Statistics for Biology and Health Series Editors K. Dietz, M. Gail, K. Krickeberg, 1. Samet, A. Tsiatis Springer-Verlag London Ltd. Statistics for Biology and Health Borchers/Buckland/Zucchini: Estimating Animal Abundance: Closed Populations. Everitt/Rabe-Hesketh: Analyzing Medical Data Using S-PLUS. Ewens/Grant: Statistical Methods in Bioinformatics: An Introduction. Hougaard: Analysis of Multivariate Survival Data. Klein/Moeschberger: Survival Analysis: Techniques for Censored and Truncated Data. Kleinbaum: Logistic Regression: A Self-Learning Text, 2nd ed. Kleinbaum: Survival Analysis: A Self-Learning Text. Lange: Mathematical and Statistical Methods for Genetic Analysis, 2nd ed. Manton/Singer!Suzman: Forecasting the Health of Elderly Populations. Salsburg: The Use of Restricted Significance Tests in Clinical Trials. Sorensen/Gianola: Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics. Themeau/Grambsch: Modeling Survival Data: Extending the Cox Model. Zhang/Singer: Recursive Partitioning in the Health Sciences. D.L. Borchers, S.T. Buckland and W. Zucchini Estimating Animal Abundance Closed Populations With 91 Figures ~Springer Series Editors K. Dietz M.GaiI K. Krickeberg Institut fiir Medizinische Biometrie National Cancer Insttitue Le Chatelet Universităt Tiibingen Rockville, MD 20892 F-63270 Manglieu West Banhofstrasse 55 USA FRANCE D-72070 Tiibingen GERMANY A. Tsiatis J. Samet Department of Statistics School of Public Health North Carolina State University Department of Epidemiology Raleigh, NC 27695 Johns Hopkins University USA 615 Wolfe Street Baltimore, MD 21205-2103 USA British Iibrary CataIoguing in Publication Data Borchers, D. L. Estirnating animal abundance : c10sed populations. - (Statistics for biology and health) 1. Animal populations - Statistical methods 2. Estimation theory 1. TitIe II. Buckland, S. T. (Stephen T.) ID. Zucchini, W. 591.7'88'0727 ISBN 978-1-84996-885-0 ISBN 978-1-4471-3708-5 (eBook) DOI 10.1007/978-1-4471-3708-5 Iibrary of Congress CataIoging-in-Publication Data Estimating animal abundance : c10sed populations I D.L. Borchers, S.T. Buckland, and W. Zucchini. p. cm. --(Statistics for biology and health) ISBN 978-1-84996-885-0 1. Animal populations - Statistical methods. 1. Borchers, D.L., 1958- II. Buckland, S.T. (Stephen T.) ID. Zucchini, W. IV. Series. QL752.E76 2002 591.7'88~c21 2002021159 Apart from any fair dea1ing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the pubJishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright licensing Agency. Enquiries conceming reproduction outside those terms should be sent to the publishers. ISBN 978-1-84996-885-0 springeronline.com © Springer-V crlag London 2002 Originally published by Springer-Verlag London Limited in 2002 Softcover reprint of the hardcover 1s t edition 2002 The use of registered names, trademarlcs, etc. in this publication does not imply, even in the absence of a specific statemenl, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard 10 the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that rnay be rnade. Typesetting: Camera-ready by authors 1213830-54321 Printed on acid-free paper SPIN 11012375 Preface We hope this book will make the bewildering variety of methods for estimat ing the abundance of animal populations more accessible to the uninitiated and more coherent to the cogniscenti. We have tried to emphasize the fun damental similarity of many methods and to draw out the common threads that underlie them. With the exception of Chapter 13, we restrict ourselves to closed populations (those that do not change in composition over the period(s) being considered). Open population methods are in many ways simply extensions of closed population methods, and we have tried to pro vide the reader with a foundation on which understanding of both closed and open population methods can develop. We would like to thank Miguel Bernal for providing the St Andrews example dataset used frequently in the book; Miguel Bernal and Jeff Laake for commenting on drafts of the book; Jeff Laake for providing Figure 10.1; NRC Research Press for allowing us to use Figures 10.2, 10.3, 10.4, 10.5, 10.6 and 10.7; the International Whaling Commission for allowing us to use Figure 12.1; Sharon Hedley for providing Figures 12.1 and 12.2. D.L.B. is eternally indebted to Carol, Alice and Aidan for their support through writing the book, and for the many evenings and weekends that it has taken from them. D.L.B. and S.T.B. acknowledge the financial support of the University of St Andrews. vi Preface W.Z. gratefully acknowledges the financial support of the University of Gottingen, Martin Erdelmeier's enormous contributions to the development of WiSP, and the boundless moral support provided by Leon and Laura. David Borchers, Stephen Buckland and Walter Zucchini Contents I Introduction 1 Introduction 3 1.1 Estimation approach 1 4 1.2 Estimation approach 2 5 1.3 Estimation approach 3 7 1.4 Heterogeneity . . . . 7 1.5 Summary ..... . 8 1.6 Outline of the book. 9 1. 7 R software . . . . . . 10 2 Using likelihood for estimation 12 2.1 An example problem .......... . 12 2.2 Maximum likelihood estimation .... . 