A note on numerical evaluation of conditional Akaike information for nonlinear mixed-effects models

TL;DR

This paper introduces two methods for evaluating the conditional Akaike information in nonlinear mixed-effects models, with one method being more robust across different data distributions, aiding model selection.

Contribution

It proposes two novel methods for computing cAI in nonlinear mixed-effects models, especially addressing cases with no restriction on cluster size and varying data distributions.

Findings

Method 1 performs well with Gaussian data.

Method 2 is robust across various distributions.

Method 2 is recommended for model selection.

Abstract

We propose two methods to evaluate the conditional Akaike information (cAI) for nonlinear mixed-effects models with no restriction on cluster size. Method 1 is designed for continuous data and includes formulae for the derivatives of fixed and random effects estimators with respect to observations. Method 2, compatible with any type of observation, requires modeling the marginal (or prior) distribution of random effects as a multivariate normal distribution. Simulations show that Method 1 performs well with Gaussian data but struggles with skewed continuous distributions, whereas Method 2 consistently performs well across various distributions, including normal, gamma, negative binomial, and Tweedie, with flexible link functions. Based on our findings, we recommend Method 2 as a distributionally robust cAI criterion for model selection in nonlinear mixed-effects models.