Ten
Capture–Recapture Methods in Prectice
10.1 Introduction
The authors of chapters 2 to 8 in this book have covered the theory and applications of capture–recapture methods from the simple two-sample, closed-population situations considered by Peterson and Lincoln, through to complex multisample, multistrata, open-population situations that can be modeled only using sophisticated computer software operating on a fast modern computer. In chapter 9, we have provided empirical examples of many methods described in earlier chapters. In this final chapter we summarize the methods that have been covered and provide some closing comments, aimed particularly at readers who are intending to use these methods for analyzing their own data for the first time.
10.2 Closed-population Models
Closed-population models are discussed by Anne Chao and Richard Huggins in chapters 2 and 4. In these chapters they present a hierarchical increase in the complexity and capability of closed-population models. They start in chapter 2 with descriptions of methodological developments before the early 1960s. These early models were limited to the analysis of two or more capture occasions, and probabilities of capture were either assumed to be constant over time or allowed to vary among sampling occasions. Chapter 2 concludes by describing the limitations of the early approaches, and provides the foundation for launching into the more advanced models of chapter 4.
Chapter 4 moves on to more recent approaches for estimation of closed populations. It begins by describing the monograph of Otis et al. (1978), wherein a set of eight models are used to account for heterogeneity in capture probabilities. Chapter 4 also covers numerous alternative approaches that have been developed since 1978 for estimating population size.
The historical context presented in chapters 2 and 4 is valuable, but the large number of analytical choices presented may be intimidating and confusing for someone needing to analyze closed population capture–recapture data for the first time. Fortunately, the method of Huggins (1989, 1991) described in chapter 4 is flexible enough to handle most closed-population estimation problems from the simple to the complex. This approach uses covariates to account for variation in capture probabilities. We recommend it as the preferred method for closed-population analyses when this variation needs to be modelled.
Our preference for the Huggins (1989, 1991) approach is based on several important factors:
1. The approach is based on maximum likelihood estimation, with the advantages that this provides in terms of variance estimation, chi-squared test for comparing the fit of different models, and the use of AIC for model selection.
2. It uses logistic regression models to relate capture probabilities to covariates, in the same way that this is often done outside the area of capture–recapture analyses. This approach is very flexible, and many biologists and managers are familiar with this type of modeling.
3. The method of using covariates is similar to the modern approach for analyzing open population capture–recapture data, as described by Nichols in chapter 5.
4. Readily available computer programs such as MARK and CARE-2 are available to do the required calculations.
Chao and Huggins conclude chapter 4 with a discussion of continuous-time capture–recapture models, where individuals can be captured at any time, rather than only at a fixed number of discrete sampling times. These continuous-time models have received less attention than the discrete-time models, but may be used more in future. Lack of computer programs for carrying out the calculations required restricts their usefulness at the present time.
10.3 Open-population Models
Chapters 3 and 5 refer specifically to situations where k samples are taken from a population for which the size may vary from one sampling occasion to the next because of one or more of births, deaths, immigration, and permanent emigration. As this is the situation with most real populations, models for estimating parameters of open populations have received a great deal of attention for more than 50 years.
In chapter 3 Kenneth Pollock and Russell Alpizar-Jara review some of the early developments in open-population modeling, and then move on to the important Jolly-Seber (JS) model for estimating population sizes, survival rates, birth numbers, and capture probabilities. That discussion then leads to a description of the Cormack-Jolly-Seber (CJS) model. The CJS model, originally conceived at about the same time as the JS model (Cormack 1964), differs from the JS model in that it deals with the marked cohort of animals only. The CJS model conditions on the first capture of each animal and follows the subsequent recapture/reobservation histories throughout the study. Unlike the JS model, the CJS construct does not deal with ratios of marked and unmarked animals and therefore is not able to provide estimates of population size using maximum likelihood theory.
One interesting aspect of the Jolly-Seber model is the assumption that has often been made that the size estimates for open populations are not affected much, if at all, by tag losses or mortality caused by tagging (e.g., Pollock et al. 1990, pp. 25 and 26). This is, in fact, not correct as we have recently demonstrated with both theoretical and simulation results (McDonald et al. 2003).
In chapter 5, James Nichols takes off from the CJS foundation built in Chapter 3 and describes the construction of a likelihood function in terms of the probabilities of survival and capture of the animals in the study. He also describes how those probabilities can be interpreted as functions of the values of covariates that are recorded during the collection of the individual capture histories. The covariates that may help explain capture histories could include the temperatures or other weather parameters at the times of animal captures or in the season prior to the capture events, the characteristics of the researcher’s capture efforts (e.g., the number of trap nights, or the numbers of hours flown by helicopter), or the characteristics of the animals themselves (e.g., the sex, age, weight, or reproductive status). The survival and capture probabilities can in this way be treated as if they are constant over the entire sampling period, or they can be assumed to vary with the specified covariates.
