Bayesian Tolerance Intervals for the Unbalanced One-Way Random Effects Model

Abstract

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 β, σ²_ε, and ν in the case of the random effects model follow easily, where β is the grand mean, σ²_ε is the variance of random errors, and ν = σ²_γ/σ²_ε 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.

• Bayesian Procedure
• Monte Carlo Simulation
• Predictive Density
• Probability Matching Prior
• Random Effects
• Reference Prior
• Tolerance Intervals

Additional information

Notes on contributors

Abraham J. Van der Merwe

Dr. Van der Merwe is a Professor in the Department of Mathematical Statistics. His e-mail address is [email protected].

Albertus L. Pretorius

Dr. Pretorius is a Senior Lecturer in the Department of Mathematical Statistics. His e-mail address is [email protected].

Johannes H. Meyer

Dr. Meyer is a Professor in the Department of Mathematics. His e-mail address is [email protected].