Parameter-expanded data augmentation for Bayesian analysis of capture–recapture models
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DOI: 10.1007/s10336-010-0619-4
- Cite this article as:
- Royle, J.A. & Dorazio, R.M. J Ornithol (2012) 152(Suppl 2): 521. doi:10.1007/s10336-010-0619-4
Abstract
Data augmentation (DA) is a flexible tool for analyzing closed and open population models of capture–recapture data, especially models which include sources of hetereogeneity among individuals. The essential concept underlying DA, as we use the term, is based on adding “observations” to create a dataset composed of a known number of individuals. This new (augmented) dataset, which includes the unknown number of individuals N in the population, is then analyzed using a new model that includes a reformulation of the parameter N in the conventional model of the observed (unaugmented) data. In the context of capture–recapture models, we add a set of “all zero” encounter histories which are not, in practice, observable. The model of the augmented dataset is a zero-inflated version of either a binomial or a multinomial base model. Thus, our use of DA provides a general approach for analyzing both closed and open population models of all types. In doing so, this approach provides a unified framework for the analysis of a huge range of models that are treated as unrelated “black boxes” and named procedures in the classical literature. As a practical matter, analysis of the augmented dataset by MCMC is greatly simplified compared to other methods that require specialized algorithms. For example, complex capture–recapture models of an augmented dataset can be fitted with popular MCMC software packages (WinBUGS or JAGS) by providing a concise statement of the model’s assumptions that usually involves only a few lines of pseudocode. In this paper, we review the basic technical concepts of data augmentation, and we provide examples of analyses of closed-population models (M_{0}, M_{h}, distance sampling, and spatial capture–recapture models) and open-population models (Jolly–Seber) with individual effects.
Keywords
Hierarchical models Individual covariates Individual heterogeneity Markov chain Monte Carlo Occupancy modelsIntroduction
Capture–recapture models with individual effects have received considerable attention in recent years. Much of this attention has focused on closed-population models, but the basic ideas have been applied in a limited sense to certain open-population models. As a working definition, models with “individual effects” are models in which parameters are functions of something that varies by individual. Several types of individual effects arise in practice. An important class of models includes those with latent (unobservable) sources of individual heterogeneity. So-called “Model M_{h}” is a standard closed-population model (e.g., Burnham and Overton 1978; Dorazio and Royle 2003; Link 2003; Pledger 2005), and similar ideas have been applied to open populations (Pledger et al. 2003; Gimenez et al. 2007; Royle 2009; Gimenez and Choquet 2010). Another useful class of models includes the individual covariate models, in which covariates are measured on each individual in the sample. Various extensions of these models accommodate imperfect information about the covariate. These include models with measurement error, time-varying covariates (Bonner and Schwarz 2006; King et al. 2008), and spatial capture–recapture (e.g., Borchers and Efford 2008; Royle and Young 2008; Royle et al. 2009). Classical “multi-state” models are also individual effects models, with a state variable that is a categorical covariate.
In this paper, we describe a general method for analyzing capture–recapture models with individual effects that uses data augmentation and model expansion to solve the problem of a variable-dimension parameter space. In this approach the observed dataset is augmented with an arbitrarily large number of all-zero encounter histories. The model of this augmented dataset is a zero-inflated version of the conventional, multinomial model for the observed data (Royle et al. 2007). Importantly, this new model may conveniently be fitted using WinBUGS, OpenBUGS (Lunn et al. 2009) or other MCMC-computing engines, such as PyMC (Patil et al. 2010). This approach, which was developed by Royle et al. (2007) and extended by Royle and Dorazio (2008), solves the problem of a variable-dimension parameter space in some generality across classes of capture–recapture models, wherein N is the object of inference (or otherwise important). The relative ease of implementation of our approach is important because the development of MCMC algorithms represents a distraction from the main task at hand (model development) and should never be the focus of effort for most researchers.
