Inferential procedures for random effects in generalized linear mixed models.

Area of Science:

• Statistics
• Computational Statistics
• Statistical Modeling

Background:

• Generalized linear mixed models (GLMMs) are widely used for complex data.
• Accurate uncertainty quantification is crucial for reliable random effects prediction in GLMMs.
• Existing prediction uncertainty measures in GLMMs require careful theoretical and empirical evaluation.

Purpose of the Study:

• To compare three common measures of uncertainty for random effects prediction in GLMMs.
• To derive asymptotic results on the consistency of these uncertainty measures.
• To resolve discrepancies between theoretical and empirical findings for a specific GLMM variance estimator.

Main Methods:

• Comparative analysis of three uncertainty measures: unconditional and conditional mean squared errors of prediction (UMSEP, CMSEP), and glmmTMB prediction gap variance.
• Derivation of asymptotic consistency results for the uncertainty estimators.
• Re-interpretation of the glmmTMB variance estimator under conditional assumptions.

Main Results:

• The three distinct theoretical uncertainty measures yield estimators that are remarkably similar in form.
• Asymptotic analysis confirms the consistency of the derived uncertainty estimators.
• A conditional re-interpretation resolves prior contradictions in the glmmTMB variance estimator's performance.

Conclusions:

• The similarity in estimators suggests practical interchangeability for certain GLMM prediction tasks.
• The derived asymptotic results provide theoretical grounding for uncertainty estimation in GLMMs.
• Findings impact the construction of prediction intervals for random effects, particularly concerning normality assumptions.