Eight
Multistate Models
8.1 Introduction
The original modeling framework for capture–recapture studies assumed homogeneous behavior among animals and was concerned with estimating parameters, such as survival or abundance, for a single uniform population. In many situations, this is unrealistic because animals within a population are not homogeneous with respect to survival and catchability, and heterogeneity in these parameters can lead to biases in estimates.
In their key paper, Lebreton et al. (1992) developed a modeling framework based on partitioning populations into homogeneous subpopulations (groups) based on fixed, unchanging attributes, such as sex. These models and the AIC model selection framework (chapter 1) start with separate parameters for each group in the population and move toward models where some parameters are common across groups, e.g., male and females having common survival rates but unequal catchabilities.
The Lebreton et al. (1992) framework assumes that membership in the groups is fixed and does not change over time, which, in a way, is similar to stratification of populations. However, in some cases, it is desirable to let group membership change over time. These changes can be divided into two categories. First, some changes are predicable. If the population is divided into age classes, the movement of animals among the age classes is predictable and regular. Second, some changes occur randomly, such as migration among geographical areas, and the movement of animals is not predictable. Movement was the prime motivation for consideration of multistate models, but these models can be generalized to any situation where animals change state in an unpredictable fashion (Lebreton et al. 1999; Lebreton and Pradel 2002).
This chapter is divided into two sections. In the first, multistate models that estimate survival, movement among strata, and capture probabilities, but not abundance, are examined. These are generalizations of the Cormack-Jolly-Seber (CJS) models described in chapters 3 and 5. Animals are marked, released into the population, and followed over time, but the process by which animals were initially marked is ignored. In the second part, models that also estimate abundance are introduced. In this part it is crucial that the process by which new animals are initially captured be properly modeled, as information from this process allows the estimation of abundance. This requires much more care in designing the study.
Figure 8.1. Close-up photo of a “Coded Wire Tag” for permanent marking of many species of fish. Tags can be cut to length upon injection into the organism. (Courtesy of Northwest Marine Technology, Inc.)
8.2 The Arnason-Schwarz Model
Introduction
Chapman and Junge (1956) and Darroch (1961) were among the first papers to take account of geographical sites in capture–recapture analyses of two-sample experiments. Arnason (1972, 1973) generalized these to three samples and Schwarz et al. (1993b) to the k-sample case. Arnason’s pioneering work remained largely unused because of the lack of computer software, and because it seems to have limited robustness with its large number of parameters (Viallefont and Lebreton 1993). However, with the development of suitable software (MSSURVIV, Brownie et al. 1993; MARK, White and Burnham 1999; M-SURGE, Choquet et al. 2003), the application of these models became more widespread. Hestbeck et al. (1991) and Brownie et al. (1993) were among the first to consider multistate models in a movement context when examining the migration patterns of Canada Geese. Further work on estimating dispersal was reported in Nichols et al. (1993), Nichols and Kaiser (1999), and Schwarz and Seber (1999). Nichols et al. (1994) generalized stratification to non-geographic cases by using these models to examine changes among weight classes, and Lebreton and Pradel (2002) examine stratified models in a more general context.
Experimental Protocol and Data Collection
The experimental protocol for the Arnason-Schwarz (AS) multistate model is an extension of that used in the CJS model. Releases of tagged animals are made in states, which are labeled as 1, 2, 3, …, S. These are followed over time, and when an animal is recaptured, its state of recapture is recorded. After recording the tag information, the animal is released back to the population, although losses on capture are allowed. There are a total of k sampling occasions.
The raw data from such an experiment consists of a generalization of the history pattern used in CJS models. The history record is of length k, h = (h1, h2, …, hk), where hj, is
| 0 | if the animal was not recaptured or released at time j; |
| 1, 2, 3, … | if the animal was initially released or was recaptured at time j in state 1, 2, …, S and was returned to the population; or |
| −1, −2, … | if the animal was recaptured at time j in state 1, 2, … and lost on captured (i.e., not returned to the population). |
The first nonzero element of the history vector indicates the stratum of first release for each animal. For example, the history vector h = (0, 2, 0, 1) would indicate an animal that was initially released in year 2 in stratum 2, not seen in the third year, and then seen in year 4 in stratum 1.
There are three sets of parameters in the AS multistate model. The first two sets are generalizations of the CJS capture and survival parameters to each stratum. The third set represents the movement of animals among strata.
As in the basic CJS model, subscripts on parameters refer to time. Superscripts will be used to represent strata. The recapture probabilities are denoted as 
, the probability of being captured in stratum s at time j given that the animal is present in stratum s. The parameter 
represents the probability that an animal alive at time j in stratum s is alive at time j + 1 and in one of the (unspecified) strata 1, …, S. It is interpreted as an average survival probability for animals alive at time j in stratum s. As in the CJS model, survival refers to apparent survival, i.e., death and permanent emigration outside the study areas cannot be distinguished. Lastly, the movement parameters 
represent the movement probabilities of an animal alive at time j in stratum s moving to stratum t at time j + 1 conditional, upon surviving until time j + 1. It is also assumed that
i.e., living animals must move to one of the strata in the study area, and animals that move off the study area are indistinguishable from those dying. Unless strong assumptions are made about behavior, it is not possible to model the timing of survival and movement. For example, one cannot distinguish between (1) movement taking place immediately after release with no real mortality during the movement, and all mortality taking place in the receiving stratum; or (2) animals remaining in the initial stratum for an extended period of time (and subject to mortality) followed by a movement just before the next sampling occasion. The movement probabilities are net; i.e., if an animal moves among many strata between time j and time j + 1, only the initial and final stratum are of interest. Lastly, there is an implicit assumption that movements prior to time j do not affect the movement choices at time j (i.e., the assumption is of Markovian movement, as discussed in chapter 7).
