baut ECOSTAT

The software ECOSTAT was developed to perform computations for the statistical

methods not readily available in standard statistical packages. It may be downloaded

free from the University of Nebraska website (http://vvww.ianr.unl,edu/ianr/biometry/ faculty/linda/lyoung.html). The zip file should be placed in a temporary folder. Then

the individual files should be extracted (unzipped) from this zip file. The temporary

folder(s) used for the zip file and the extracted files should NOT be ca lied ECOSTAT. Once the files have been extracted, ECOSTAT must be installed on the machine before

it can be run. To do so, run SETUP.EXE. The downloaded zip file and the files that were extracted from the zip file may then be deleted from the temporary folder(s).

ECOSTAT was developed for IBM-compatibles and a Windows95 environ­

ment. At least 16 megabytes of RAM are needed to run the full program, although some portions require less memory. The program is written in Microsoft's Visual Basic.

ECOSTAT has seven parts. Probability Distributions has the set of programs that complement the first three chapters of the book. Sequential Sampling contains the programs for both the fixed and sequential methods in Chapters 4-6. Spatial Statistics is

the portion relating to Chapters 7 and 8. Capture-Recapture is the program correspond­

ing to Chapters 9 and 10. Transect Sampling may be used for the material covered in

Chapter 11. The temperature data of Chapter 12 can be analyzed using Temperature

Models. The portion. of the program relating to Chapter 13 is in Life Stage. An eighth

program, DISCRETE (Gates 1989), has been induded with ECOSTAT for use in Chapter 2. ECOSTAT is able to help the user identify some problems in running the

program. However, at the present time, it is not designed to detect errors in the input file. These indude the errors that arise because the wrong format was used for the data. Care should be taken at this point.

Should you encounter an error, please contact us at the following e-mail address: UYoung@unl.edu

....

Linda J. Young Biometry Department

University of Nebraska-Lincoln

Jerry H. Young Entomology Department (Emeritus)

Oklahoma State University

" Springer Science+Business Media, LLC

tt E1ectronic Services < http://www.wkap.nl >

Library of Congress Cataloging-in-Publication

Young. Linda J. Statistical ecology : a population perspective / Linda J. Y oung. J erry H. Y oung

p. cm. Includes bibliographical references and index.

ISBN 978-1-4757-2831-6 ISBN 978-1-4757-2829-3 (eBook) DOI 10.1007/978-1-4757-2829-3

1. Ecology--Statistical methods. I. Young. Jerry H. 11. Tide.

2. Population biology--Statistical methods

QH541.l5.S72Y68 1998 577' .01 '5195--dc21

British Library Cataloguing in Publication Data available

97-25603 CIP

Copyright c 1998 by Springer Science+Business Media New York. Third Printing 2002.

Originally published by Kluwer Academic Publishers in 1998.

Softcover reprint of the hardcover 1 st edition 1998

Cover design: Curtis Tow Graphics

This printing is a digital duplication of the original edition.

All rights reserved. No part of this publication may be reproduced. stored in a retrieval system or transmitted in any form or by any means. mechanical. photo-copying. recording. or otherwise. without the prior written permission of the publisher. Springer Science+ Business Media, LLC

Printed on acid-free paper.

To lohn, Shamar, and RaQwin

Table of Contents

Preface

Acknowledgments

Chapter 1: Probability Distributions Introduetion Diserete Distributions Negative Binomial Distribution

Expected Frequencies Geometrie Distribution

Expected Frequencies Binomial Distribution

Expected Frequencies Poisson Distribution

Expected Frequencies Confidenee Intervals Continuous Distributions Normal Distribution Lognormal Distribution Exponential Distribution Gamma Distribution Weibull Distribution Summary Exereises

Chapter 2: Goodness-of-Fit Tests Introduetion Pearson's Chi-Squared Test

vii

xiii

xv

1 2 7 8

13 14 17 18 22 23 24 26 27 33 34 35 37 38 39

42 42 44

viii / Statistical Ecology

Likelihood Ratio Test Freeman-Tukey Chi-Squared Test Power Divergence Statistic Nass Test Kolmogrov-Smimov Test Summary Exercises Appendix

Chapter 3: Models and Sampling Introduction Binomial Models Poisson Models Negative Binomial Models Bose-Einstein Versus Maxwell-Boltzmann Statistics Stochastic Immigration Model Modeling Within Field Movement Restrietions on Carrying Capacity Sampling Concepts Simple Random Sampling Stratified Random Sampling Systematic Sampling Ratio Estimation Summary

Chapter 4: Sequential Estimation Introduction Sampie Sizes Required to Control CV(X) Sampie Sizes Required to Set Confidence Intervals

