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Every day, biologists in parkas, raincoats, and rubber boots go into the field to capture and mark a variety of animal species. Back in the office, statisticians create analytical models for the field biologists' data. But many times, representatives of the two professions do not fully understand one another's roles. This book bridges this gap by helping biologists understand state-of-the-art statistical methods for analyzing capture-recapture data. In so doing, statisticians will also become more familiar with ...
Every day, biologists in parkas, raincoats, and rubber boots go into the field to capture and mark a variety of animal species. Back in the office, statisticians create analytical models for the field biologists' data. But many times, representatives of the two professions do not fully understand one another's roles. This book bridges this gap by helping biologists understand state-of-the-art statistical methods for analyzing capture-recapture data. In so doing, statisticians will also become more familiar with the design of field studies and with the real-life issues facing biologists.
Reliable outcomes of capture-recapture studies are vital to answering key ecological questions. Is the population increasing or decreasing? Do more or fewer animals have a particular characteristic? In answering these questions, biologists cannot hope to capture and mark entire populations. And frequently, the populations change unpredictably during a study. Thus, increasingly sophisticated models have been employed to convert data into answers to ecological questions. This book, by experts in capture-recapture analysis, introduces the most up-to-date methods for data analysis while explaining the theory behind those methods. Thorough, concise, and portable, it will be immensely useful to biologists, biometricians, and statisticians, students in both fields, and anyone else engaged in the capture-recapture process.
BRYAN F. J. MANLY, TRENT L. McDONALD, AND STEVE C. AMSTRUP
In September of 1802, Pierre Simon Laplace (1749-1827) used a capture-recapture type of approach to estimate the size of the human population of France (Cochran 1978; Stigler 1986). At that time, live births were recorded for all of France on an annual basis. In the year prior to September 1802, Laplace estimated the number of such births to be approximately X = 1,000,000. These newly born individuals constituted a marked population. Laplace then obtained census and live birth data from several communities "with zealous and intelligent mayors" across all of France. Recognizing some variation in annual birth rates, Laplace summed the number of births reported in these sample communities for the three years leading up to the time of his estimate, and divided by three to determine that there were x = 71,866 births per year (marked individuals) in those communities. The ratio of these marked individuals to the total number of individuals in the sampled communities, y = 2,037,615 was then the estimate
p = 71, 866 / 2, 037, 615 = 0.0353
of the proportion of the total population of France that was newly born.On this basis, the one million marked individuals in the whole of France is related to the total population N as
Np [approximately equal to] 1,000,000
N [approximately equal to] 1, 000, 000 / 0.0353 = 28, 328, 612
This estimation procedure is equivalent to the Lincoln-Peterson capture-recapture estimator described in chapter 2.
Although Laplace is commonly thought of as the first to use the capture-recapture idea, he was preceded by almost 200 years by John Graunt in his attempts to use similar methods to estimate the effect of plague and the size of populations in England in the early 1600s (Hald 1990). The theories and applications of capture-recapture have moved far beyond the concepts of John Graunt and Pierre Laplace in the ensuing centuries. Current methods do, however, share the basic concept, of ratios between known and unknown values, that guided those pioneers.
Our purpose in this book is to provide a guide for analyzing capture-recapture data that can lead the naive reader through basic methods, similar to those used by the earliest of workers, to an understanding of modern state of the art methods. This handbook is intended primarily for biologists who are using or could use capture-recapture to study wildlife populations. To the extent practicable, therefore, we have kept mathematical details to a minimum. We also have, beginning with this first chapter, attempted to explain some of the mathematical details that are necessary for a complete conceptual understanding of the methodologies described. Also, authors of each chapter have been encouraged to provide all the references that are necessary to enable readers to obtain more details about the derivations of the methods that are discussed. Therefore, this book also will be a useful introduction to this subject for statistics students, and a comprehensive summary of methodologies for practicing biometricians and statisticians.
The book is composed of three sections. Section 1 is this chapter, which is intended to set the scene for the remainder of the book, to cover some general methods that are used many times in later chapters, and to establish a common notation for all chapters. Section 2 consists of seven chapters covering the theory for the main areas of mark-recapture methods. These chapters contain some examples to illustrate the analytical techniques presented. Section 3 consists of two chapters in which we explicitly describe some examples of data sets analyzed by the methods described in chapters 2 to 8. When useful throughout the book, we discuss computing considerations, and comment on the utility of the different methods.