In an experiment with k sample times and S strata, there are a total of (k − 1)S average survival parameters, , j = 1,…, k − 1 and s = 1,…, S; (k − 1 )(S2 − S) movement parameters, , j = 1,…, k − 1 and s = 1,…, S, but with 
capture parameters, 
, j = 2, …, k and s = 1, …, S; and finally (k − 1)S loss-on-capture parameter. This gives a total of (k − 1)(S2 + 2S) parameters, compared to the 3(k − 1) parameters in the CJS model.
The key step in any modeling framework is to express the probability of the observed history data in terms of the parameters of interest. Here, this step proceeds similarly to the illustration in chapter 1 for CJS models except that capture and survival parameters are stratum specific, and movement among strata must be accounted for. The probability expression for a particular history vector can be complex. It is fairly straightforward to write the contribution when an animal is sighted at two adjacent sample times. The complications arise when dealing with (1) a sighting gap (e.g., 2, 0, 1), (2) the fate of the animal after the last capture (e.g., . . ., 1, 2, 0, 0, 0), and (3) losses on capture (e.g.,…, 1, 0, –2, 0, 0, 0). Generally, when an animal is unobserved at time j the stratum of membership is also unknown and so all possible movement paths between the observed locations must be modeled. For example, assuming that only two strata are present, the probability of the history h = (1, 2, 0, 1) is
where 
is the probability that the animal released in stratum 1 at time 1 will survive to time 2 and be present in stratum 2, 
is the probability that the animal alive at time 2 and in stratum 2 will be recaptured, and
is the overall probability that an animal alive in stratum 2 at time 2 will be alive at time 4 and in stratum 1. The last term must account for the two possible paths from stratum 2 to stratum 1 between times 2 and 4, which are unobserved. Finally, 
is the probability that the animal alive at time 4 in stratum 1 was recaptured.
Once the probability for each history is defined, the overall probability of observing the data (the likelihood) can be determined using a multinomial distribution in much the same way as in simple models shown in chapter 1.
This model makes a number of assumptions in addition to those made for the CJS model:
1. all data are recorded without error with respect to the stratum of recovery, resight, or recapture;
2. there are equal movement probabilities for all animals in stratum s at time j;
3. the movement probability and recapture probabilities do not depend on the past history of the animal; and
4. recoveries are instantaneous.
Schwarz et al. (1993a) developed movement models for use when recoveries take place over a period of time such as in a fishery that are generalizations of Brownie et al. (1985).
Model Fitting and Selection
No simple formula exists for the maximum likelihood estimators, but Schwarz et al. (1993b, appendix A) present moment estimators. Standard numerical procedures can be used to maximize the likelihood function, to obtain estimates of the parameters, and to obtain estimated standard errors. Three software packages that can analyze the multistate model are MSSURVIV (Brownie et al. 1993), MARK (White and Burnham 1999), and M-SURGE (Choquet et al. 2003). Cooch and White (2001) give a detailed description on how to use MARK to fit multistrata models. Choquet et al. (2003) also has a manual demonstrating the use of M-SURGE.
It should be noted that there is a substantial risk of converging to local maxima of the likelihood (Lebreton and Pradel 2002). Choosing several different initial starting points and ensuring that these converge to the same values will serve as a check of the final model estimates. Similarly, any large change in the estimates of movement or catchability parameters between models with similar structure is an indication of lack of convergence.
As in the CJS model, not all parameters may be identifiable. In the most general model with capture, survival, and movement probabilities time dependent, the final survival, movement, and recapture probabilities are all confounded. Some parameters may also not be estimable because of sparse data, e.g., when there are no recoveries in any stratum at time i or no releases in a stratum at time i. Gimenez et al. (2003) have created a software package that identifies nonidentifiable parameters using a symbolic algebra package following earlier work by Catchpole and Morgan (1997) for the CJS model. Kendall and Nichols (2002) examined identifiability of the transition probabilities when some states are completely unobservable, such as nonbreeders.
Pradel et al. (2003) and Wintrebert (1998) have devised goodness-of-fit tests for the multistate model that are similar to those available for the CJS model and have prepared a computer program (UCARE) to perform these tests. One set of contingency tables extends the Test3 contingency tables for the CJS models and compares subsequent recaptures of subgroups of animals that are all seen and released at stratum s at time i. A new contingency table (TestM) tests if animals are mixing by comparing previous sightings for animals seen at stratum s at time j. Finally, a third set of contingency tables tests the memoryless property. Because of the large number of possible states, the number of counts in each cell of the contingency tables may be small, so rules are suggested for extensive pooling.
In cases where k and S are small, another possibility is to enumerate all the possible capture histories, compute the expected values under the final model, and perform a classical chi-squared goodness-of-fit test (Hestbeck et al. 1991). This approach is impractical in larger problems because the number of possible histories grows very rapidly and the data are then very sparse. Given these problems, a step-up approach starting from a simple model and moving to a more complex model is a good strategy.
Following Lebreton et al. (1992), the above modeling process can be generalized to multiple groups. For example, males and females may be assured to have separate movement, survival, and capture probabilities. The data demands for the most general model allowing for group, time, and stratum specific estimates are considerable.
