A diagnostic test for the mixing distribution in a generalised linear mixed model

We introduce a diagnostic test for the mixing distribution in a generalised linear mixed model. The test is based on the difference between the marginal maximum likelihood and conditional maximum likelihood estimators of a subset of the fixed effects in the model. We derive the asymptotic variance of this difference, and propose a test statistic that has a limiting chi-squared distribution under the null hypothesis that the mixing distribution is correctly specified. This strategy uses an idea presented by Hausman (1978), who considered analogous tests for the linear mixed model. An important advantage of the methods outlined here is that the resulting diagnostic test is easily implemented in commercial software. We illustrate the method by applying it to data from a clinical trial investigating the effect of hormonal contraceptives in women.