Identification of the variance components in the general two-variance linear model

Bayesian analyses frequently employ two-stage hierarchical models involving two-variance parameters: one controlling measurement error and the other controlling the degree of smoothing implied by the model's higher level. These analyses can be hampered by poorly identified variances which may lead to difficulty in computing and in choosing reference priors for these parameters. In this paper, we introduce the class of two-variance hierarchical linear models and characterize the aspects of these models that lead to well-identified or poorly identified variances. These ideas are illustrated with a spatial analysis of a periodontal data set and examined in some generality for specific two-variance models including the conditionally autoregressive (CAR) and one-way random effect models. We also connect this theory with other constrained regression methods and suggest a diagnostic that can be used to search for missing spatially varying fixed effects in the CAR model.