Bayesian analysis of random-effect models in the analysis of variance

Abstract

Bayesian methods are utilized to analyse a one-way random effect model yij = μ+ai+eij in which the errors eij are assumed to follow a first-order autoregressive series eij = ρei(j-1)+εij. For given value of ρ, the model can be reduced to that considered in our earlier paper (1965). It is shown that inferences about the variances σα2 = var(ai), and σ2 = var (εij) can be very sensitive to changes in the value of ρ assumed. In particular, in a set of data generated from a population with ρ = − 0·5, σ = 1·5 and σα2 = 1, we find that if the independence assumption ρ = 0 were made, the classical unbiased estimator of σα2 would be negative and the posterior distribution of σα2 would have its modal value at the origin. Uncertainty in the inferences about (σ2, σα2⁠) is then removed by considering the posterior distribution of ρ. Some sampling theory considerations of the behaviour of certain quantities in the posterior distributions of (σ2, σα2⁠, ρ) are also discussed.