Measuring complexity for hierarchical models using effective degrees of freedom

Hierarchical models can express ecological dynamics using a combination of fixed and random effects, and measurement of their complexity (effective degrees of freedom, EDF) requires estimating how much random effects are shrunk toward a shared mean. Estimating EDF is helpful to (1) penalize complexity during model selection and (2) to improve understanding of model behavior. I applied the conditional Akaike Information Criterion (cAIC) to estimate EDF from the finite-difference approximation to the gradient of model predictions with respect to each datum. I confirmed that this has similar behavior to widely used Bayesian criteria, and I illustrated ecological applications using three case studies. The first compared model parsimony with or without time-varying parameters when predicting density-dependent survival, where cAIC favors time-varying demographic parameters more than conventional Akaike Information Criterion. The second estimates EDF in a phylogenetic structural equation model, and identifies a larger EDF when predicting longevity than mortality rates in fishes. The third compares EDF for a species distribution model fitted for 20 bird species and identifies those species requiring more model complexity. These highlight the ecological and statistical insight from comparing EDF among experimental units, models, and data partitions, using an approach that can be broadly adopted for nonlinear ecological models.