Frequentist Conditional Variance for Nonlinear Mixed-Effects Models

Nonlinear mixed-effects models are commonly used in fisheries and ecological studies to account for complex relationships and dependencies in data. These models involve both fixed parameters to estimate and random-effects (REs) to predict. This paper addresses the inferential setting involving repeated sampling of the data but conditional on the unknown REs. This setting is more appropriate when the focus is on statistical inferences based on the specific values of REs that generated the data. Assuming the Laplace approximation is appropriate to derive the marginal likelihood and following a frequentist framework, this work derives RE-conditional bias approximations of maximum likelihood parameter estimators and empirical Bayes RE predictors, as well as the conditional covariance and mean squared error (MSE) among parameter estimators and RE predictors. It is shown that the RE-conditional MSE can be approximated with the unconditional MSE. Simulation studies demonstrate that the variance and MSE approximations are reasonably accurate for relevant sample sizes. Considering the finite-sample RE-conditional biases in the parameter estimates and RE predictions, the MSE is more appropriate for constructing confidence intervals (CIs), and the CI coverage of REs should be interpreted as the average coverage over a range of REs or over repeated generation of REs.