1.2 Overview of Chapters 2 to 8
Chapters 2 to 8 cover the main methods available for the analysis of capture-recapture models. For those who are unfamiliar with these methods the following overviews of the chapters should be useful for clarifying the relationships between them. Figure 1.2 contains a flowchart of the capture-recapture methods described in this section of the book. This flowchart may help to clarify the relationship between analyses, and will indicate the chapter (or section) containing methods appropriate for a particular data set.
A closed population is one in which the total number of individuals is not changing through births, deaths, immigration, or emigration. The first applications of capture-recapture methods were with populations that were assumed to be closed for the period of estimation. It is therefore appropriate that the first chapter in section 2 of this book should describe closed-population models. In practice, most real populations are not closed. Sometimes, however, the changes over the time period of interest are small enough that the assumption of closure is a reasonable approximation, and the effects of violating that assumption are minimal. For this reason, the analysis of capture-recapture data from closed populations continues to be a topic of interest to biologists and managers.
In chapter 2, Anne Chao and Richard Huggins begin by discussing some of the early applications of the capture-recapture method with one sample to mark some of the individuals in a population, and a second sample to see how many marked animals are recaptured. The data obtained from the two samples can be used to estimate the population size.
A natural extension of the two-sample method, which can be traced back to Schnabel (1938), involves taking three or more samples from a population, with individuals being marked when they are first caught. The analysis of data resulting from such repeated samples, all within a time period during which the population is considered closed, is also considered in chapter 2. The goal still is estimation of the population size, but there are many more models that can be applied in terms of modeling the data. Chao and Huggins therefore conclude chapter 2 by noting the need for more general models.
The discussion is continued by Chao and Huggins in chapter 4. There they consider how the probability of capture can be allowed to vary with time, the capture history of an animal, and different animals, through the Otis et al. (1978) series of models. Other topics that are covered by Chao and Huggins in chapter 4 are the incorporation of covariates that may account for variation in capture probabilities related to different types of individuals (e.g., different ages or different sexes) or different sample times (e.g., the sampling effort or a measure of weather conditions), and a range of new approaches that have been proposed for obtaining population size estimates.
Basic Open-population Models
An open population is one that is (or could be) changing during the course of a study, because of any combination of births, deaths, immigration, or emigration. Because most natural wildlife populations are affected in this way, the interest in using capture-recapture data with open populations goes back to the first half of the 20th century when ecologists such as Jackson (1939) were sampling populations that were open, and developing methods for estimating the changing population sizes, the survival rates, and the number of individuals entering the populations between sample times.
A major achievement was the introduction of maximum likelihood estimation for the analysis of open-population capture-recapture data by Cormack (1964), Jolly (1965), and Seber (1965). This led to the development of what are now called the Cormack-Jolly-Seber (CJS) and the Jolly-Seber (JS) models. The CJS model is based solely on recaptures of marked animals and provides estimates of survival and capture probabilities only. The JS model incorporates ratios of marked to unmarked animals and thereby provides estimates of population sizes as well as survival and capture probabilities. The fundamental difference between the two is that the JS model incorporates the assumption that all animals are randomly sampled from the population and that captures of marked and unmarked animals are equally probable. The CJS model, on the other hand, does not make those assumptions and examines only the recapture histories of animals previously marked.
The CJS and JS models are the main topics of chapter 3 by Kenneth H. Pollock and Russell Alpizar-Jara. For the JS model, equations are provided for estimates of population sizes at sample times, survival rates between sample times, and numbers entering between sample times. In addition, there is a discussion of versions of this model that are restricted in various ways (e.g., assuming constant survival probabilities or constant capture probabilities) or generalized (e.g., allowing parameters to depend on the age of animals). The CJS model, which utilizes only information on the recaptures of marked animals, is then discussed. As noted above, this model has the advantage of not requiring unmarked animals to be randomly sampled from the population, but the disadvantage that this allows only survival and capture probabilities to be estimated. Population sizes, which were the original interest with capture-recapture methods, cannot be directly estimated without the random sampling, which allows extrapolation from the marked to the unmarked animals in the population.
