Goodness of fit statistics for mixed effect logistic regression models

Mixed effects logistic regression models have become widely used statistical models to model clustered binary responses. However, assessing the goodness of fit (GOF) in these models, when the cluster sizes and the number of clusters are small, is not clear. In this research, three GOF statistics are proposed, and their performance in terms of Type I error rate and power is examined via simulation study. The proposed GOF statistics are the logit residual, log-transformed residual and the absolute residual GOF statistics. The simulation study was applied on different cases of number of clusters, cluster sizes and types of predictors. The simulation results showed the performance of the logit residual and the log-transformed residual GOF statistics to be poor. The absolute residual GOF statistic performed well over most cases of the simulation. It gave proper Type I error rates and high power for most cases and it is recommended to use for mixed effects logistic regression models as long as the number of clusters is at least 10 and the cluster sizes are 10 or more. However, the absolute residual GOF statistic can be affected by extremely small or large estimated probabilities and further research is recommended to avoid or reduce this restriction