Inference about the fixed and random effects in a mixed-effects linear model: an approximate Bayesian approach
An approximate Bayesian analysis is considered for data that follow a mixed-effects linear model of the form y = X[beta] + Zs + e, where X and Z are known matrices, [beta] is a vector of unknown parameters, and s and e are statistically independent random vectors whose distributions are multivariate normal with null mean vectors and variance-covariance matrices [sigma][subscript]sp12 I and [sigma][superscript]2 I, respectively. It is assumed that a priori [beta], [sigma][subscript]sp12, and [sigma][superscript]2 are statistically independent, that the distribution of [beta] is noninformative, and that 1/[sigma][subscript]sp12 and 1/[sigma][superscript]2 have gamma distributions;The problem considered is that of inferences about a linear combination of the fixed and random effects, say w = [lambda][superscript]'[beta] + [delta][superscript]' s. Specifically, consideration is given to inferences about w that based on a normal approximation to the posterior density p([beta], s ǁ y) of [beta] and s. The use of this normal approximation (in making inferences about w) does not require numerical integration, whereas the use of the exact posterior distribution of w requires numerical integration;Two approaches to the approximation of p([beta], s ǁ y) are discussed. One is centered at ([beta],\ s) where \ s is the maximum of p( s ǁ y) and [beta] is the maximum of p([beta] ǁ s = \ s, y), and the other is centered at the values of [beta] and s that maximize p([beta], s ǁ y). The possible multimodality of p( s ǁ y) and of p([beta], s ǁ y) is discussed. The numerical problem of finding the maxima of p( s ǁ y) or p([beta], s ǁ y) is considered. It is shown that by adopting an approach that is similar in spirit and technique to the ridge analysis of a response surface, this problem can be reduced to a one-dimensional problem. A "linearized" version of Newton's method is proposed for solving the latter problem, and appropriate starting values for this and other iterative algorithms are described;The approximate and exact posterior distributions of w are compared for some real data sets. The approximate posterior distribution was found to perform relatively well when the prior information is reasonably consistent with the information provided by the data;Some extensions to mixed-effects linear models with more than two variance components are considered.