Abstract

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.

Disciplines

Biostatistics | Longitudinal Data Analysis and Time Series | Numerical Analysis and Computation | Statistical Methodology | Statistical Models | Statistical Theory