One-Sided Tolerance Limits for Balanced and Unbalanced Random Effects Models

Abstract

A simple analytical approach for constructing one-sided β-content, γ-confidence tolerance limits is proposed for general random effects models with normal data in both balanced and unbalanced data scenarios. The approach is based on an approximation to the noncentral t distribution and modified large sample methods for constructing confidence bounds on functions of variance components. An alternative bootstrap-adjusted limit is also proposed. The performance of the analytical and bootstrap-adjusted limits is assessed via simulation techniques. The results indicate that the analytical limit is generally somewhat conservative, but is often less conservative than an existing analytical approach and may provide substantially shorter interval lengths, particularly for smaller sample sizes and larger values of γ. The bootstrap-adjusted limit generally maintains the nominal confidence level and yields shorter interval lengths, though it can be anticonservative for small sample sizes. This article has supplementary material online.