A family of tests to detect misspecifications in the random-effects structure of generalized linear mixed models

Estimation in generalized linear mixed models for non-Gaussian longitudinal data is often based on maximum likelihood theory, which assumes that the underlying probability model is correctly specified. It is known that the results obtained from these models are not always robust against misspecification of the random-effects structure. Therefore, diagnostic tools for the detection of this misspecification are of the utmost importance. Three diagnostic tests, based on the eigenvalues of the variance-covariance matrices for the fixed-effects parameters estimates, are proposed in the present work. The power and type I error rate of these tests are studied via simulations. A very acceptable performance was observed in many cases, especially for those misspecifications that can have a big impact on the maximum likelihood estimators.