Versions of the CJS model with survival or capture rates constant across groups or time have proved useful for single-stratum studies and the motivation to use simpler models is even greater in multistate models because of the large number of parameters. Using a generalization of the notation of Lebreton et al. (1992) and of Lebreton et al. (1999), a collection of models can be considered. For example, using a similar notation as in CJS models, the model {pg*s*t, ϕg*s*t*, ψg*s*t} has capture probabilities that are group, stratum, and time specific; survival probabilities that are group, stratum, and time specific; and movement probabilities that are group, stratum, and time specific. This is the most general model possible. Two examples of simpler models are the model {pg*s*t, ϕg*s*t, ψg*s} with capture probabilities that are group, stratum, and time specific; survival probabilities that are group, stratum, and time specific; and movement probabilities that are group and stratum specific but constant over time. The model {pg*s*t, ϕ, ψ} has capture probabilities that are group, stratum, and time specific; survival probabilities that are constant over group, time, and stratum; and stratum movement probabilities that are constant over group, time, and stratum.
Model selection among the set of candidate models fit to the data can be based on the Akaikie Information Criterion (AIC) as described in chapter 1. Then the model with the lowest AIC represents the best choice in the trade-off between goodness of fit (more parameters) and parsimony (fewer parameters). Estimates from models with similar AIC can also be used in the model averaging procedures as described by Burnham and Anderson (1998). The greatest difficulties that will likely be encountered in the modeling exercise are determining the number of identifiable and estimable parameters and dealing with the effects of estimates on the boundary of the allowable range of values. Automatic parameter counting methods are under development (Reboulet et al. 1999; Catchpole and Morgan 1997) but these can be unreliable with sparse data. There are no simple ways to deal with parameter estimates on the boundary, because standard likelihood theory is no longer applicable.
Dupuis (1995) and Dupuis (2002) have developed Bayesian formulations of the Arnason-Schwarz model. These are likely to be too computationally intense for many problems, but provide a nice way of dealing with some of the technical problems encountered using standard likelihood methods.
Finally, because the number of parameters increases as the square of the number of strata, models with fewer strata can lead to more precise estimates from the same set of data. It is tempting to try and use the model selection methods to choose the number of strata. However, this must be done by constraining parameters in a general model rather than simply pooling strata. The latter involves comparing different models on different data sets (despite being derived from the same data), and may give misleading results.
Example—Movement of Canada Geese
Hestbeck et al. (1991) considered a study with multiple-recapture data of Canada Geese (Branta canadensis), which was further analyzed in Brownie et al. (1993). The geese were captured in three geographic strata (the mid-Atlantic, the Chesapeake, and the Carolinas) from 1984 to 1989. Only the 1986 to 1989 data will be used in this example. The capture histories are shown in table 8.1.
This dataset has only a single group so the group subscript g will be dropped in the remainder of this section. The most general model, model {ϕs*t, ψs*t, ps*t} was initially considered where all parameters vary with stratum and time. In this model, the final capture probabilities are confounded with the final survival probability and so cannot be separately estimated. The former were therefore arbitrarily constrained to be 1.0. The estimates from this general model are shown in table 8.2. They seem to indicate an overall yearly survival rate of about 65–70% that may be constant over time, but varies among strata. The capture rates seem to vary among strata and possibly over time. The migration rates show strong fidelity to the area captured and appear to be roughly constant in the first two years of the study but with a large change in year 3 for stratum 3.
Based on the preliminary estimates, a series of models were considered, as summarized in table 8.3. The most appropriate model is model {ϕs, ψs*(t1 = t2, t3), ps*(t1 = t2, t3)}, where the survival rates were constrained to be constant over time (but vary among strata), and the migration and capture rates were constrained to be equal in the first two years, with possible changes in the third year. This model has a high AIC weight. Estimates from the final model are shown in table 8.4. Some other simpler models may also be appropriate, such as one with the survival probabilities being equal for all strata, but further models were not considered. The estimates have remarkably small standard errors, but because the study observed over 10,000 birds this is not surprising.
Brownie et al. (1993) also fitted a non-Markovian model to the data and showed that there was evidence that the models considered here may not be fully appropriate. Their paper should be consulted for more details.
Other Uses for Multistate Models
Although the Arnason-Schwarz multistrata model was originally developed in the context of physical movement, the multistate model can also be used in a number of other contexts.
TABLE 8.1
Raw data (1986–1989) for the Canada goose study (provided by J. Hines, Patuxent Wildlife Center)
TABLE 8.2
Estimates from the goose data from the model {ϕs*t, ψs*t, ps*t}
TABLE 8.3
Summary of model fitting to the Canada goose data
Live and Dead Recoveries
Lebreton et al. (1999) noted that the ordinary CJS model can be recast into a multistate model, with the two strata corresponding to live and dead. In this case, recaptures are possible only from live animals, and movement between live and dead states is unidirectional. In a similar fashion, the band-recovery models of Brownie et al. (1985) can also be cast into multistate framework, where the three strata correspond to live, newly dead, and previously dead. Only a single recapture is possible when the animal is recovered dead; and movement is again unidirectional. Models with both live recaptures and releases and dead recoveries (chapter 7) are also easily constructed. Now recaptures occur locally on a smaller study area while recoveries of newly dead animals can occur on a larger geographical scale. Lebreton et al. (1999) also show how to model radio-tracking data where dead animals are detectable from their functioning radio collars.