Recent Developments with Open-population Models
Since the derivation of the original CJS and JS models there have been many further developments for modeling open populations, which are covered by James D. Nichols in chapter 5. These developments are primarily due to the increasing availability of powerful computers, which make more flexible, but also much more complicated, modeling procedures possible. Parameter values can be restricted in various ways or allowed to depend on covariates related either to the individuals sampled or to the sample time.
The flexible modeling makes it possible to consider very large numbers of possible models for a set of capture-recapture data, particularly if the animals and sample times have values of covariates associated with them. The larger number of possible models that can be considered with modern computerized approaches elevates the importance of objective model selection procedures that test how well each model fits the data. It always has been necessary to assess whether models were apt, how well they fit the data, and which of the models should be considered for final selection. Our greater ability now to build a variety of models is accompanied by a greater responsibility among researchers and managers to perform the comparisons necessary so that the best and most appropriate models are chosen.
The methodological developments in chapter 5 were motivated primarily by biological questions and the need to make earlier models more biologically relevant. This underlying desire to generalize and extend the CJS model resulted in several new models. These methods, covered in chapter 5, include reverse-time modeling, which allows population growth rates to be estimated; the estimation of population sizes on the assumption that unmarked animals are randomly sampled; models that include both survival and recruitment probabilities; and the robust design in which intense sampling is done during several short windows of time (to meet the assumption of closure) that are separated by longer intervals of time during which processes of birth, death, immigration, and emigration may occur. Population size estimates are derived from capture records during the short time periods of the robust design, and survival is estimated over the longer intervals between periods.
The tag-recovery models that are discussed by John M. Hoenig, Kenneth H. Pollock, and William Hearn in chapter 6 were originally developed separately from models for capture-recapture data. These models are primarily for analyzing data obtained from bird-banding and fish-tagging studies. In bird-banding studies, groups of birds are banded each year for several years and some of the bands are recovered from dead birds, while in fish-tagging studies, groups of fish are tagged and then some of them are recovered later during fishing operations. The early development of tag-recovery models was started by Seber (1962), and an important milestone was the publication of a handbook by Brownie et al. (1978) in which the methods available at that time were summarized.
The basic idea behind tag-recovery models is that for a band to be recovered during the jth year of a study, the animal concerned must survive for j - 1 years, die in the next year, and its band be recovered. This differs from the situation with capture-recapture data where groups of animals are tagged on a number of occasions and then some of them are recaptured later while they are still alive.
Joint Modeling of Tag-recovery and Live-recapture or Resighting Data It is noted above that the difference between standard capture-recapture studies and tag-return studies is that the recaptures are of live animals in the first case, while tags are recovered from dead animals in the second case. In practice, however, the samples of animals collected for tagging do sometimes contain previously tagged animals, in which case the study provides both tag-return data and data of the type that comes from standard capture-recapture sampling.
If there are few recaptures of live animals, they will contribute little information and can be ignored. If there are many live recaptures, however, it is unsatisfactory to ignore the information they could contribute to analyses, leading to the need for the consideration of methods that can use all of the data. This is the subject of chapter 7 by Richard J. Barker, who considers studies in which animals can be recorded after their initial tagging (1) by live recaptures during tagging operations, (2) by live resightings at any time between tagging operations, and (3) from tags recovered from animals killed or found dead between tagging occasions. In addition to describing the early approaches to modeling these types of data, which go back to papers by Anderson and Sterling (1974) and Mardekian and McDonald (1981), Barker also considers the use of covariates, model selection, testing for goodness of fit, and the effects of tag loss.
Multistate Models The original models of Cormack (1964), Jolly (1965), and Seber (1965) for capture-recapture data assumed that the animals in the population being considered were homogeneous in the sense that every one has the same probability of being captured when a sample was taken, and the same probability of surviving between two sample times. Later, the homogeneity assumption was relaxed, with covariates being used to describe different capture and survival probabilities among animals. However, this still does not allow for spatial separation of animals into different groups, with random movement between these groups. For example, consider an animal population in which members move among different geographic locales (e.g., feeding, breeding, or molting areas). Also consider that survival and capture probabilities differ at each locale. Covariates associated with the individual animals or sample times are insufficient to model this situation, and the movement between locations must be modeled directly.