The whole approach is very flexible because if available covariates can help explain capture histories, then restrictive assumptions about survival and capture probabilities are not necessary. Also, the maximum likelihood method allows the fit of nested models (models including covariates that are subsets of a more complete cast of covariates) to be compared using chi-squared tests. Further, the appropriateness of any assortment of alternative models can be compared using Akaike’s information criterion (AIC) and related methods. For all of these reasons, we recommend the CJS approach, incorporating explanatory covariates, for analyzing capture–recapture data from open populations involving more than three samples.
Chapter 5 also covers some recent extensions to the basic covariate approach to the CJS model. Reverse-time modeling, temporal symmetry models, and the robust design all are theoretically appealing extensions of the basic approach of CJS, and all can employ information from covariates. Application of these extensions in the real world, however, has been rare due primarily to their recent development.
As noted above, an important limitation with the CJS approach is that it does not directly provide estimates of population sizes. At one time the prevailing opinion among statisticians seemed to be that this was not an important shortcoming. In reality, however, biologists and managers are usually very interested in estimating population sizes and trends. An important innovation therefore is the calculation of Horvitz-Thompson estimates to overcome this limitation (McDonald and Amstrup 2001; see also chapter 9). This method requires only the assumption that the animals captured at any time are representative of the population of animals alive at that time. Because this assumption is fundamental to all capture–recapture methods that draw inferences to the population being studied it is not a serious limitation. With the Horvitz-Thompson method, the estimated population size at the time of a capture occasion is obtained by estimating the capture probability for each of the animals captured at that occasion and adding up the reciprocals of these probabilities. We therefore recommend, the fitting of CJS model for open populations followed by Horvitz-Thompson estimation when population sizes at each occasion are desired.
10.4 Tag-recovery Models
Models for analyzing tag-recovery data were originally developed separately from models for mark–recapture data. The situations of data collection were different from the normal capture and recapture paradigms in that tagged animals usually were recovered dead. Classic tag-recovery data derived from individual animals (most commonly birds or fish) that were tagged and released, and then harvested at a later time. In modeling terms this is not the same as tagging individuals and then recapturing some of them at later times while they are still alive.
In chapter 6 John Hoenig, Kenneth Pollock, and William Hearn describe the analysis of tag-recovery data with the pioneering methods of Brownie et al. (1985). They then explore the situation where an estimate of the exploitation rate (the fraction of animals harvested) explains only a portion of the recovery rate (which also depends on the probability of tag loss, and the probability of a tag on a harvested animal being reported). The chapter concludes with a discussion of methods for testing whether the assumptions of tag-recovery models apply for the data being considered, and the remedies that are available if this is not the case.
The tag-recovery models and methods are particularly appropriate for many fisheries studies where a good estimate of the exploitation rate is crucial for assessing the status of stocks subjected to commercial and or recreational fisheries. Another important application has been waterfowl harvest monitoring. Many species of waterfowl can be marked in great numbers as flightless chicks or during flightless periods that accompany molt. Then, despite extensive migrations, the nationwide network of sport hunters can provide returns in large numbers.
10.5 Other Models
Many wildlife and fish studies have data available from both live recaptures and dead recoveries. For example, waterfowl tagged in the spring are subject to an autumn hunt. Subsequently, birds not harvested may be re-observed on wintering grounds or when they return the next season to the spring nesting and molting areas. To address these cases Richard Barker in chapter 7 describes an extension of the general CJS approach that can incorporate tag returns. He points out that the alternative to modeling the combined data, analyzing each type of data separately, may be considerably less efficient than using the combined approach. Unless it happens that almost all of the data collected in a study are of one type or the other, we recommend the methods described by Barker for studies that produce data from both harvests and live recoveries.
Carl Schwarz’s final chapter on methods for analyzing capture–recapture data describes multistrata models, which are more commonly now called multistate models. These models apply where the captures and recaptures made on a population can occur at different geographic locations or “strata,” or “states.” States may be different geographic locales or behavioral or reproductive conditions (e.g., hibernation, occurrence above or below the surface of the soil, snow, or water, or different reproductive conditions), and where the animals may move between those states from one capture occasion to the next. The CJS model is extended to this situation by allowing capture and survival probabilities to vary with the different states, and by introducing parameters that describe the probabilities of movement among states. Needless to say, this makes the model considerably more complicated than the usual CJS model. Because estimation of the probabilities of movement among states depends on the numbers of observations (captures) in all of them, realistic estimates of all interchange rates among states cannot usually be obtained unless many (often thousands of) animals are tagged, and the numbers of reobservations in each state are very high. With smaller data sets, constraining certain interchange rates to be equal is often necessary, and if done usually results in reliable estimates of the constrained parameters.
In addition to the extension of the CJS model to cases with two or more states, Schwarz also covers closed-population situations, with two samples in particular. He notes, however, that there are several practical problems involved with these two sample situations, including again the need for very large sample sizes.
