Fixed Size Confidence Regions for the Parameters of the Mixed Effects Logistic Regression Model

We develop fixed size confidence regions for estimating the fixed and random effects parameters of the mixed effects logistic regression model. This model applies to, among others, the study of the effects of covariates on a dichotomous response variable when subjects are sampled in clusters. Two sequential procedures are developed to estimate with a prescribed accuracy (confidence level) and fixed precision the set of fixed and random effects parameters and linear transformations of these parameters, respectively. We show that the two procedures are asymptotically consistent (i.e., the coverage probability converges to the nominal confidence level) and asymptotically efficient (i.e., the ratio of the expected random sample size to the unknown best fixed sample size converges to 1) as the width of the confidence region converges to 0. Suggestions to improve the performance of the procedures are provided based on Monte Carlo simulation and illustrated through a longitudinal clinical trial data.