Local Recruitment of Breeders
The age of first reproduction is important in population regulation and life-history theory. Clobert et al. (1994) developed a model for mark–recapture data where animals are marked as young, but are not resightable until they commence to breed. Schwarz and Arnason (2000) considered the same problem and recast it as a Jolly-Seber experiment. Pradel and Lebreton (1999) examined the problem from a multistate perspective. Here, the two states are defined as nonbreeder or breeder. All animals are tagged at age 0 as nonbreeders, both breeders and nonbreeders share the same survival probabilities, initial breeding depends only on age, and experienced breeders breed systematically (i.e., do not skip a breeding season). Lebreton et al. (2003) have extended this approach to dispersal–recruitment over several sites. Kendall et al. (2003) used multistate models along with Pollock’s robust design to adjust estimates of recruitment for misclassification of the animal’s state with an application to estimate the change in breeding status for Florida manatees (Trichechusmanatus latirostris).
TABLE 8.4
Estimates from goose data from the final model {ϕs, ψs*(t1 = t2, t3), Ps*(t1 = t2, t3)}
Individual Covariates That Change over Time
Nichols et al. (1994) used multistate models where states were defined by individual covariates (weight classes of meadow voles) that changed over time, rather than geographical sites. This is very general, and any continuous covariate could be stratified. One area that requires further research is the choice of class boundaries and the comparison of models with different class boundaries. States can also be defined based on discrete covariates such as reproductive success, e.g., breeders vs. nonbreeders. These have been used to study trade-offs between different reproductive strategies (Nichols et al. 1994).
8.3 The Jolly-Seber Approach
Despite the focus of current capture–recapture experiments and models for estimating survival and movement parameter, the original motivation for these methods was on estimating abundance. For example, Jolly (1965) and Seber (1965) were primarily interested in estimating the number of animals alive at each sampling occasion. The general class of models dealing with abundance and survival (JS models) is a generalization of the CJS models that estimated only survival and capture probability.
Far less work has been done on multistate models that include estimating abundance, for several reasons. First, the experimental protocol necessary to estimate abundance is more rigorous than that to estimate survival or movement. In the latter cases, the process by which animals are first captured and tagged is relatively unimportant, as long as the subsequent survival and movement of the marked animals is representative of the larger population. In contrast, estimating abundance requires that the process by which new animals are captured gives a random selection from the entire population. As well, the definition of the relevant population is also important. If captures take place at a single geographical location of a highly mobile animal, then it is not clear whether the relevant population is defined as those within 1, 5, or 20 km of the sampling site.
Second, heterogeneity in capture probabilities can lead to biases when estimating abundance. At the extreme, animals with very low capture probabilities may be missed entirely in the estimates. Recent work in the CJS framework of trying to estimate population changes rather than abundance looks promising, and Schwarz (2001) has shown that that heterogeneity in capture probabilities causes little bias in estimates of population change in the simple JS model.
Third, temporary emigration also leads to severe biases in the estimates and makes it difficult to clearly define the relevant population in terms of whether the estimate of population size should include temporary migrants or exclude them. Kendall et al. (1995, 1997), Schwarz and Stobo (1999), and others have shown how to use Pollock’s (1982) robust design to estimate both components of the population.
The latter two concerns have been the primary motivation of work in estimating abundance using stratified methods. In order of prevalence of use, these can be broadly classified as
• two-sample experiments to estimate abundance, where the focus has been on reducing the biases in the simple Petersen estimator caused by heterogeneity in capture probabilities;
• multisample closed populations, where the focus has been on modeling movement to reducing biases caused by heterogeneity of capture and temporary absences from a single sampling location; and
• multisample open populations, where the focus has been modeling movement to reduce biases caused by heterogeneity of capture and temporary absences from a single sampling location.
Two-sample Stratified Experiments
In a two-sample experiment, animals are captured, tagged, and released in the first sample and allowed to mix with the remaining population. Then a second sample is taken and inspected. The simplest estimator of abundance is then the Lincoln-Petersen estimator
where n1 is the number of animals seen at time 1, n2 is the number of animals seen at time 2, and m2 is the number of animals seen at times 1 and 2. Small-sample, bias-corrected versions are also available.
The Lincoln-Petersen estimator is a consistent estimator of the population size when either or both of the samples are a simple random sample so that either all animals in the population have the same probability of being tagged or all animals have the same probability of being captured in the second sample. It is also assumed that the population is closed, there is no tag loss, the tagging status of each animal is determined without error, and tagging has no effect on the subsequent behavior of the animals (chapter 2).
There are many cases where the capture probabilities are not homogeneous over the animals in either sample. For example, in surveys of spawning populations, fish are caught and tagged as they pass a common point prior to the spawning grounds, over the course of several days or weeks. It is unlikely that the tagged fish are a simple random sample. This would require, at a minimum, that a constant proportion of the fish be tagged each day. Similarly, the recaptures may be obtained by searching carcasses on selected spawning area. The recapture probabilities are likely to vary by recovery area and over time, and some spawning areas may not be sampled at all. Lastly, it is unlikely that fish passing the tagging site early in the run mix completely with fish that pass the tagging site late in the run. Under these circumstances, the simple Petersen estimator formed by pooling over time and space all of the tags applied and all of the recoveries may be biased and this bias can be considerable (Arnason et al. 1996b). Schwarz and Taylor (1998) derived an approximation to the size of the bias and showed that it was related to the correlation between the two capture probabilities. Substantial bias is likely to occur if there is great variability in both the capture and recapture probabilities (e.g., through sex or size effects), or if there is little mixing of the animals as they move from the tagging to the recovery locations.
One method of reducing this bias is to stratify by time, space, or other characteristic. The captures at time 1 are placed into S nonoverlapping strata while captures at time 2 are placed into T nonoverlapping strata at the time of recovery. Unlike the AS models, the two-sample stratified experiment allows for different number of strata at each sample time. For example, in geographic stratification, tagging strata may be different stocks of fish caught on different breeding areas, and recovery strata may be different fishing areas. In temporal stratification, tagging and recovery may take place over several weeks, and individual strata correspond to time intervals (e.g., weeks). Stratification can also be by permanent attributes (e.g., sex) or changing attributes (weight class).
