Link misspecification in generalized linear mixed models with a random intercept for binary responses

We present in this paper tests for link misspecification in generalized linear mixed models with a random intercept for binary responses. To facilitate model diagnosis, we consider two types of grouped responses induced from the original responses and also stochastically create reclassified binary responses. Maximum likelihood estimators based on the original observed data and the counterpart estimators based on different induced data sets are investigated in the presence of link misspecification. Results from this investigation motivate four diagnostic tests for assessing the adequacy of an assumed link, which can provide information on how the true link differs from an assumed symmetric link when the proposed tests reject the assumed link. The performance of these tests is illustrated via simulation studies in comparison with an existing method for checking link assumptions. These tests are applied to a data set from a longitudinal study that was analyzed in the existing literature using logistic regression.