In the following sections, we review the technical formulation of data augmentation and illustrate its use in fitting closed-population models, including Models M_{0} and M_{h}, distance sampling, and spatially explicit capture–recapture models. We then describe how data augmentation may be used in the analysis of open populations. Along the way, we discuss the duality between models for estimation of population size and so-called “occupancy models.” In particular, the Jolly–Seber (JS) type model under data augmentation is precisely equivalent to a type of metapopulation “occupancy” model that allows for local extinction and colonization. Specifically, the JS model is a constrained occupancy model wherein an individual cannot be recruited once it has died (whereas in metapopulation type models, sites or patches can be recolonized).
Parameter-expanded data augmentation (PX-DA)
The general concept behind data augmentation is to augment the observed data with “latent data” with the intent of simplifying the analysis, usually by MCMC. Here, latent data may include missing observations, parameter values, or values of sufficient statistics (Tanner and Wong 1987; Tanner 1996). Royle et al. (2007; henceforth RDL) developed a general form of DA to simplify the analysis of capture–recapture models wherein N, the multinomial sample size or index parameter, is unknown. RDL’s development was motivated by the analysis of models with individual effects in which the dimension of the parameter space (i.e., the number of unknowns) is itself unknown. RDL’s idea was to fix the dimension of the parameter space by embedding the “complete data” (wherein N is known) into a larger dataset of fixed dimension and to analyze this larger, augmented dataset with a new model, one which provides a reparameterization of N in the conventional model of the observed (unaugmented) data. This new model of the augmented data is a zero-inflated version of the conventional known-N model and is easily fitted using standard methods of MCMC sampling (e.g., Gibbs sampling). In many cases, even classical (i.e., non-Bayesian) methods may be used to fit the model of augmented data.
Because inference is based on a new model that results from augmenting the data, RDL’s proposal is not simply data augmentation. Rather, it is what Liu and Wu (1999) refer to more formally as parameter-expanded data augmentation (PX-DA), because the conventional model of the complete data must be expanded to account for—and to estimate—the number of non-sampling zeros in the augmented data. Some basic conditions must be satisfied for the model expansion to be innocuous with respect to the inference problem (Liu and Wu 1999). In the Appendix, we show that RDL’s expanded model of the augmented data satisfies these conditions.
We use the term PX-DA in this paper to clarify the methodological context of RDL and also distinguish it from other uses of the term “data augmentation” in capture–recapture models, such as Schofield and Barker (2008) and Wright et al. (2009), in which the dataset is augmented with all-zero encounter histories, but not in a manner that yields a new model upon which inference is based.
Computational benefits
An important benefit of RDL’s use of PX-DA is that it solves the problem of a variable dimension parameter space in some generality across classes of capture–recapture models. No specialized algorithms, such as reversible-jump MCMC (Green 1995), are needed to fit these models because the dimension of the parameter space is fixed. Furthermore, by augmenting the data with all-zero capture histories, we eliminate the need to evaluate π(0), the probability that an individual in the population is not captured in any sample. In complex capture–recapture models (e.g., those with individual effects), π(0) can be difficult to calculate because it is a function of individual attributes such as distance, age, sex, or other things that influence detection or survival probabilities. These attributes are unobserved for the missing component of the population (that is, for the uncaptured individuals), which leads to what statisticians refer to as nonignorable missingness. The adjective, nonignorable, implies that the probability that an observation is “missing” depends on what is missing. A good example is distance sampling—observations far away from the observer are less likely to appear in the dataset. Thus, individuals that are not in the sample (and their distances) will tend to be farther away than those in the sample. It is certainly possible to conduct a Bayesian analysis of capture–recapture data without DA; however, such analyses require explicit calculation of π(0), which is generally difficult in complex models. See Dorazio and Royle (2005) for an especially challenging problem wherein π(0) is computed.
PX-DA versus RJ-MCMC
Data augmentation can be used alone (that is, without model expansion) to conduct a Bayesian analysis of complex, capture–recapture data. In this approach, capture histories of the n observed individuals are augmented with only N − n all-zero capture histories (and possibly with N individual-level parameters) to account for unobserved individuals in the population. While this approach avoids the difficult calculation of π(0), it also requires a specialized algorithm, such as reversible-jump MCMC (RJ-MCMC), to solve the problem of the variable-dimension parameter space because N is an unknown parameter in the model. There are several examples of capture–recapture analyses using RJ-MCMC (King and Brooks 2008; Schofield and Barker 2008; Wright et al. (2009).