Capture histories can again be constructed to represent the data as in the previous section. The history vectors take the forms {s, t}, s = 1, … , S, t = 1, … , T for animals captured in stratum s at time 1 and in stratum t at time 2; {0, t}, t = 1, … , T for animals not captured at time 1 but captured in stratum t at time 2; and {s, 0}, s = 1, … , S for animals captured in stratum s at time 1 and not captured at time 2. The number of animals with these histories is often placed in a rectangular array as shown in table 8.5.
TABLE 8.5
Summary statistics for female pink salmon tagged at Glen Valley and recovered in the Fraser River Main Stem arranged in a rectangular array
The parameters of the model are the capture probabilities at times 1 and 2, 
, s = 1, …, S and 
, t = 1, …, T; the movement parameters among strata between times 1 and 2, ψst, s = 1, …, S, t = 1, …, T; and the population sizes in the strata at time 1 and time 2, 
and 
. The model allows “leakage” between the two sampling occasions, i.e., 
, which could be caused by mortality or movement out of the study area. This is unlike the AS models where the movement probabilities must sum to 1.
The usual assumptions for capture–recapture experiments are made and it is additionally assumed that
1. animals behave independently of one another in regard to moving among strata;
2. all tagged animals released in a stratum have the same probability distribution of movement to the recovery strata;
3. all animals in a recovery stratum behave independently with regard to being caught and all have the same probability of being caught;
4. no tags are lost; and
5. all animals that are recovered are correctly identified as to the tagging status, and if tagged, the tag number is correctly recorded, although Schwarz et al. (1999) extend this situation to cases where some of the tags in the recovery strata are counted but not read to identify the stratum of release.
In addition, one or both of the following assumptions is usually made depending whether the goal of the study is to estimate the number of animals in the tagging or recovery strata:
6a. the movement pattern, death, and migration rates are the same for tagged and untagged animals in each tagging stratum (required to estimate the total number of animals in the tagging strata); or
6b. the population is closed with respect to movement among strata (required to estimate the total number of animals at the recovery strata).
Model Fitting and Estimation
In the most straightforward case there is a one-to-one correspondence between the tagging and recovery strata. This will occur, for example, with stratification based on permanent attributes (e.g., sex) where animals cannot move from one stratum to another. In these cases, the problem reduces to s independent Petersen estimates. The total abundance is found by summing these independent estimates and the overall standard error is easily found.
Three common estimators are used in the case of the more general stratified samples. The pooled Petersen is the simplest estimator and is formed by ignoring the stratification completely and computing a simple Petersen estimate based on the total releases and total recaptures:
where ns+ = ns0 + ns1 + ns2 + … + nsT and n+t = n0t + n1t + n2t+ … + nSt.
Darroch (1961) and Seber (1982), summarized three common circumstances under which the pooled-Petersen remains a consistent estimator:
1. the tagging probabilities are the same in all release strata;
2. the recovery probabilities are the same in all recovery strata, and the same degree of closure of movement occurs for each release stratum, i.e., the same fraction of each release stratum is subject to recovery in the recovery strata; and
3. there is complete mixing of animals from all the release strata before animals are recovered, and the movement pattern for tagged and untagged animals is the same.
Darroch (1961) discussed formal statistical tests that can be used to examine whether the pooled-Petersen estimate is consistent. In particular, he discussed two simple contingency table methods that have been implemented by Arnason et al. (1996a).
Schaefer (1951) developed an estimator that has been rediscovered numerous times (e.g., Macdonald and Smith 1980; Warren and Dempson 1995). This takes the form
Surprisingly, although the Schaefer estimator has been used in the literature in three common disguises, its properties were not systematically investigated until recently (Schwarz et al. 2002). In fact, the Schaefer estimator has essentially the same performance (in terms of bias and standard error) and requires the same conditions for consistency as the pooled-Petersen estimator. Consequently, there is no advantage to using the Schaefer estimator over the pooled-Petersen estimator, and it is no longer recommended for use.
Lastly, a Petersen type of estimator can be constructed that accounts for the stratified nature of the experiment and models movement. As in the AS model, the probability of each history vector must be expressed in terms of the parameters. However, it is more convenient to express the expected counts in terms of the parameters to give
and
A likelihood function can be constructed based on these expected counts, assuming a multinomial model from each release. Plante et al. (1998) and Banneheka et al. (1997) show that all of the parameters cannot be individually estimated. However, certain functions of the parameters may be estimated under each of two different scenarios.
If the number of tagging strata is less than or equal to the number of recovery strata (S ≤ T), and if the assumption of the same movement patterns for tagged and untagged animals holds, but not necessarily closure, then Plante et al. (1998) and Banneheka et al. (1997) showed that the number of animals in the population at the time of tagging can be estimated. If the number of tagging strata is greater than or equal to the number of recovery strata (S ≥ T), and there is closure of the population with respect to movement to the recovery strata, then the population size at the time of recovery can be estimated. When S = T and both assumptions 6a and 6b above are satisfied, then the population sizes at the time of tagging and at the time of recovery are identical, and both can be estimated. The case of S = T was fully developed by Darroch (1961) and summarized by Seber (1982).
