Inference about Variance Components in the One-Way Model

Abstract The estimation of variance components in the one-way model is considered from a subjective Bayesian point of view. The situation in which the classical unbiased estimate of the between variance component is negative is explored in some detail. Exact and approximate posterior distributions are obtained in both the balanced and unbalanced case. Common sense aspects of the problem are emphasized, and some contrasts with other approaches. For example, Bayesianly speaking, a large negative unbiased estimate of the between variance component indicates an uninformative experiment in which the effective likelihood for that variance component is extremely flat, instead of strong evidence that the variance component is nearly zero.