Computational aspects of N‐mixture models

• Published in: Biometrics Vol. 71; no. 1; pp. 237 - 246
• Main Authors: Dennis, Emily B, Morgan, Byron J.T, Ridout, Martin S
• Format: Journal Article
• Language: English
• Published: England International Biometric Society, etc. 01.03.2015; Blackwell Publishing Ltd; International Biometric Society; BlackWell Publishing Ltd
• Subjects: Abundance estimation; Algorithms; Animals; BIOMETRIC PRACTICE; Biometrics; biometry; Biometry - methods; Census of Population; Censuses; Computer Simulation; covariance; Data Interpretation, Statistical; Environmental Monitoring; France; Method of moments; Models, Statistical; Multivariate negative binomial; Multivariate Poisson; Optimal design; Population Dynamics; probability; Sampling; Temporal replication; Testudo; tortoises; Turtles - physiology; France; Multivariate Poisson; Temporal replication; Multivariate negative binomial; Abundance estimation; Sampling; Method of moments; Optimal design
• ISSN: 0006-341X; 1541-0420; 1541-0420
• DOI: 10.1111/biom.12246

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.