Bayesian analysis for random coefficient regression models using noninformative priors
type: Item
instance of: scholarly article
title: Bayesian analysis for random coefficient regression models using noninformative priors
zbMATH Open document ID: 0843.62031
author: Ruoyong Yang
author: Ming-Hui Chen
published in: Journal of Multivariate Analysis
publication date: 1996-08-21T00:00:00Z
review text: Consider a random coefficient regression (RCR) model \[ \mathbf {y}_i = X_i {\mathbf \beta}_i + \varepsilon_i, \tag{1} \] where \({\mathbf y}_i\) is a \(t_i \times 1\) vector of observations, \(X_i\) is a \(t_i \times p\) constant design matrix, \({\mathbf \beta}_i\) is a \(p \times 1\) vector of random coefficients for the \(i\) th experimental subject, and \(\varepsilon_i\) is a vector of errors for \(i = 1,2,\dots, n\). Furthermore, we assume that \(({\mathbf \beta}_i, \varepsilon_i, i = 1,2,\dots, n)\) are independent and \[ {\mathbf \beta}_i \sim MVN ({\mathbf \beta}, \Sigma), \quad \varepsilon_i \sim MVN ({\mathbf 0}, \sigma^2 I_i), \] where \(\mathbf \beta\) is a \(p \times 1\) vector of the mean of the \({\mathbf \beta}_i\) vectors and \(I_i\) is the \(t_i \times t_i\) identity matrix. The RCR model (1) arises from that the probability distributions for responses of different individuals belong to a single family and that the regression parameters vary across individuals because of random effects. It is widely used in growth curves, medical studies, repeated measures, or longitudinal studies. A lot of experiments are performed for the study of the population parameters, \({\mathbf \beta}\), \(\sigma^2\), and \(\Sigma\). Unfortunately, the inference concerning \(\mathbf \beta\) for this model (1) is a notoriously difficult problem, especially when the data is unbalanced or incomplete, which is the typical case in clinical trials. We study the general RCR model (1) using a Bayesian approach through noninformative priors. The outline of this paper is as follows: Section 2 contains several matrix results and the derivation of the Fisher information matrix of the RCR model (1). In Section 3, we derive the Jeffreys prior, reference priors, and bounds for these priors. In Bayesian analysis, it is important to know if the posteriors are proper or not when the noninformative priors are used. In Section 4, we study the sufficient conditions for the proper of the poste…
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62J99
zbMATH DE Number: 844614
zbMATH Keywords: random coefficient regression model
zbMATH Keywords: Gibbs sampler
zbMATH Keywords: repeated measures
zbMATH Keywords: noninformative priors
zbMATH Keywords: Fisher information matrix
zbMATH Keywords: Jeffreys prior
zbMATH Keywords: reference priors
zbMATH Keywords: bounds
zbMATH Keywords: posterior moments
zbMATH Keywords: hybrid Markov chain sampling scheme
zbMATH Keywords: posterior expectations
zbMATH Keywords: simulation
MaRDI profile type: Publication
full work available at URL: jmva.1995.1080
OpenAlex ID: W2135165016
DOI: 10.1006/JMVA.1995.1080
description: scientific article; zbMATH DE number 844614