Assessing the predictive ability of a multilevel binary regression model

An adaptation of the Brier score and the concordance probability is proposed for the two-level and the three-level random intercept binary regression model. This results in 2 different Brier scores and 3 different C-indices for the two-level binary regression model and 4 different Brier scores and 7 different C-indices for the three-level binary regression model. The ensemble of these measures offers a better view on how the different elements of the random effects model, i.e. the covariates and the random effects, affect the predictive ability of the model separately, evaluated on a within-cluster, between-cluster and global level. For all measures, an estimation procedure using Bayesian and likelihood estimation methods was developed, including a percentile and a BCa non-parametric bootstrap step to construct credible/confidence intervals. In a simulation study, the likelihood estimation procedure showed difficulties in estimating unbiasedly the predictive ability of the random effects, while the Bayesian estimation procedure resulted in good estimation properties for all of the developed measures. The BCa non-parametric bootstrap method resulted in confidence/credible intervals with better coverage properties than the percentile non-parametric bootstrap method. The proposals are applied to a real-life binary data set with a three-level structure using the Bayesian estimation procedure.