Random-intercept misspecification in generalized linear mixed models for binary responses

We study properties of maximum likelihood estimators of parameters in generalized linear mixed models for a binary response in the presence of random-intercept model misspecification. Further exploiting the test proposed in an existing work initially designed for detecting general random-effects misspecification, we are able to reveal how the true random-intercept distribution deviates from the assumed. Besides this advance compared to the existing methods, we also provide theoretical insights on when and why the proposed test has low power to identify certain forms of misspecification. Large-sample numerical study and finite-sample simulation experiments are carried out to illustrate the theoretical findings.