For most experiments, the assumptions of the same movement patterns for tagged and untagged animals, but not necessarily closure over recovery strata, may be more tenable than the assumption of closure with respect to movement among strata. For example, in a typical spawning escapement survey, the population is not closed because there is mortality between the tagging site and the spawning areas, fish leave the population because of spawning in other than the recovery sites, fish may arrive and spawn before or after recovery takes place, and fish may be removed through a fishery that takes place during the run. For this reason, the stratified-Petersen analysis with S ≤ T is the most common to estimate the number of animals in the population at the first sampling occasion.
The usual goodness-of-fit test comparing the observed and expected counts for each capture history can be used to assess the fit of a model.
Plante et al. (1998) developed general maximum likelihood theory for the cases where the number of release strata did not match the number of recapture strata. Banneheka et al. (1997) developed a least-squares estimator and found its properties. Arnason et al. (1996a, b) examined conditions under which stratified estimators were superior to the usual Petersen estimator, and have developed a user-friendly computer package that implements all of these recent advances. Finally, Schwarz and Taylor (1998) provided a comprehensive development and review of the use of the stratified-Petersen estimator, with an application to estimating the number of salmon returning to spawn.
Fitting models like these to real data is challenging because of two common problems:
1. values of nst close to or equal to zero; and
2. rows of the {nst} matrix that are near or exact multiples of each other or, more generally, rows that are exactly or nearly linearly dependent (i.e., some linear combination of the rows sum to zero).
Elements of nst that are exactly equal to 0 are a problem because sampling zeros (values of nst = 0 that occur just by chance) cannot be distinguished from structural zeros (physically impossible movements, such as recoveries occurring before releases in time stratification). In both cases, the maximum likelihood estimate of the movement probability is zero. This is potentially a problem for sampling zeros because there was some movement of the population that, just by chance, was not detected.
Linear dependency among the rows of the {nst} matrix is a problem because the resulting estimators are no longer consistent (Darroch 1961; Banneheka et al. 1997), although linear dependency among the columns of the matrix is not a problem as long as the rank of the matrix is equal to the number of rows. In all cases, the stratified data may have to be modified by either dropping rows or columns, or by pooling two or more rows or columns, and then refitting the model with the new stratification. Schwarz and Taylor (1998) examine in detail the effects of pooling or dropping both rows and columns, and the resulting bias in the estimates. There is not as yet any objective way to determine the optimal pooling of rows or columns.
Example
Fraser River pink salmon (Oncorhynchus gorbuscha) swim up the Fraser River in mid-September to spawn in more than 60 different tributary streams. Here we describe a tag-recovery study where approximately 22,000 female pink salmon were tagged and released between 10 Sep 91 and 8 Oct 91 at Duncan Bar (Glen Valley) downstream of the spawning areas. Tag recovery was done by searching spent carcasses (called dead pitching) found on the stream banks after spawning. Approximately 300,000 female carcasses were dead-pitched from 2 Oct to 12 Nov 1991.
Schwarz and Taylor (1998) examined the recoveries for both sexes from five spawning areas. In this example data on the female releases at Glen Valley and recoveries on the Fraser River Main Stem spawning areas are examined in order to estimate the number of fish passing the tagging site.
The data were too sparse to stratify on a daily basis. Schwarz and Taylor (1998) showed that pooling of recovery strata may be done in a fairly arbitrary fashion without affecting the consistency of the estimates. They found it necessary to stratify releases into 5-day intervals (10–14 Sep 1991, 15–19 Sep 1991, etc.) and recoveries into 6-day intervals (8–13 Oct 1991, 14–19 Oct 1991, etc.). The resulting 6 × 5 matrix of recoveries on the Fraser River Main Stem with summary statistics is shown in table 8.5.
The estimate from the pooled-Petersen method is 7.70 million fish, with a standard error of 0.59 million. However, the contingency table tests for the consistency of the pooled-Petersen method showed some evidence that complete pooling may not be appropriate.
Before, using the stratified-Petersen, it was noticed that no recoveries were obtained from the release group dated 5 Oct 1991 and so this row was deleted. Because of the small number of fish tagged in this release group, these observed zeroes may not be structural zeroes. Fish from this release group may spawn in the Fraser Main Stem, but there may have been too few tagged to detect such spawning. As well, it was necessary to pool the first two and last three rows of the data to obtain nonnegative estimates.
Final results are shown in table 8.6. A chi-squared goodness-of-fit test of the stratified-Petersen model to the pooled data shows some evidence of a lack of fit in two cells, but given the relatively small counts the lack-of-fit is assessed not to be serious. The estimated number of fish to pass the tagging sites in the first five release periods is 8.35 million with an standard error of 0.82 million fish. The pooled-Petersen estimator is 8% less than the stratified estimate.
Discussion
The stratified-Petersen estimator should be used when estimates of the individual strata population sizes are of direct interest. Even if the individual strata are not of interest, it can significantly reduce the bias of a pooled-Petersen estimator when there is substantial variation in and correlation among the initial-capture and final-recapture probabilities. In many escapement studies, the natural pattern of returns induces a positive correlation between these two probabilities and this results in a negative bias in the pooled-Petersen and Schaefer estimates.
However, numerous practical problems often make it difficult to use the stratified-Petersen estimator:
• Recovery data are relatively sparse. Even with the application of over 40,000 tags and pitching over half a million carcasses in the salmon example, the number of tags returned was often small after stratification by sex, time of tagging, spawning ground, and time of recovery.
• Release strata are not completely distinct. In many cases, migration patterns are slowly changing over time and a release stratum consists of fish with migration patterns from the previous and the next stratum. This causes near linear dependencies to exist in the observed and expected recovery matrices.
