The Schwarz criterion and related methods for normal linear models

In this paper we derive Schwarz's information criterion and two modifications for choosing fixed effects in normal linear mixed models. The first modification allows an arbitrary, possibly informative, prior for the parameter of interest. Replacing this prior with the normal, unit-information, prior of Kass & Wasserman (1995) and the generalised Cauchy prior of Jeffreys (1961) yields the usual Schwarz criterion and a second modification, respectively. Under the null hypothesis, these criteria approximate Bayes factors using the corresponding priors to increased accuracy. In regression, the second modification also corresponds asymptotically to the Bayes factors of Zellner & Siow (1980) and O'Hagan (1995), and is similar to the Bayes factor of Berger & Pericchi (1996). In mixed models, the effective sample size term in Schwarz's formula is ambiguous because of correlation between observations. We propose an appropriate generalisation of Schwarz's approximation and apply our results to evaluate a large class of models for repeated neuron area measurements in alcoholic and suicidal patients.