Why a time effect often has a limited impact on capture‐recapture estimates in closed populations

The author is concerned with log-linear estimators of the size N of a population in a capture-recapture experiment featuring heterogeneity in the individual capture probabilities and a time effect. He also considers models where the first capture influences the probability of subsequent captures. He derives several results from a new inequality associated with a dispersive ordering for discrete random variables. He shows that in a log-linear model with inter-individual heterogeneity, the estimator N is an increasing function of the heterogeneity parameter. He also shows that the inclusion of a time effect in the capture probabilities decreases N in models without heterogeneity. He further argues that a model featuring heterogeneity can accommodate a time effect through a small change in the heterogeneity parameter. He demonstrates these results using an inequality for the estimators of the heterogeneity parameters and illustrates them in a Monte Carlo experiment Pourquoi un effet temporel a souvent peu d'impact sur les estimateurs de capture-recapture dans des populations ferme′es L'auteur s'interesse aux estimateurs log-lineaires de la taille N d'une population dans une experience de capture-recapture u les probabilites de capture varient en fonction du temps et des individus. Il aborde aussi le cas des modeles u la premiere capture change la probabilite de captures ulterieures. Il deduit plusieurs resultats d'une nouvelle inegalite liee a un ordre dispersif pour des variables aleatoires discretes. Il montre que dans un modele log-Iineaire avec heterogeneite inter-individus, l'estimateur N est une fonction croissante du parametre d'heteroge. Il montre aussi que l'inclusion de variations temporelles dans les probabilites de capture diminue N dans un modele sans heterogeneite. Il explique de plus comment un petit ajustement du parametre d'heteroge du modele permet d'accommoder un effet temporel. D deduit ces resultats d'une inegalite pour les estimateurs des parametres d'heterogeneneite et les illustre au moyen d'une etude de Monte-Carlo.

[1]  Moshe Shared,et al.  On Mixtures from Exponential Families , 1980 .

[2]  S E Fienberg,et al.  A three-sample multiple-recapture approach to census population estimation with heterogeneous catchability. , 1993, Journal of the American Statistical Association.

[3]  Ward Whitt,et al.  Uniform conditional variability ordering of probability distributions , 1985 .

[4]  Richard M. Huggins,et al.  An examination of the effect of heterogeneity on the estimation of population size using capture-recapture data , 2005 .

[5]  Louis-Paul Rivest,et al.  Improved log‐linear model estimators of abundance in capture‐recapture experiments , 2001 .

[6]  W. Link.  Nonidentifiability of Population Size from Capture‐Recapture Data with Heterogeneous Detection Probabilities , 2003, Biometrics.

[7]  Louis-Paul Rivest,et al.  Loglinear Models for the Robust Design in Mark–Recapture Experiments , 2004, Biometrics.

[8]  K H Pollock,et al.  Robust estimation of population size in closed animal populations from capture-recapture experiments. , 1983, Biometrics.

[9]  J. Andrew Royle,et al.  Mixture Models for Estimating the Size of a Closed Population When Capture Rates Vary among Individuals , 2003, Biometrics.

[10]  Anne Chao,et al.  Estimating population size for sparse data in capture-recapture experiments , 1989 .

[11]  Louis-Paul Rivest,et al.  Applications and extensions of Chao's moment estimator for the size of a closed population. , 2007, Biometrics.

[12]  B. Lindsay.  Exponential family mixture models (with least-squares estimators) , 1986 .

[13]  A. Agresti.  Simple capture-recapture models permitting unequal catchability and variable sampling effort. , 1994, Biometrics.

[14]  J. Norris,et al.  NONPARAMETRIC MLE UNDER TWO CLOSED CAPTURE-RECAPTURE MODELS WITH HETEROGENEITY , 1996 .

[15]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[16]  Richard M. Huggins,et al.  A note on the difficulties associated with the analysis of capture-recapture experiments with heterogeneous capture probabilities , 2001 .

[17]  A. Agresti,et al.  The Use of Mixed Logit Models to Reflect Heterogeneity in Capture‐Recapture Studies , 1999, Biometrics.

[18]  L. Rivest,et al.  Rcapture: Loglinear Models for Capture-Recapture in R , 2007 .

[19]  R. L. Sandland,et al.  Statistical inference for Poisson and multinomial models for capture-recapture experiments , 1984 .

[20]  P. Jupp,et al.  Inference for Poisson and multinomial models for capture-recapture experiments , 1991 .

[21]  M. Conroy,et al.  Analysis and Management of Animal Populations , 2002 .

[22]  S. Pledger.  Unified Maximum Likelihood Estimates for Closed Capture–Recapture Models Using Mixtures , 2000, Biometrics.

[23]  R. Cormack.  Loglinear Models for Capture-Recapture Experiments on Open Populations , 1980 .

[24]  S. Fienberg.  The multiple recapture census for closed populations and incomplete 2k contingency tables , 1972 .

[25]  Peter McCullagh,et al.  Exponential mixtures and quadratic exponential families , 1994 .

[26]  G. Seber,et al.  The estimation of animal abundance and related parameters , 1974 .

[27]  Alan E. Gelfand,et al.  A Note on Overdispersed Exponential Families , 1990 .

[28]  A Chao,et al.  Estimating population size for capture-recapture data when capture probabilities vary by time and individual animal. , 1992, Biometrics.

[29]  Bruce G. Lindsay,et al.  Tests and diagnostics for heterogeneity in the species problem , 2003, Comput. Stat. Data Anal..

[30]  K. Burnham,et al.  Estimation of the size of a closed population when capture probabilities vary among animals , 1978 .