Length Proportional to the Parameter o/Interest Length Fixed

Sequential Estimation Sequential Estimation for the Negative Binomial

Parameter k Unknown Estimating Parameter k Parameter k Known

Sequential Estimation for the Geometrie Sequential Estimation for the Poisson Sequential Estimation for the Binomial Sequential Estimation Based on Iwao's Patchiness Regression Sequential Sampling Based on Taylor's Power Law Summary Exercises

50 52 54 55 58 65 66 70

75 75 75 76 76 78 82 84 85 87 92 93 96 97 97

99 99

100 105 106 108 110 112 112 114 123 129 131 133 136 144 148 149

Table of Contents / ix

Chapter 5: Sequential Hypothesis Testing 153 Introduction 153 Wald's Sequential Probability Ratio Test 153 SPRT for the Negative Binomial Distribution 159 SPRT for the Poisson Distribution 162 SPRT for the Binomial Distribution 165 Operating Characteristic and Average SampIe Number Functions 167 The 2-SPRT 173 Summary 186 Exercises 188

Chapter 6: Sequentially Testing Three Hypotheses 191 Introduction 191 Ecologists' Sequential Test 192 Sobel and Wald's Method 199 Armitage's Method 204 Testing Composite Hypotheses 209

Iwao's Method 209 Armitage's Methods 212

Summary 213 Exercises 213

Chapter 7: Aggregation and Spatial Correlation 215 Introduction 215 Measures of Aggregation 216

Variance-to-Mean Ratio and Index of Dispersion 217 Index of Clumping 217 Mean Crowding and Mean Patchiness 218 Comparison of Indices 219

Spatial Correlation 222 Moran's land Geary's c 224 Geostatistics 231

Intrinsic Stationarity 243 Median Polishing 244 The Semivariogram 250

Summary 268 Exercises 269

Chapter 8: Spatial Point Patterns 272 Introduction 272 Complete Spatial Randomness 273 K(h) amd L(h) Functions 276 Monte Carlo Tests 284

x / Statistical Ecology

Nearest Neighbor Techniques Summary Exercises Appendix

Chapter 9: Capture-Recapture: Closed Populations Introduction Lincoln-Petersen Model

Confidence Intervals Sampie Size Considerations Assumptions

Multiple Recapture Models Model Mo: Constant Capture Probabilities Model M,: Capture Probabilities Val}' With Time

Tests for the Model Model Mb: Behavioral Response to Capture

Tests for the Model Model M h: Heterogeneity of Capture Probabilities

Tests for the Model Model Mbh: Heterogeneity of Capture Probabilities

and Trap Response Tests for the Model

Models Mb,' Mh,. Mbh, Model Selection Confidence Intervals

Removal and Catch Effort Models Change-in-Ratio or Selective Removal Models Density Estimation Summary Exercises

Chapter 10: Capture-Recapture: Open Populations Introduction Jolly-Seber Model Adult Band and Tag Recovery Models Summary Exercises

Chapter 11: Transect Sampling Introduction Strip Transects and Circular Plots Line and Point Transects

Ungrouped Data

288 292 293 295

297 297 298 300 303 305 310 314 315 317 318 320 322 324

325 326 327 330 331 335 338 347 347 348

357 357 358 375 383 384

390 390 391 397 407

Grouped Data Clustered Populations

Design Summary Exercises

Chapter 12: Degree-Day Models Introduction Assumptions Calculating Degree-Days Summary Exercises

Chapter 13: Life-Stage Analysis Introduction Life Tables Key Factor Analysis

Varley and Gradwell's Method Regression olK against kj Values Manly Method

Multi-Cohort Stage-Frequency Data Kiritani-Nakasuji-Manly Method KNM Iterative Method Kempton Method Bellows and Birley Model

Single Cohort Stage-Frequency Data Analysis Using Multi-Cohort Methods Nonparametric Estimation

Matrix Models for Reproducing Populations Bernardelli-Leslie-Lewis Model Lefkovitch Model Usher Model

Summary Exercises

Chapter 14: Probit and Survival Analysis Introduction Probit Analysis Nest Survival Analysis

Mayfield Method Pollock Method

Analysis of Radiotelemetry Data Trent and Rongstad Method

Table 01 Contents / xi

412 414 416 417 417

421 421 423 426 433 433

440 440 443 446 447 451 453 460 462 472 480 482 483 483 486 490 490 491 492 493 494

506 506 507 514 514 517 519 519

xii / Statistical Ecology

Survival Analysis Model Summary Exercises

Chapter 15: Chaos Introduction Population Models Chaos Summary

References

Index

520 522 522

524 524 524 528 530

531

555

Preface

This book is a collection of fonnulae, techniques, and methods developed for use in field ecology. We try to iIIustrate treatment of ecological data, from sampling through modeling, for single-species populations. The material has been chosen because of its frequent application by researchers and workers in pest manage­ment, forestry, wildlife, plant protection and environmental studies. These meth­ods are not always the strongest ones statistically because ease of application is a primary consideration. At times, the statistical properties of well-accepted pro­cedures are unknown. By giving an awareness of the statistical foundation for existing methods, we hope that biologists will become more aware of the strengths, and possible weaknesses, of the procedures and that statisticians will more fully appreciate the needs of the field ecologist.