14 2.2.1 Known detection probability (p) 14 2.2.2 Unknown detection probability 14 2.2.3 Basics of the method ...... . 15 2.2.4 Likelihood function ...... . 16 2.3 Estimator uncertainty and confidence intervals 19 2.3.1 What are confidence intervals? . 19 2.3.2 Constructing confidence intervals 22 2.4 Approximate confidence intervals 24 2.4.1 Asymptotic normality ..... . 24 viii Contents 2.4.2 Profile likelihood 25 2.4.3 Bootstrap 27 2.5 Summary 30 2. 6 Exercises . . . . 31 II Simple Methods 3 Building blocks 37 3.1 State and observation models 37 3.1.1 State models ..... 39 3.1.2 Survey design and observation models 43 3.2 Design and model. . . . . . . . . . . . . . 46 a 3.2.1 What is survey design? . . . . . 47 3.2.2 Design- vs. model-based inference. 47 3.2.3 Can we tell if the model is wrong? 49 3.2.4 Design-based vs. Model-based: pros and cons 51 3.2.5 Relevance of design for likelihood-based inference . 52 3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 Plot sampling 55 4.1 Introduction ..... 55 4.2 A simple plot survey . 56 4.3 Estimation by design 57 4.3.1 Horvitz-Thompson estimator of abundance 57 4.4 Maximum likelihood estimation 59 4.4.1 Point estimation ..... 60 4.4.2 Interval estimation . . . . 60 4.4.3 A more realistic example 63 4.5 Effect of violating assumptions 66 4.6 Summary 67 4. 7 Exercises . . . . . . . . . . . . 67 5 Removal, catch-effort and change-in-ratio 72 5.1 Introduction ...... . 72 5.2 Removal method .......... . 74 5.2.1 Point estimation ...... . 74 5.2.2 Simple removal method MLE 76 5.2.3 Interval estimation 77 5.2.4 Heterogeneity 83 5.3 Catch-effort ...... . 88 5.3.1 Likelihood .... . 90 5.3.2 Removal models and model selection . 91 5.3.3 A word on CPUE as an index of abundance . 93 5.4 Change-in-ratio . . . . . . . . . . . . . . . . . . . . . 94 Contents ix 5.4.1 Full likelihood ....... . 96 5.4.2 Conditional likelihood ... . 97 5.5 Effect of violating main assumptions 99 5.6 Summary 99 5. 7 Exercises . . . . . . . . . . . . . . . 101 6 Simple mark-recapture 104 6.1 Introduction ..... . 104 6.2 Single recapture and some notation . 106 6.2.1 St Andrews data example . . 107 6.3 A two-sample mark-recapture likelihood 108 6.3.1 First capture occasion . . . . . . 109 6.3.2 Second capture occasion . . . . . 109 6.3.3 Putting the two capture occasions together 111 6.3.4 Interval estimation . . 111 6.3.5 Heterogeneity . . . . . . . . 113 6.4 Related methods and models . . . 116 6.4.1 Unknown marking process . 116 6.4.2 Removal method likelihood 116 6.4.3 Hypergeometric models . . 117 6.4.4 Single mark, multiple captures 117 6.5 Multiple occasions: the "Schnabel census" 117 6.5.1 A likelihood for multiple capture occasions 118 6.5.2 Capture histories and individual identification . 118 6.6 Types of mark-recapture model . . . . . . . 119 6.6.1 Classification by observation model . . . . 119 6.6.2 Models Mo, Mt, Mb, Mtb . . . . . . . . . 121 6.6.3 Likelihoods for models Mo, Mt, Mb, Mtb . 121 6. 7 Examples . . . . . . . . . . . . . . . . . . . . . 123 6.7.1 St Andrews example data revisited . . . . 123 6.7.2 No animal heterogeneity; true model Mt . 125 6.7.3 No heterogeneity; true model Mt; small n 126 6.8 Effect of violating main assumptions 127 6.9 Summary 128 6.10 Exercises . . . . . . . . . . . . . . . 128 7 Distance sampling 131 7.1 Introduction .......... . 131 7. 2 Line transect sampling . . . . . . . . 132 7.2.1 A simple line transect survey 133 7.2.2 Maximum likelihood estimation . 137 7.2.3 Horvitz-Thompson: estimation partly by design 143 7.2.4 Populations that occur in groups 145 7.2.5 Interval estimation . . . . . . 145 7.2.6 An example with heterogeneity . 147 x Contents 7.3 Point transect sampling . . . . . . . . . . . . . . . . . . 149 7.3.1 Maximum likelihood estimation . . . . . . . . . . 150 7.3.2 Horvitz-Thompson: estimation partly by design 154 7.3.3 Populations that occur in groups 155 7.3.4 Interval estimation . . . . . 155 7.4 Other distance sampling methods . 157 7.4.1 Cue counting . . . . . . . . 157 7.4.2 Trapping webs . . . . . . . 158 7.4.3 Indirect distance sampling surveys 159 7.5 Effect of violating main assumptions 160 7.6 Summary 161 7. 7 Exercises . . . . . . . . . . . . . . . 162 8 Nearest neighbour and point-to-nearest-object 165 8.1 Introduction .......... . 165 8.2 Maximum likelihood estimation . 167 8.3 Interval estimation 168 8.4 Example . 169 8.5 Summary 171 8.6 Exercises 172 III Advanced Methods 9 Further building blocks 177 9.1 Introduction .... . 177 9.2 State models ... . 178 9.3 Observation models. 180 10 Spatial/temporal models with certain detection 182 10.1 Introduction ............. . 182 10.2 Temporal modelling: migration counts . 184 10.2.1 Estimation by design ...... . 184 10.2.2 Maximum likelihood estimation . 185 10.2.3 Inference with "Unbinned" data 187 10.3 Spatial modelling: plot sampling 188 10.4 Spatia-temporal modelling . 192 10.5 Other methods 193 10.6 Summary 196 10.7 Exercises . . . 197 11 Dealing with heterogeneity 199 11.1 Introduction .................... . 199 11.1.1 Combining state and observation models. 200 11.2 Distance sampling with covariates . . . . . . . . 201