Computational aspects of N-mixture models.

Martin S. Ridout,Byron J.T. Morgan,Emily B. Dennis

doi:10.1111/biom.12246

Abstract

The N-mixture model is widely used to estimate the abundance of a population in the presence of unknown detection probability from only a set of counts subject to spatial and temporal replication (Royle, 2004, Biometrics 60, 105–115). We explain and exploit the equivalence of N-mixture and multivariate Poisson and negative-binomial models, which provides powerful new approaches for fitting these models. We show that particularly when detection probability and the number of sampling occasions are small, infinite estimates of abundance can arise. We propose a sample covariance as a diagnostic for this event, and demonstrate its good performance in the Poisson case. Infinite estimates may be missed in practice, due to numerical optimization procedures terminating at arbitrarily large values. It is shown that the use of a bound, K, for an infinite summation in the N-mixture likelihood can result in underestimation of abundance, so that default values of K in computer packages should be avoided. Instead we propose a simple automatic way to choose K. The methods are illustrated by analysis of data on Hermann's tortoise Testudo hermanni.

Highlights

Estimating the abundance of a population is an important component of ecological research
Multivariate Poisson Distribution For general T, let Xi denote the set of all possible values xi,s of the random variables Xi,s, s ∈ S such that nit =
Fitting the multivariate Poisson model to simulated data created under comparable scenarios for T = 2, 3, 4 produced a second peak in the sampling distribution for λ, but as described in Section 3.2, the estimates were substantially greater in the absence of the limiting value K in the N-mixture model

Summary

Introduction

Estimating the abundance of a population is an important component of ecological research. We investigate computational aspects of fitting N-mixture models, in particular via a simulation study for scenarios where detection probability is low and/or the number of sampling occasions is small This may be important for the study of cryptic species, and have implications for sample design: many applications to date have made only three visits, whereas in Royle (2004a) simulations were tested for five visits and an application made to data with 10 visits. We use this formulation to show that infinite estimates of abundance may arise, and provide a simple diagnostic to identify such cases.

The N-Mixture Model

Equivalence of the Poisson N-Mixture Model

Performance of the Multivariate Poisson Model

Explicit Form for the Bivariate Negative-Binomial Case

Automatic Choice of K

Moment Estimation for a Mixed-Poisson N-Mixture Model

Incorrect Estimates due to the Choice of K

Moment Estimation

Performance of the Multivariate Negative-Binomial Model

Findings

Application to Hermann’s Tortoise Data

Discussion and Recommendations