Weighted cumulative sum tests for random effect models with binary responses

Correlated binary responses are very commonly encountered in many disciplines like, for example, medical studies. The development of goodness-of-fit tests is essential for examining the adequacy of the fitted models. The objective of this article is to provide weighted modifications of cumulative sums or moving cumulative sums of residuals for testing goodness-of-fit of random effects logistic regression models. The proposed weights can be interpreted as the residuals of a weighted linear regression of an omitted covariate on the covariates already included in the fixed part of the model. These processes lead to supremum statistics whose null distribution is derived using simulation. Results from a simulation study suggest better performance of the weighted when compared to the unweighted supremum statistics. The proposed tests are illustrated using a real data example.