Conceptually, multistate models seem to fit many biological situations. Models that acknowledge movements of animals among states are intuitively appealing, because we know animals do so. It is theoretically possible to estimate the population sizes in different states at different times for an open population, but because the data sets required to do so are quite large, a more effective approach may be to fit a CJS type of model, extended to take into account the stratification. The population size in each state and at each occasion can then be estimated using the Horvitz-Thompson approach of adding up the reciprocals of capture probabilities for all of the animals captured in that stratum at that time (McDonald and Amstrup 2001). In general, the data requirements and the computational difficulties of multistate modeling of capture–recapture data mean that biologists should think carefully before embarking on studies of this type.
10.6 Model Selection
A few words of caution are in order about automatic methods of model selection such as AIC and its variations. For example, although Manly et al. (1999) found that model selection using this approach usually chose a model that was close to being correct, this was not always the case.
It may be that one or more of a series of plausible models should not be used for estimation, although they may apparently be the best choice using an AIC type of criterion. This was discovered by Boyce et al. (2001) with a simulation study of the estimation of the number of female bears with cubs of the year in Yellowstone National Park. Six plausible models were defined based on the assumption that the numbers of females with cubs observed in any year has a negative binomial distribution. The simulation study demonstrated clearly that the two most complicated models gave such poor estimates of true population sizes that neither of them should ever be used, even when they are known to be the correct model. Similarly, Amstrup et al. (2001) chose not to use the model with the lowest value for an AIC criterion when analyzing mark–recapture data on polar bears because of the poor estimates of population sizes produced by this model.
Our recommendation in terms of model selection by AIC and other similar types of criterion is that these should be regarded as useful guides for choosing models, but that the final decision on a model to use should take into account other factors, such as what management decisions will be based on the model, the apparent quality of estimates of quantities derived indirectly from the estimated parameters in the fitted model, and the degree to which estimates agree with other known information about the populations in question as well as the biologist’s intuition regarding the situation, etc. In other words, it is not safe to assume that mathematics alone will always guide you to the correct choice of models.
10.7 Known Ages
We conclude this final chapter with a brief description of a method that has recently been developed for the estimation of open-population sizes from mark–recapture data on animals of known ages (Manly et al. 2003). Age information is used only indirectly, if at all, with the methods described in earlier chapters. For example, in an open-population model survival rates can be assumed to depend on age through the use of a covariate. However, the age information is not used to determine when the animal was in the population, and hence possibly available for capture before the first actual capture time.
There are many cases of long-lived animals where the ages of the individuals can be and routinely are accurately determined. For example, the age of polar bears (Ursus martimus) can be determined by removing and sectioning a vestigial premolar tooth. Hence, if a polar bear is captured for the first time and found to be ten years old, then it is known that this bear was in the population and available for capture, provided that the bear did not immigrate into the population after it was born, in each of the previous nine years, even if it was not captured in those years.
This idea can be combined with the Manly and Parr (1968, see chapter 3) concept of estimating the probability of capture by defining a group of animals known to be alive at the time of a sample, and finding what fraction of these animals were actually captured. Age information allows the size of the group to be enlarged by the inclusion of animals captured for the first time in later samples that must have been alive when the sample in question was taken.
A likelihood function can be constructed using age information and conditioning on the last capture time of animals. This turns out to involve capture probability parameters, but not survival probability parameters, so that it is relatively simple in comparison with other likelihood functions for open-population data. It also turns out that the capture probability parameters can easily be interpreted as functions of covariates, in a similar way to the models discussed by James Nichols in chapter 5. There are three considerable advantages of the analysis using age data. First, all the calculations with can be carried out using an ordinary program for logistic regression. The usual specialized software for fitting capture–recapture models is therefore no longer required. This is important because most biologists and managers are familiar with regression concepts and the programs available to them. It is also important because the outputs of such programs are easily interpreted. Second, it is possible to estimate the population size at the time of the first sampling occasion, which is not possible with the Jolly-Seber method, for example. This means that it is possible to estimate population size in only two occasions rather than the minimum of three normally required for open populations. Because funding for long-term studies is always more difficult to obtain, this benefit offers the promise of estimates that otherwise may not have been obtained due to absence of protracted funding commitments. Third, estimates of population size should have better precision than those obtained from other methods that do not use the age information.
A likelihood function can also be constructed conditioning on the known time when captured animals entered the population based on their ages. By treating this time as a first “capture” it becomes possible to model both capture and survival probabilities, using the computer program MARK, for example. This is more complicated than using the logistic regression approach for modeling capture probabilities only, but this may be worthwhile because of the extra information obtained from the times of last captures of animals.
The strengths and weaknesses of this approach to population size estimation are still being developed. Simulation results and recommendations for appropriate application situations should be published in the coming years.
Figure 10.1. Double ear-tagged black bear (Ursus americanus) cub watches mom from an elevated perch, Boise National Forest, Idaho, 1973. (Photo by Steven C. Amstrup)