A review of N-mixture models
N-mixture models were born in 2004 of the necessity to model animal population size from point counts with imperfect detection of individuals, where capture-recapture methods are infeasible. Initially developed for applications where population size was assumed constant, N-mixture models were extended in 2011 to include population dynamics, allowing application to populations whose size fluctuates during the study. A further extension in 2014 accommodates populations with multiple “states” such as age class or sex. More recent extensions model spatial movement of animals among habitat patches or the spatial spread of infectious disease in a human population. The core idea underlying this class of models is a hierarchical structure, where the observation model is defined conditional on the model for true abundance. This hierarchy allows researchers to incorporate information about observation and abundance processes, while permitting distinct inferences about elements affecting detection and those affecting abundance. Another benefit of the hierarchical approach is the ability to accommodate many existing sampling protocols such as removal sampling and distance sampling. One drawback to N-mixture models is that since they estimate both abundance and detection from replicated but unmarked counts, model parameters may not be clearly identifiable. A second drawback is that when observed counts are large, calculating the N-mixture likelihood is computationally infeasible. This difficulty motivated an approximate likelihood based on the normal approximation to the binomial. The normal approximation provides a diagnostic of parameter estimability based on the closed-form expression of the Fisher information matrix for a multivariate normal likelihood.