A practical impediment to applications of RJ-MCMC is that implementations are model specific. For every set of candidate models, a set of full conditionals and proposal distributions must be derived for “jumping” among models. In the context of capture–recapture models, the RJ-MCMC algorithm must specify how to add or subtract parameters when different values of N (vis-a-vis different models) are visited by the sampler. To our knowledge, RJ-MCMC algorithms suitable for capture–recapture models have not been implemented in popular software programs for Bayesian analysis. Therefore, applications of RJ-MCMC are currently inaccessible to many ecologists because the details of implementation require a level of statistical and computational expertise not common among them. In contrast, applications of PX-DA are relatively simple to implement in popular software programs, such as WinBUGS or JAGS (Royle and Dorazio 2008; Link and Barker 2010). We provide several examples of such applications in later sections of this paper.
Link and Barker (2010, p. 210) assert that RDL’s use of data augmentation (i.e., PX-DA) “is equivalent to a reversible jump algorithm \(\ldots\) with a discrete uniform prior for N.” A referee also suggested that PX-DA is nothing more than a specific manifestation of the RJ-MCMC algorithm. However, our development of PX-DA for use in capture–recapture problems did not originate from considerations of RJ-MCMC (Royle et al. 2007). RJ-MCMC refers to a specific class of MCMC algorithms that are designed to analyze models with variable-dimension parameter spaces (Green 1995). Conversely, PX-DA is not an algorithm for doing MCMC. Rather, it is a reformulation of the model intended to simplify analysis by providing a straightforward, conventional implementation of MCMC. In our case, capture–recapture models reformulated by PX-DA can be analyzed seamlessly using Gibbs sampling directly from full-conditional distributions or using the Metropolis–Hastings algorithm with default proposal distributions. Consequently, PX-DA is useful because of the hierarchical construction of the model that is produced and the resulting simplicity of MCMC implementations for that model, not because it corresponds to an implementation of RJ-MCMC. Furthermore, in many cases, the expanded model of the augmented data is also relatively easy to fit using classical methods, such as maximum likelihood.
To summarize, we use PX-DA to facilitate Bayesian analysis of various classes of capture–recapture models in which N is unknown and is important to inference. Other investigators have also proposed DA for fitting these models, but their analyses are made conditional on N and therefore require RJ-MCMC to deal with the problem of a variable-dimension parameter space.
PX-DA for model M_{0}
An augmented dataset with n = 6 observed individuals and J = 5 samples
indiv i | Sample occasion | y_{i} | z_{i} | ||||
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |||
1 | 1 | 0 | 0 | 1 | 0 | 2 | 1 |
2 | 0 | 1 | 0 | 0 | 1 | 2 | 1 |
3 | 1 | 0 | 0 | 1 | 0 | 2 | 1 |
4 | 1 | 0 | 1 | 0 | 1 | 3 | 1 |
5 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
n = 6 | 1 | 0 | 0 | 0 | 0 | 1 | 1 |
n + 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
\(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) |
N | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
N + 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) |
M | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
We emphasize that M is fixed in the new model of the augmented data. As a result, MCMC is a relatively simple proposition using standard Gibbs Sampling. We have to update {z_{i}}, p, and ψ. Updating each z_{i} parameter is easily done by computing a random draw from a Bernoulli distribution (Royle et al. 2007). Moreover, this is true for every class of capture–recapture models. Thus, PX-DA casts every capture–recapture model as a zero-inflated version of the original multinomial model, which is the main reason for the versatility of PX-DA. In contrast, analysis by RJ-MCMC is based on analyzing the complete-data likelihood (i.e., the original multinomial model) associated with an augmented set of “complete data” whose size N is unknown (King and Brooks 2008; Schofield and Barker 2008). In PX-DA, these complete data are included in a larger dataset of known size M and an entirely different model—the zero-inflated multinomial—is therefore needed in the analysis.
