Hierarchical models are statistical models containing random components in addition to or instead of the usual fixed parameter values.  In ecology, specialized applications of hierarchical models are increasing rapidly and take such varied forms as generalized linear models with mixed random and fixed effects, structured population state-space models with observational and process variability, and capture recapture models with randomly varying capture probabilities.  To date, fitting such models to data using maximum likelihood (ML) estimation has been difficult, because the likelihood function is an intractable integral for all but the simplest models (usually based on the normal distribution).  Instead, the most practical choice of model fitting methods has been the Bayesian approach, for which the integrals can be cleverly bypassed with the Markov chain Monte Carlo (MCMC) simulation algorithms.

This presentation introduces a new statistical computing method to calculate ML estimates and their standard errors for hierarchical statistical models. Although the method uses the Bayesian framework and exploits the computational simplicity of the MCMC algorithms, it provides valid frequentist inferences such as the ML estimates and their standard errors. The inferences are completely invariant to the choice of the prior distributions and therefore avoid the inherent subjectivity of the Bayesian approach. The method is easily implemented using standard MCMC software ("WinBUGS", etc.).

The method is illustrated with two ecological examples for which ML estimation has formerly been problematic.  The two models are non-linear population dynamics models containing process noise as well as observation error.