The book is designed as a reference or entry level text for biologists or stat­isticians that are developing a better understanding of statistical ecology. It is assumed that readers have an understanding of basic ecological principles and a statistical foundation in estimation, hypothesis testing, and regression. The ECOS­

TAT software that accompanies the text will hopefully permit the focus to change from the computations to the concepts underlying the methods.

Most of the equations are presented in calculator or computer programming fonn. The notations are minimal and are taken, for the most part, from biology. An effort has been made to relate the notation used to that of standard statistics, mathematics and physics. Extensive examples for most methods have been included.

These materials have been used in statistical ecology courses taught at Oklahoma State University and the University of Nebraska-Lincoln. Numerous students from these courses have aided in the development of this manual. Several graduate students from the Departments of Statistics and Entomology at Oklahoma State and the Biometry Department at the University of Nebraska-

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Lineoln have researehed key areas in this book. We have attempted to note their eontributions where appropriate.

The treatment of Bose-Einstein Statisties in Chapter 3 is a rather sharp depar­ture from the traditional view of population dynarnies and should be read to give the reader an insight into the philosophy of the authors.

The ehapters are arranged in a natural progression of identifying the applieable distributions, developing sampling programs, and modeling populations. An ef­fort has been made to make eaeh ehapter eomplete, allowing workers to readily foeus on particular interests.

Acknowledgments

Statistical ecology is an exciting area of study from both the statistical and eco­logical perspectives. The enormous variation encountered in nature enhances its beauty and challenges those who wish to quantify its characteristics. The disci­pline has benefitted from the intense efforts of pioneers in this field. These indi­viduals have made great inroads into our understanding and have been generous in suggesting avenues they believe will lead to further progress. We have made an effort to identify at least some of these leaders within each chapter.

This text grew out of a statistical ecology dass developed at Oklahoma State University and later taught at the University of Nebraska-Lincoln. Our graduate students and students taking the dass have added greatly to this work. Bob Hill and Bill Ruth helped census the insects on 11 quarter-acres of cotton. This work shaped our views of insect distribution. Michelle Strabala investigated the per­formance of normal-based methods applied to discrete distributions. Be-ny Wu conducted a study of goodness-of-fit tests for discrete distributions. The models of insect movement in ECOSTAT are based on an initial program written by Alan Stark. Katherine Seebeck developed software for studying the properties of the sequential probability ratio test. Lim Siew with the input of Madhuri Mulekar produced thtl foundational software for the 2-SPRT. Our views of deciding among three hypotheses were shaped by Mark Payton's work.

In developing this book, we benefitted from the unselfish help and support from many colleagues. From the beginning, Bill Drew provided strong support and encouragement. Nitis Mukhopadhyay introduced us to sequential analysis. Igo Kotlarski provided insight and references leading to the models in Chapter 3. Colleagues and students in the Departments of Statistics and Entomology at Oklahoma State University and the Department of Biometry at the University of Nebraska-Lincoln shaped our views through frequent discussions and questions. They also provided data for a number of the examples and exercises. Charles Gates provided us with the DISCRETE program that we have induded with

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ECOSTAT. Noel Cressie was always encouraging and pointed out the Power Series family of test statistics. Carol Gotway read and gave detailed suggestions for strengthening Chapters 7 and 8. Svata Louda provided ideas for improving the presentation of Chapter 8 that were useful in other chapters as weH. Ken Bumham gave insightful comments on Chapters 9, 10, and 11. Leon Higley not only read and commented on Chapters 12 and 13 but made significant changes in Chapter 12 that greatly improved it. Chris Heinzle helped us make the transition to Visual Basic from our DOS-based programs. Others offered encouragement and help.

Bea Shube provided great insight and useful suggestions in the early stages of development. We still review materials she provided. Henry Flesh, Kendall Har­ris, Lisa LaMagna, and Deslie B. Lawrence were instrumental in this book reach­ing completion.

Patsy Lang helped collect materials from the library, developed figures for some later chapters, and provided general support. Linda Pavlish, Daryll Trav­nicek and Leona Barratt developed a number of the figures in the first six chapters.

Friends and neighbors have given us invaluable help and support, especially during the last stages of writing. They have driven for us in the car pools, taken our children to numerous events, and treated our children as their own. At the same time, they have encouraged us to persevere. We are truly blessed.

John, Shamar, and RaQwin helped us keep a perspective on the truly important aspects of life. They were understanding and helped keep our household func­tioning during this process.