Model M_{0} in BUGS
Model M_{0} described in the BUGS model specification language
p ~ dunif (0,1) |
psi ~ dunif (0,1) |
# nind = number of individuals captured at least once |
# nz = number of uncaptured individuals added for PX-DA |
for(i in 1: (nind+nz)) { |
z[i] ~ dbern (psi) |
mu[i] ← z[i]*p |
y[i] ~ dbin (mu[i],J) |
} |
N ← sum(z[1:(nind+nz)]) |
Example: tiger data
Heuristic development of PX-DA based on occupancy models
There is a formal duality between “occupancy” models and closed-population models that provides a nice heuristic motivation for PX-DA. This duality was developed in Royle et al. (2007) and in Royle and Dorazio (2008) for both closed- and open-population models. The basic idea originates from J.D. Nichols (see Nichols and Karanth 2002).
Hypothetical occupancy dataset (left), capture–recapture data in standard form (center), and capture–recapture data augmented with all-zero capture histories (right)
Occupancy data | Capture–recapture | Augmented C-R | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Site | k = 1 | k = 2 | k = 3 | Ind | k = 1 | k = 2 | k = 3 | Ind | k = 1 | k = 2 | k = 3 |
1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
2 | 1 | 0 | 1 | 2 | 1 | 0 | 1 | 2 | 1 | 0 | 1 |
3 | 0 | 1 | 0 | . | 0 | 1 | 0 | 3 | 1 | 0 | 1 |
4 | 1 | 0 | 1 | . | 1 | 0 | 1 | 4 | 1 | 0 | 1 |
5 | 0 | 1 | 1 | . | 0 | 1 | 1 | 5 | 1 | 0 | 1 |
. | 0 | 1 | 1 | . | 0 | 1 | 1 | . | 0 | 1 | 1 |
. | 1 | 1 | 1 | . | 1 | 1 | 1 | . | 0 | 1 | 1 |
. | 1 | 1 | 1 | . | 1 | 1 | 1 | . | 1 | 1 | 1 |
1 | 1 | 1 | . | 1 | 1 | 1 | . | 1 | 1 | 1 | |
n | 1 | 1 | 1 | n | 1 | 1 | 1 | n | 1 | 1 | 1 |
. | 0 | 0 | 0 | . | 0 | 0 | 0 | ||||
. | 0 | 0 | 0 | . | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | 0 | ||||||
0 | 0 | 0 | 0 | 0 | 0 | ||||||
0 | 0 | 0 | 0 | 0 | 0 | ||||||
0 | 0 | 0 | N | 0 | 0 | 0 | |||||
. | 0 | 0 | 0 | . | 0 | 0 | 0 | ||||
. | 0 | 0 | 0 | 0 | 0 | 0 | |||||
M | 0 | 0 | 0 | . | 0 | 0 | 0 | ||||
. | . | . | . | ||||||||
. | . | . | . | ||||||||
. | . | . | . | ||||||||
M | 0 | 0 | 0 |
Closed-population models
PX-DA is useful primarily for its generality and extensibility across different classes of models with unknown N. The main MCMC problem is focused on imputing the missing z values as a step in the MCMC algorithm. For example, in closed population models with individual effects the MCMC algorithm will typically involve 2 main steps: (1) imputing z_{i} for each \(i=1,2,\ldots,M\) and then (2) imputing any of the missing individual effects, say x_{i}, which can be achieved easily in most cases using simple Gibbs sampling or the Metropolis–Hastings algorithm. In addition to these two steps, we require updates of the basic structural parameters of the model (e.g., hyper-parameters). An important practical benefit of PX-DA is that it allows one to formulate and fit a large variety of models using WinBUGS and other MCMC computing engines. Royle and Dorazio (2008) provided several illustrations of data augmentation applied to closed populations, and some of those examples are repeated here.
