Bayesian analysis for random coefficient regression models using noninformative priors
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 posterior distributions corresponding to the Jeffreys prior and reference priors and the existence of the posterior moments. In Section 5, a hybrid Markov chain sampling scheme, which is used for computing posterior expectations, is developed. A small-scale simulation study and a real data example are also given therein. Comments and generalizations are given in Section 6.
scientific article; zbMATH DE number 844614