Bayesian tolerance intervals for the unbalanced one-way random effects model

Statistical intervals, properly calculated from sample data, are likely to be substantially more informative to decision makers than obtaining a point estimate alone and are often of paramount interest to practitioners and thus management (and are usually a great deal more meaningful than statistical significance or hypothesis tests). In this note, a simulation-based approach for determining Bayesian tolerance intervals in an unbalanced one-way random effects model is illustrated. Reference and probability matching priors are first derived for a more general mixed linear model from which the priors for beta, sigma(2)(epsilon), and nu in the case of the random effects model follow easily, where beta is the grand mean, sigma(2)(epsilon) is the variance of random errors, and nu = sigma(2)(gamma)/sigma(2)(epsilon) is the ratio of the random effect to noise variances. A tensile-strength example illustrates the flexibility and unique features of the Bayesian simulation method for the construction of tolerance intervals. Although this example has only one random effect, the method can be applied similarly to other unbalanced data sets and models with multiple variance components. In the last section, a procedure is discussed to obtain Bayesian tolerance intervals for the three-component balanced hierarchical design model.