Model M_{h}: individual heterogeneity
Model M_{h} assuming that \(\hbox{logit}(p)\) has a normal distribution
p0 ~ dunif (0,1) # prior distributions |
mup ← log (p0/(1-p0)) |
taup ~ dgamma (.1,.1) |
psi ~ dunif (0,1) |
for(i in 1: (nind+nz)){ |
z[i] ~ dbern (psi) # zero inflation variables |
lp[i] ~ dnorm (mup,taup) # individual effect |
logit(p[i]) ← lp[i] |
mu[i] ← z[i]*p[i] |
y[i] ~ dbin (mu[i],J) # observation model |
} |
N ← sum(z[1: (nind+nz)]) |
Parameter estimates for Model M_{h} (logit-normal model for p) fitted to the Nagarahole tiger camera-trapping data
Parameter | Mean | SD | 2.5% | Median | 97.5% |
---|---|---|---|---|---|
N | 113.641 | 48.280 | 65.000 | 99.000 | 248.000 |
μ | −3.175 | 0.597 | −4.673 | −3.021 | −2.393 |
p_{0} | 0.046 | 0.020 | 0.009 | 0.046 | 0.084 |
ψ | 0.154 | 0.066 | 0.082 | 0.135 | 0.334 |
σ | 0.673 | 0.342 | 0.245 | 0.580 | 1.484 |
Spatial capture–recapture
In Bayesian analysis of these models, it is necessary to prescribe \(\mathcal{S}\) because the analysis proceeds by simulating realizations of \(s_{1},\ldots,s_{N}\) from the required posterior distribution. Individuals have to reside somewhere. In developing the problem this way, N is sensitive to the choice of \(\mathcal{S}\)—as its area increases, so too does N and vice versa. However, as \(\mathcal{S}\) becomes large, then p(s_{i}) diminishes to zero rapidly (under well-behaved models), and additional increases in \(\mathcal{S}\) are inconsequential. In particular, N will continue to increase, but density will become invariant to the size of \(\mathcal{S}\), which is a consequence of the model (which implies a constant density of individuals). This same phenomenon is relevant to the likelihood-based analyses of Borchers and Efford (2008). In particular, they parameterize the model in terms of density, and observe that it is invariant as long as integration occurs over a sufficiently large state-space.
Description of spatial capture–recapture model in the BUGS language
a ~ dnorm(0,.1) |
b ~ dunif(0,10) |
psi ~ dunif(0,1) |
for(i in 1:(nind+nz)){ |
z[i] ~ dbern(psi) |
sx[i] ~ dunif(Xl,Xu) |
sy[i] ~ dunif(Yl,Yu) |
for(j in 1:ntraps){ |
dist2[i,j] ← (pow(sx[i]-grid[j,1],2)+pow(sy[i]-grid[j,2],2)) |
cloglog(p[i,j]) ← a - b*dist2[i,j] |
muy[i,j] ← z[i]*p[i,j] |
y[i,j] ~ dbin(muy[i,j],12) |
} |
} |
N ← sum(z[1:(nind+nz)]) |
D ← N/area |
Parameter estimates for the spatial capture–recapture model fitted to the Nagarahole tiger camera trapping data
Parameter | Mean | SD | 2.5% | 50% | 97.5% |
---|---|---|---|---|---|
N | 185.377 | 36.198 | 128.000 | 181.000 | 271.000 |
D | 12.771 | 2.494 | 8.818 | 12.469 | 18.669 |
ψ | 0.418 | 0.084 | 0.280 | 0.408 | 0.614 |
σ | 0.336 | 0.084 | 0.214 | 0.322 | 0.538 |
λ_{0} | 0.015 | 0.004 | 0.008 | 0.015 | 0.025 |
We close this section by summarizing our analyses of the tiger data. We expect that the estimator of N under model M_{0} is biased in the presence of heterogeneity in detection probability, which we know must be present given the spatial context and thus one that is not even sensible to formally test (Johnson 1999). We might approximate the existing heterogeneity using a standard model M_{h} (but see Link 2003). The basic problem with that approach is that we do not know the area for which the parameter N of that model applies. Finally, we can use a formal spatial model—a closed population model, extended hierarchically with a point process model to describe the juxtaposition of individuals with traps thereby accounting implicitly for heterogeneity induced by the spatial organization of traps and individuals. This model fixes the state-space of the underlying point process, so that the area over which the N individuals inhabit is well defined. The spatial model then provides an estimate of N that applies to that prescribed state-space, and hence density can be computed. In addition, the number or density of individuals in any subset of \(\mathcal{S}\) can be obtained by summarizing the posterior samples of the individual activity centers.
