian inference for the regression coefficients and the spline. We propose a correction using a post-processing technique. We show that under mild conditions the posterior is proper under the proposed USPs, a flat prior for the fixed effects parameters, and an improper prior for the residual variance. We illustrate the proposed approach using a longitudinal hormone dataset, and carry out extensive simulation studies to compare its finite sample performance with that of the existing methods.
Biostatistics | Longitudinal Data Analysis and Time Series | Numerical Analysis and Computation | Statistical Methodology | Statistical Models | Statistical Theory
Li, Yisheng; Lin, Xihong; and Mueller, Peter, "Bayesian inference in semiparametric mixed models for longitudinal data" (August 2007). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 37.
Biostatistics Commons, Longitudinal Data Analysis and Time Series Commons, Numerical Analysis and Computation Commons, Statistical Methodology Commons, Statistical Models Commons, Statistical Theory Commons