Some Bayesian and non-Bayesian procedures for the analysis of comparative experiments and for small-area estimation: computational aspects, frequentist properties, and relationships
Let y represent an n x 1 observable random vector that follows the linear model y = X[beta] + Zs + e. Here X and Z are give matrices, [beta] is a vector of unknown parameters, and s and e are statistically independent random vectors that have multivariate normal distributions with mean vectors equal to 0 and covariance matrices [sigma][subscript]spe2 I and [sigma][subscript]sps2 I, respectively; so that E(y) = X[beta] and var(y) = [sigma][subscript]spe2( I + [gamma] ZZ[superscript]'), where [gamma] = [sigma][subscript]sps2/[sigma][subscript]spe2. The problem of interest is the prediction of the realization of a random variable of the form w = [lambda][superscript]'[beta] + [delta][superscript]' s--we refer to this problem as the general prediction problem. Many inference problems, including the estimation of a treatment contrast and the estimation of a small-area mean, can be regarded as special cases of the general prediction problem;We consider both traditional (frequentist) and Bayesian approaches to the point and interval prediction of w. Our coverage of frequentist methodology includes the ordinary least squares approach to point prediction, the estimation of the variance components [sigma][subscript]spe2 and [sigma][subscript]sps2 (including restricted maximum likelihood estimation), and the use of variance-component estimates to obtain generalized least squares point predictors. Some exact and approximate prediction interval procedures, based on ordinary or generalized least squares point predictors, are also considered, and the computational aspects of their implementation is discussed;In applying the Bayesian approach to the general prediction problem, we specify a general class of prior distributions, derive the corresponding posterior distributions of w given y, and describe point and interval characterizations of the posteriors that can be used as predictors for w. We show how, by taking advantage of computational results for frequentist predictors, the normally severe computational requirements of the Bayesian approach can be reduced;The frequentist and Bayesian approaches to the mixed model are essentially equivalent to the hierarchical Bayes and empirical Bayes approaches as applied to the fixed model obtained by regarding s as a vector of unknown parameters rather than as a random vector. Thus, the frequentist predictors can be viewed as approximations to the various Bayesian predictors. We present the results of a Monte Carlo study of the frequentist properties of both the traditional and the Bayesian predictors as applied to inference about treatment contrasts and small-area means. The results suggest that the Bayesian approach produces point and interval predictors whose overall performance compares favorably with that of the frequentist predictors, and that there are applications where the Bayesian predictors should be used in preference to the frequentist predictors.