[Paper Review] Conditional Model Selection in Mixed-Effects Models with cAIC4

Benjamin Säfken, David Rügamer+2 more|arXiv (Cornell University)|Mar 15, 2018

Statistical Methods and InferenceMathematics15 references48 citations

TL;DR

The paper introduces the R package cAIC4 for computing the conditional Akaike Information Criterion (cAIC) in mixed models estimated with lme4 and gamm4, and provides an automated stepwise model selection scheme with practical examples.

ABSTRACT

Model selection in mixed models based on the conditional distribution is appropriate for many practical applications and has been a focus of recent statistical research. In this paper we introduce the R-package cAIC4 that allows for the computation of the conditional Akaike Information Criterion (cAIC). Computation of the conditional AIC needs to take into account the uncertainty of the random effects variance and is therefore not straightforward. We introduce a fast and stable implementation for the calculation of the cAIC for linear mixed models estimated with lme4 and additive mixed models estimated with gamm4 . Furthermore, cAIC4 offers a stepwise function that allows for a fully automated stepwise selection scheme for mixed models based on the conditional AIC. Examples of many possible applications are presented to illustrate the practical impact and easy handling of the package.

Motivation & Objective

• Motivate model selection in mixed models from a conditional perspective (focusing on cluster/subject-specific predictions).
• Provide a fast, stable implementation of the conditional AIC (cAIC) that accounts for random effects uncertainty.
• Extend cAIC computation to Gaussian, Poisson, and Bernoulli responses within lme4/gamm4 frameworks.
• Introduce an automated stepwise selection procedure based on cAIC and illustrate with real data examples.

Proposed method

• Define cAIC as a corrective criterion for the conditional likelihood, accounting for estimation of fixed effects and covariance parameters.
• Provide analytic bias corrections for Gaussian responses that incorporate unknown variance and boundary issues (Greven 2010).
• Derive Poisson-specific bias correction (BC) based on log-derivative identities, enabling direct computation without full bootstrap.
• Offer bootstrap-based (or equivalent) Bernoulli corrections for binary responses to obtain cAIC.
• Implement fast computation by adapting Greven and Kneib’s approach to lmerMod objects and enabling automated boundary handling.
• Include a stepwise, automated model selection function using the conditional AIC for mixed models.

Experimental results

Research questions

• RQ1 How can the conditional AIC be computed efficiently for linear, Poisson, and Bernoulli mixed models within the lme4/gamm4 framework?
• RQ2 How should the bias correction terms for cAIC be computed when random effects variances are on the boundary or when variance is estimated?
• RQ3 Does a conditional AIC-based stepwise selection yield practical and reliable model choices for mixed models with random effects and penalized spline components?

Key findings

• The package cAIC4 provides fast and stable computation of the conditional AIC for mixed models estimated with lme4 and GAMM via gamm4.
• It distinguishes and implements the appropriate bias corrections for Gaussian, Poisson, and Bernoulli responses to obtain cAIC.
• For Poisson responses, the BC is reformulated to a computable analytic form, enabling efficient cAIC calculation.
• For Bernoulli responses, a bootstrap-based estimate of the bias correction is implemented to obtain cAIC.
• A fully automated stepwise conditional variable selection scheme is introduced and demonstrated on real data examples, highlighting practical utility.