Frequentist delta-variance approximations with mixed-effects models and TMB

Measures of uncertainty are investigated for estimates and predictions using nonlinear mixed-effects models including state–space models in particular. These nonlinear mixed-effects models include fixed parameters and random effects. Maximum likelihood estimation of the parameters and conditional mean predictors of random effects are commonly used to estimate important quantities for a wide spectrum of applications. These quantities of interest may be functions of the parameters and random effects. In this case, software packages such as TMB and glmmTMB use a generalized delta method to provide standard errors and statistical inference. In the frequentist framework, it is clarified that these packages actually provide estimates of mean squared errors (MSE’s) based on a multivariate normal approximation of the distribution of the random effects given data. It is further shown that the MSE’s are not the variance of estimates due to repeated sampling of the data and the random effects. Equations are provided for that variance, including orders of approximations. In many cases the MSE’s will be more appropriate to use for statistical inference, but not always, and this is demonstrated for a simple random-walk state–space model example.