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List of Illustrations ix
List of Tables xi
Chapter One: Introduction to the Handbook by Bryan F. J. Manly, Trent L. McDonald, and Steven C. Amstrup 1
1.1 Introduction 1
1.2 Overview of chapters 2 to 8 3
1.3 Maximum Likelihood with Capture-Recapture Methods 9
1.4 Model Selection Procedures 17
1.5 Notation 19
Chapter Two: Classical Closed-population Capture-Recapture Models by Anne Chao and Richard M. Huggins 22
2.1 Introduction 22
2.2 Structure of Capture-Recapture Experiments and Data 23
2.3 Early Models and Estimators 26
2.4 Limitations of Early Models and the Motivation for More General Models 34
2.5 Chapter Summary 35
Chapter Three: Classical Open-population Capture-Recapture Models by Kenneth H. Pollock and Russell Alpizar-Jara 36
3.1 Introduction 36
3.2 The Original Jolly-Seber Model 38
3.3 The Jolly-Seber Likelihood Components 44
3.4 Restrictions and Generalizations of the Jolly-Seber Model 45
3.5 Age-dependent Models 46
3.6 Goodness-of-Fit and Model Selection Issues 47
3.7 Examples 48
3.8 Conclusions 55
3.9 Chapter Summary 55
Chapter Four: Modern Closed-population Capture-Recapture Models by Anne Chao and Richard M. Huggins 58
4.1 Introduction 58
4.2 Discrete-time Models with Unequal Catchabilities 58
4.3 Continuous-time Models 78
4.4 Computing Considerations 85
4.5 Chapter Summary 86
Chapter Five: Modern Open-population Capture-Recapture Models by James D. Nichols 88
5.1 Introduction 88
5.2 Conditional Single-age Models 89
5.3 Conditional Multiple-age Models 102
5.4 Reverse-time Models 107
5.5 Unconditional Models 109
5.6 The Robust Design 116
5.7 Discussion 120
5.8 Chapter Summary 121
Chapter Six: Tag-recovery Models by John M. Hoenig, Kenneth H. Pollock, and William Hearn 124
6.1 Introduction 124
6.2 Assumptions of Brownie Models 128
6.3 Interpretation of the Tag-recovery Rate Parameter 128
6.4 Functional Linkage Between the Exploitation Rate and the Survival Rate 131
6.5 Instantaneous Rate Models for Estimating Harvest and Natural Mortality 131
6.6 Diagnostics and Tests of Assumptions 132
6.7 Preventing and Dealing with Failures of Assumptions 134
6.8 Chapter Summary 140
Chapter Seven: Joint Modeling of Tag-recovery and Live-resighting Data by Richard J. Barker 142
7.1 Introduction 142
7.2 Data Structure 144
7.3 Simple Models 145
7.4 More General Models 156
7.5 Model Fitting and Assessment 157
7.6 Tag Misreads and Tag Loss 161
7.7 Computing Considerations 161
7.8 Chapter Summary 163
Chapter Eight: Multistate Models by Carl J. Schwarz 165
8.1 Introduction 165
8.2 The Arnason-Schwarz Model 166
8.3 The Jolly-Seber Approach 177
8.4 Multisample Stratified Closed Populations 187
8.5 Multisample Stratified Open Populations 192
8.6 Chapter Summary 194
Chapter Nine: Examples by Trent L. McDonald, Steven C. Amstrup, Eric V. Regehr, and Bryan F. J. Manly 194
9.1 Introduction 196
9.2 Open-population Analyses of Data on the European Dipper 198
9.3 The Huggins Closed-population Model Applied to the European Dipper Data 231
9.4 Assessing Goodness-of-Fit 236
9.5 Horvitz-Thompson Open-population Size Estimates 241
9.6 A Multistate (Multistrata) Model 245
9.7 Polar Bears in the Southern Beaufort Sea 247
9.8 Dead Recoveries of Mallard Ducks 254
9.9 Chapter Summary 263
Chapter Ten: Capture-Recapture Methods in Practice by Bryan F. J. Manly, Steven C. Amstrup, and Trent L. McDonald 266
10.1 Introduction 266
10.2 Closed-population Models 266
10.3 Open-population Models 267
10.4 Tag-recovery Models 269
10.5 Other Models 270
10.6 Model Selection 271
10.7 Known Ages 272
A.1 Capability Matrix for Common Capture-Recapture Software Packages 275
A.2 General and Contact Information for Common Capture-Recapture Software Packages Listed in Table A.1 277
Contributor's Notes 301