Use of the Ratio Plot in Capture–Recapture Estimation

Abstract

Statistical graphics are a fundamental, yet often overlooked, set of components in the repertoire of data analytic tools. Graphs are quick and efficient, yet simple instruments for preliminary exploration of a dataset to understand its structure and to provide insight into influential aspects of inference such as departures from assumptions and latent patterns. In this article, we present and assess a graphical device for choosing a method for estimating population size in capture–recapture studies of closed populations. The basic concept is derived from a homogeneous Poisson distribution where the ratios of neighboring Poisson probabilities multiplied by the value of the larger neighbor count are constant. This property extends to the zero-truncated Poisson distribution, which is of fundamental importance in capture–recapture studies. In practice, however, this distributional property is often violated. The graphical device developed here, the ratio plot, can be used for assessing specific departures from a Poisson distribution. For example, simple contaminations of an otherwise homogeneous Poisson model can be easily detected and a robust estimator for the population size can be suggested. Several robust estimators are developed and a simulation study is provided to give some guidance on which one should be used in practice. More systematic departures can also easily be detected using the ratio plot. In this article, the focus is on Gamma-mixtures of the Poisson distribution that leads to a linear pattern (called structured heterogeneity) in the ratio plot. More generally, the article shows that the ratio plot is monotone for arbitrary mixtures of power series densities. This article has online supplementary materials.

Key Words:

• Chao and robust and generalized Chao estimator
• Closed population
• Generalized Turing estimator
• Ord plot
• Poisson-Gamma model
• Ratio plot
• Robust Turing estimator
• Structured heterogeneity
• Turing estimator

SUPPLEMENTARY MATERIAL

The following supplementary material is available online.

Grizzly Bears: The data file contains the frequency distribution of sightings for female grizzly bears with cubs-of-the-year, separately for 1996, 1997, and 1998 (GrizzlyBears.dat).

Scrapie in Great Britain: The data file contains the frequency distribution of the numbers of detections of scrapie per holding, separately for 2002, 2003, and 2004 (ScapieInGreatBritain.dat).

Dystrophin: The data file contains the frequency distribution of the count of antibodies per dystrophin epitop (Dystrophin.dat).

Drug Use California: The data file contains the frequency distribution of the count of contacts (episodes) per drug user within the California Drug Abuse Data System in 1989 (DrugUseCalifornia.dat).

Drug Use Scotland: The data file contains the frequency distribution of the count of contacts (episodes) in a needle exchange program in Scotland in 1997 (DrugUseScotland.dat).

Appendix: The file contains the proofs of all theorems of the article.

R-code: The file contains the R-code for computing the upper truncation point k (optimal cut-off value k) for a given data file (the program uses “sample.dat” that can be changed if needed) consisting of counts x with associated frequency f_x (robust_estimate.R).

Additional information

Notes on contributors

Dankmar Böhning

Dankmar Böhning is at the School of Mathematics, and Southampton Statistical Sciences Research Institute, University of Southamption, Southampton SO17 1BJ, England (E-mail: [email protected]). M. Fazil Baksh is at the Department of Mathematics and Statistics, School of Mathematical and Physical Sciences, University of Reading, Reading RG6 6BX, England (E-mail: [email protected]). Rattana Lerdsuwansri is at the Department of Mathematics and Statistics, Thammasat University, Bangkok 10200, Thailand (E-mail: [email protected]). James Gallagher is at the Statistical Services Centre, School of Mathematical and Physical Sciences, University of Reading, Reading RG6 6FN, England (E-mail: [email protected]).

M. Fazil Baksh

Dankmar Böhning is at the School of Mathematics, and Southampton Statistical Sciences Research Institute, University of Southamption, Southampton SO17 1BJ, England (E-mail: [email protected]). M. Fazil Baksh is at the Department of Mathematics and Statistics, School of Mathematical and Physical Sciences, University of Reading, Reading RG6 6BX, England (E-mail: [email protected]). Rattana Lerdsuwansri is at the Department of Mathematics and Statistics, Thammasat University, Bangkok 10200, Thailand (E-mail: [email protected]). James Gallagher is at the Statistical Services Centre, School of Mathematical and Physical Sciences, University of Reading, Reading RG6 6FN, England (E-mail: [email protected]).

Rattana Lerdsuwansri,

Dankmar Böhning is at the School of Mathematics, and Southampton Statistical Sciences Research Institute, University of Southamption, Southampton SO17 1BJ, England (E-mail: [email protected]). M. Fazil Baksh is at the Department of Mathematics and Statistics, School of Mathematical and Physical Sciences, University of Reading, Reading RG6 6BX, England (E-mail: [email protected]). Rattana Lerdsuwansri is at the Department of Mathematics and Statistics, Thammasat University, Bangkok 10200, Thailand (E-mail: [email protected]). James Gallagher is at the Statistical Services Centre, School of Mathematical and Physical Sciences, University of Reading, Reading RG6 6FN, England (E-mail: [email protected]).

James Gallagher

Dankmar Böhning is at the School of Mathematics, and Southampton Statistical Sciences Research Institute, University of Southamption, Southampton SO17 1BJ, England (E-mail: [email protected]). M. Fazil Baksh is at the Department of Mathematics and Statistics, School of Mathematical and Physical Sciences, University of Reading, Reading RG6 6BX, England (E-mail: [email protected]). Rattana Lerdsuwansri is at the Department of Mathematics and Statistics, Thammasat University, Bangkok 10200, Thailand (E-mail: [email protected]). James Gallagher is at the Statistical Services Centre, School of Mathematical and Physical Sciences, University of Reading, Reading RG6 6FN, England (E-mail: [email protected]).