• The animals from different release strata appear to mix on their way to the recovery strata. This implies that the recovery matrices will not have an upper-triangular structure and allowed linear dependencies to occur.
TABLE 8.6
Detailed results from analyzing female Glen Valley releases recovered in the Fraser Main Stem only—final pooling
In many cases, extensive pooling will be required to obtain admissible estimates, which partially defeats the purpose of stratification.
The pooled-Petersen and stratified-Petersen provide methods at opposite ends of the spectrum with regards to assumptions about capture and movement. The pooled-Petersen assumes that all animals have identical capture and movement probabilities, while the stratified-Petersen lets each release stratum have its own distinct movement pattern. While this may be realistic for geographical stratification, it is, paradoxically, too flexible for temporal stratification. As was seen with the pink-salmon data, migration patterns are likely to change slowly over time, leading to situations where near linear dependencies can occur. Intermediate models have been developed. For example, Schwarz and Dempson (1994) model the movement process of tagged outgoing salmon smolt. This leads to a dramatic decrease in the number of parameters, fewer problem with sparse data, and very precise estimates.
8.4 Multisample Stratified Closed Populations
Heterogeneity in capture probabilities is a well-known problem in closed-population models. The Otis et al. (1978) suite of models (M0, Mt, Mb, Mh, …) allow for heterogeneity in capture probabilities caused by time, behavior, or intrinsic animal characteristics and much modeling effort has been devoted to these models (chapter 2). The motivation for considering multisite, closed-population models is to account for heterogeneity among animals that can be identified with strata. These strata can be quite general (e.g., weight class, geographical location, type of trap).
The survey protocol is similar to that of the AS model. At each of k sampling times, animals are captured, tagged if necessary, and released. At each capture occasion, the stratum of capture (s = 1, … , S) is also recorded. As noted earlier, the key difference in the survey protocols when estimating abundance is that the process by which animals are first tagged must be carefully designed. A key assumption is that untagged animals are a simple random sample from the untagged animals in the population with the same capture probabilities as tagged animals. This experiment also differs from the two-sample case in that the number of strata (S) is the same at all sampling times.
The raw data from the experiment consist of a history vector of length k for each of the captured animals in the same format as in the AS model of section 8.1. The key difference in interpreting the history vector from the interpretation in section 8.1 is that the animal is known to be alive both prior to its initial capture and following its last capture.
The parameters of this model are the total population size over all strata (N), the initial distribution of these animals among the strata (χs), the capture probabilities at each time point 
, the movement probabilities between sampling times j and j + 1 
, and possibly loss-on-capture parameters. The latter three parameter sets have an identical interpretation to those of the AS model. Because the population is closed, there are no survival parameters.
In an experiment with k sample times and S strata, there are a total of (k − 1 )(S2 − S) movement parameters 
, j = 1, . . ., k − 1 and s = 1, . . ., S, but 
for each s and j; kS capture parameters 
, j = 1, …, k and s = 1, … , S; kS loss-on-capture parameters; one population size parameter; and S − 1 parameters for the initial distribution among the strata, with X1 + X2 + … + Xs = 1.
In a similar fashion to section 8.1, the parameters can be used to model the probability of a particular capture history. However, it is more instructive to write out the expected counts for each history, since this explicitly includes the abundance term. For example, assuming only 2 strata, the expected number of animals with the history vector h = (1, 2, 0, 1) is
where N is the initial population size, 
is the probability that the animal started in stratum 1 at time 1, 
is the probability that the animal in stratum 1 at time 1 was captured, 
is the probability that the animal released in stratum 1 at time 1 will be present in stratum 2 at time 2, 
is the probability that the animal in stratum 2 at time 2 will be captured,
is the overall probability that an animal present in stratum 2 at time j = 2 will be in stratum 1 at time 4 (a term that must account for the two possible paths from stratum 2 to stratum 1 between times 2 and 4), and 
is the probability that the animal stratum 1 at time 4 was captured.
Note that after the first capture of an animal, the subsequent modeling for the capture history is identical to the AS model. The complexities in writing out the expressions for each capture history are the same as in the AS model, but with the additional complexity that the all possible movements from time 1 to the first capture must be included because it is known that the animal is alive from the start of the experiment. As well, it is also possible to write out the probability for the animals never seen in the experiment.
The likelihood is constructed as the product of probabilities—one for each animal captured and one for the animals never captured but known to be alive. All parameters are identifiable, though some may be non-estimable in cases with sparse data. No closed-form expressions exist for the maximum likelihood estimators, but Schwarz and Ganter (1995) have written a SAS program to compute the values.
The model makes a number of assumptions in addition to those made for the CJS model:
• all data are recorded without error with respect to stratum of capture;
• there are homogeneous transition probabilities for all animals in stratum s at time j; and
• there is a lack of memory, i.e., the transition probability and recapture probabilities do not depend on the past history of the animal.
The models can also be generalized to multiple groups or simplified using models with survival or capture rates that are constant across groups or time. Notation for these models can be generalized from the previous chapter. Model selection among a set of candidate models for the data may be based on the Akaike information criterion (chapter 1).
Example
During the spring migration of a major population of barnacle geese (Branta leucopsis), the birds stage (to prepare for migration) in five areas of the Schleswig-Holstein region of northern Germany for several weeks. Numerous researchers have banded many geese on their breeding and wintering areas using colored leg bands that are visible during the staging period. There is little mortality during the staging period. Consequently, the population is nearly closed during this period (Ganter 1995).
