Computational Aspects of N-Mixture Models

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.

[1]  N. L. Johnson,et al.  Discrete Multivariate Distributions , 1998 .

[2]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[3]  J. Andrew Royle,et al.  Accounting for non‐independent detection when estimating abundance of organisms with a Bayesian approach , 2011 .

[4]  J. Andrew Royle.  N‐Mixture Models for Estimating Population Size from Spatially Replicated Counts , 2004, Biometrics.

[5]  B. Lindsay,et al.  A Penalized Nonparametric Maximum Likelihood Approach to Species Richness Estimation , 2005 .

[6]  Marc Kéry,et al.  Estimating Abundance From Bird Counts: Binomial Mixture Models Uncover Complex Covariate Relationships , 2008 .

[7]  J. Andrew Royle,et al.  MODELING AVIAN ABUNDANCE FROM REPLICATED COUNTS USING BINOMIAL MIXTURE MODELS , 2005 .

[8]  M. Kéry,et al.  Population trends of brown hares in Switzerland: The role of land-use and ecological compensation areas , 2011 .

[9]  Katherine C. Kendall,et al.  Linking landscape characteristics to local grizzly bear abundance using multiple detection methods in a hierarchical model , 2011 .

[10]  An evaluation of the Bayesian approach to fitting the N-mixture model for use with pseudo-replicated count data , 2012 .

[11]  P. Holgate.  Estimation for the bivariate Poisson distribution , 1964 .

[12]  F. Weckerly,et al.  Reliability of Occupancy and Binomial Mixture Models for Estimating Abundance of Golden-Cheeked Warblers (Setophaga chrysoparia) , 2012 .

[13]  R. Cormack.  Log-linear models for capture-recapture , 1989 .

[14]  M. Cheylan,et al.  Estimating abundance and population trends when detection is low and highly variable: A comparison of three methods for the Hermann's tortoise , 2013 .

[15]  J. Hilbe.  Negative Binomial Regression: Preface , 2007 .

[16]  L. Madsen,et al.  Models for Estimating Abundance from Repeated Counts of an Open Metapopulation , 2011, Biometrics.

[17]  Doron Zeilberger,et al.  A symbolic computation approach to a problem involving multivariate Poisson distributions , 2010, Adv. Appl. Math..

[18]  J. Andrew Royle,et al.  Trend estimation in populations with imperfect detection , 2009 .

[19]  D. Karlis.  An EM algorithm for multivariate Poisson distribution and related models , 2003 .

[20]  E. A. Catchpole,et al.  On the Near‐Singularity of Models for Animal Recovery Data , 2001, Biometrics.

[21]  J. Andrew Royle,et al.  Modelling occurrence and abundance of species when detection is imperfect , 2005 .

[22]  Richard B. Chandler,et al.  unmarked: An R Package for Fitting Hierarchical Models of Wildlife Occurrence and Abundance , 2011 .

[23]  Byron J. T. Morgan,et al.  Two-Stage Bayesian Study Design for Species Occupancy Estimation , 2014 .

[24]  Abdelhak M. Zoubir,et al.  Estimating the Parameters of the Multivariate Poisson Distribution Using the Composite Likelihood Concept , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[25]  T. Quinn,et al.  Empirical and simulation evaluations of an abundance estimator using unmarked individuals of cryptic forest-dwelling taxa , 2012 .

[26]  Byron J. T. Morgan,et al.  Models for species‐detection data collected along transects in the presence of abundance‐induced heterogeneity and clustering in the detection process , 2012 .

[27]  R. Dorazio,et al.  Estimating abundance while accounting for rarity, correlated behavior, and other sources of variation in counts. , 2013, Ecology.

[28]  J. Andrew Royle.  Generalized estimators of avian abundance from count survey data , 2004 .

[29]  Byron J. T. Morgan,et al.  Design of occupancy studies with imperfect detection , 2010 .