Bootstrap estimation of uncertainty in prediction for generalized linear mixed models

In the framework of Mixed Models, it is often of interest to provide an estimate of the uncertainty in predictions for the random effects, customarily defined by the Mean Squared Error of Prediction (MSEP). To address this computation in the Generalized Linear Mixed Model (GLMM) context, a non-parametric Bootstrap algorithm is proposed. First, a newly developed Bootstrap scheme relying on random weighting of cluster contributions to the joint likelihood function of the model and the Laplace Approximation is used to create bootstrap replicates of the parameters. Second, these replicates yield in turn bootstrap samples for the random effects and for the responses. Third, generating predictions of the random effects employing the bootstrap samples of observations produces bootstrap replicates of the random effects that, in conjunction with their respective bootstrap samples, are used in the estimation of the MSEP. To assess the validity of the proposed method, two simulation studies are presented. The first one in the framework of Gaussian LMM, contrasts the quality of the proposed approach with respect to: (i) analytical estimators of MSEP based on second-order correct approximations, (ii) Conditional Variances obtained with a Bayesian representation and (iii) other bootstrap schemes, on the grounds of relative bias, relative efficiency and the coverage ratios of resulting prediction intervals. The second simulation study serves the purpose of illustrating the properties of our proposal in a Non-Gaussian GLMM setting, namely a Mixed Logit Model, where the alternatives are scarce.