From mid-January to early April, members of a research team visited the five areas. During each visit, team members recorded the band number of any banded goose spotted. For reasons outlined in Schwarz and Ganter (1995), the data were grouped into five two-week intervals with interval 1 starting 30 Jan 1990. The capture histories for this experiment are given in table 8.7.
Even after grouping, areas 1 and 5 had no sighting effort in some intervals. Hence, the full model cannot be used, and only models assuming constant transition probabilities over time can be considered (Schwarz and Ganter 1995). The model {ps*t, ψs} was considered for the complete data. Information on movement from area 5 is obtained from releases at time 2 while information on movement from area 1 is obtained from releases at times 3 and 5. A summary of the fitted model is shown in table 8.8. The goodness-of-fit test failed to reject this model and residual plots also showed few problems. A simpler model where resighting rates were constant over time, {ps, ψs}, was strongly rejected although results are not shown. This is again not surprising given the wide range in the number of visits made to the areas. Another simpler model, {pf(visits), ψs}, where the sighting probabilities increase with the number of visits made to an area, was also examined, but had little support. The final estimates are shown in table 8.9. These indicated a high fidelity to area 2 and 5, but substantial movement among areas 1-2, 3-2, 3-5, 4-2, and 4-5. The relatively strong fidelity to sites 2 and 5 may be due to the physical conditions of the sites as described in Ganter (1995).
TABLE 8.7
Capture histories for the Barnacle goose experiment
TABLE 8.8
Results from multistate multisample closed-model fitting to all areas for the Barnacle goose data
Discussion
Many studies of animal movement have been fragmented (e.g., effort has not been expended simultaneously in all areas in all times). This implies that many general models cannot be fitted because of nonidentifiability of the parameters. However, in some cases, subsets of the data are more uniform. An analysis of a subset of the data may provide some information about likely models that may be appropriate for a population. Under a somewhat restrictive model of equal movement rates over time, even fragmented studies can yield important information.
Finally, as noted by Brownie et al. (1993), multisite models allow the modeling of heterogeneity in capture and survival rates. For example, model Mb of Otis et al. (1978) assumes that once an animal is captured, its subsequent capture probabilities remain fixed. An even richer class of models can be fitted if we allow the behavior modification to be probabilistic in nature by defining two strata corresponding to behaving as if not captured and behavior modified because of capture. Movement parameters can be defined to model the fraction of animals that forget they have been captured in each time period. This would allow the full capture history of the animal to be used, rather than discarding all subsequent recaptures after the first as is done in model Mb of Otis et al. (1978).
8.5 Multisample Stratified Open Populations
This is a generalization of the Jolly-Seber model to the case of multistrata. Again, the motivation for this model is the estimation of movement and to account for heterogeneity in the capture probabilities among the different strata. This was first considered in detail by Arnason (1972, 1973) who developed moment estimates for the case of three sample times. Schwarz et al. (1993b) developed moment estimates for the general k-sample case. Surprisingly, very little additional theoretical work has been done on this model, presumably because of the very exacting experimental protocols required.
Fortunately, the results of earlier sections can be used for a large part of the analysis. As in the ordinary Jolly-Seber model, the probability for each capture history will consist of components describing the first capture (which is used to estimated abundance), and then the subsequent recapture history of marked animals. The latter will be identical to that in the AS model of movement discussed in section 8.1. The first component is likely to be very complicated and strategies similar to that used by Schwarz and Arnason (1996) and Schwarz and Ganter (1995) will likely be necessary to describe new animals entering the population using a complex multinomial distribution.
However, based on experience with abundance models, a close approximation to the fully maximized likelihood can be obtained by fitting the AS model of section 8.1 to the recapture data only. Then the abundance at each time point can be estimated using the moment equations
where the subscripts refer to the estimate in stratum s at time j. As in the simple Jolly-Seber model, estimates of recruitment to each stratum can be obtained from the moment estimates
where losses on capture have been ignored. As in the Jolly-Seber model, this estimator may result in estimates of “births” less than zero, suggesting shortcomings in this approach to evaluate recruitment. Because of the large data requirements and strict protocols required for abundance estimation, there are few (if any) examples in the literature. Dupuis and Schwarz (in preparation) have developed a Bayesian approach to estimate abundance, survival, and movement of walleye in Mille Lac, Minnesota.
TABLE 8.9
Estimates from model {ps*t, ψs} (equal movement over time, unequal resighting rates over time) for the Barnacle goose example
Figure 8.2. Radio-collared polar bear (Ursus maritimus) and new cub on the sea ice north of Kaktovik, Alaska, April 1986. (Photo by Steven C. Amstrup)
8.6 Chapter Summary
• Motivation is provided for considering models where animals are in several groups (such as geographical areas), and these groups change with time.
• The Arnason-Schwarz model that allows animals to move between states is reviewed, including the earlier work that led to this model, the data collection procedures, and model fitting and selection. The model is illustrated using some data on the captures and recaptures of Canada geese.
• Applications of the model are described with live and dead recoveries of animals, the recruitment of breeders, and covariates that change with time.
• The Arnason-Schwarz model does not allow the direct estimation of population sizes. Models of the Jolly-Seber type that do allow this estimation are discussed, starting with the two-sample, closed-population case, which is illustrated by an example on the recaptures of male and female pink salmon from five spawning areas.
• The multisample closed-population case is discussed, with an example involving the movement of barnacle geese.
• There is a brief discussion of the multisample case for an open population. It is recommended that for the estimation of population sizes the Arnason-Schwarz model should be fitted to the recapture data, and then reciprocals of estimated capture probabilities can be used to estimate the sizes.
This work was supported by a Research Grant from the Canadian Natural Science and Engineering Research Council (NSERC).