A Goodness-of-Fit Test for Mixed-Effects Logistic Regression

TL;DR

This paper introduces a new goodness-of-fit test for mixed-effects logistic regression models with random slopes, addressing limitations in sparse cluster settings and demonstrating reliable performance through simulations.

Contribution

It extends a grouping-based Wald test to models with random slopes, including a data-driven method for selecting the number of groups, and provides implementation in Stata.

Findings

Maintains nominal Type I error in small and three-level models.

High power to detect fixed-effects misspecification, especially nonlinearity and interactions.

Data-driven G selection outperforms fixed G in sparse balanced designs.

Abstract

Mixed-effects logistic regression is widely used for binary outcomes in hierarchical data, yet formal goodness-of-fit tests remain limited to random-intercept models and do not address sparse cluster settings. We extend a grouping-based Wald test to mixed-effects logistic models with random slopes. The procedure groups observations by predicted probabilities within clusters, augments the model with pooled group indicators, and tests their joint significance using a Wald statistic. To accommodate small clusters, we introduce a data-driven rule for selecting the number of groups, G=min(10,n_min), where n_min is the smallest cluster size, ensuring feasible estimation. Simulation studies across 24 null scenarios show that the test maintains nominal Type I error in three-level random slope models, including at smaller sample sizes than previously studied. The test exhibits increasing power to detect fixed-effects misspecification: power against omitted nonlinearity rises from 0.07 to 1.00 across effect sizes, and power against omitted interactions reaches 0.87. As expected, the test has no power to detect omission of a clustering level, reflecting its focus on residual structure in predicted probabilities. In sparse balanced designs, fixing G=10 leads to complete test failure, whereas the data-driven rule performs reliably. The method is implemented in the Stata program mlm_gof.