Individual-covariate models and distance sampling
Examples of individual-covariate models can be found in Royle and Dorazio (2008, Chap. 6), including a standard closed population with the individual covariate, “body mass”, which is thought to influence detectability. Other examples are provided by models of aerial survey data wherein detection probabilities of observers are specified as a function of the individual covariate, “group size” (Royle and Dorazio 2008; Royle 2009; Langtimm et al. 2010).
Distance sampling model in WinBUGS, using a “half-normal” detection function
b ~ dunif (0,10) |
psi ~ dunif (0,1) |
for(i in 1: (nind+nz)){ |
z[i] ~ dbern (psi) # DA Variables |
x[i] ~ dunif (0,B) # B=strip width |
p[i] ← exp(logp[i]) # DETECTION MODEL |
logp[i] ← ((x[i]*x[i])*b) |
mu[i] ← z[i]*p[i] |
y[i] ~ dbern(mu[i]) # OBSERVATION MODEL |
} |
N ← sum(z[1: (nind+nz)]) |
D ← N/strip area # area of transects |
Open-population models
Open populations are susceptible to the demographic processes of mortality and recruitment over time and thus are parameterized in terms of survival probabilities ϕ and various parameters that describe recruitment. Cooch and White (2001) recognize at least 5 distinct parameterizations of the recruitment process, which we will not summarize here. We refer to models for open populations as JS-type models to encompass any possible parameterization of the basic demographic processes, following the initial technical developments of Jolly (1965) and Seber (1965).
Other recent implementations of JS models exist that also condition on N (Dupuis and Schwarz 2007; Schofield and Barker 2008). Unlike these implementations, we use PX-DA to fix the dimension of the parameter space, and analyze a model that is unconditional on N. Conversely, Dupuis and Schwarz (2007) and Schofield and Barker (2008) retain N in the model and devise specific MCMC algorithms to solve the problem of a variable-dimension parameter space. Both authors use DA to formulate the model for missing state variables but not to fix the dimension of the parameter space. That is, they augment up to N, not M [see “Parameter-expanded data augmentation (PX-DA)”]. Durban and Elston (2005) devised an MCMC implementation for closed capture–recapture models in which they also fixed the dimension of the dataset by adding a collection of all-zero encounter histories. However, they did not exploit the fact that the model for the augmented data can be simplified to yield a convenient MCMC implementation. Finally, the analysis of open populations presented by Link and Barker (2010, Chap. 11, p. 263) uses PX-DA and exploits the resulting model for the augmented data as in Royle and Dorazio (2008, Chap. 10) and described subsequently.
Data structure
JS datasets under the Robust design with T = 6 years and J = 5 secondary samples
Hierarchical model development
In open-population models, two distinct processes combine to produce individual encounter histories: (1) the detection/encounter process, which describes how individuals appear in the sample, and (2) population dynamics (survival/recruitment), which dictate when individuals can appear in the sample. In what follows, we provide the hierarchical formulation of the JS-type model conditional-on-N that was given in Royle and Dorazio (2008, Chap. 10). This formulation begins by defining the two random variables. One is y(i, t), the observed number of captures of individual i out of J samples in year t. The other is a latent state variable, z(i, t), for the “alive state” of individual i in year t (i.e., z = 1 if alive; 0 of not alive).
Unknown N
Definition of a basic Jolly-Seber type model with fixed super-population size, N_{super}, and “conditional entrance probabilities” (γ_{t})
for (t in 1:T){ |
phi[t] ~ dunif (0,1) |
p[t] ~ dunif (0,1) |
gamma[t] ~ dunif (0,1) |
N[t] ← sum (z[1:Nsuper,t]) |
} |
for (i in 1:Nsuper){ |
z[i,1] ~ dbin (gamma[1],1) |
mu[i] ← z[i,1]*p[1] |
y[i,1] ~ dbin (mu[i],J[1]) |
r[i,1] ← 1 |
for(t in 2:T){ |
survived[i,t] ← phi[t]*z[i,t-1] |
r[i,t] ← r[i,(t-1)]*(1-z[i,t-1]) |
muz[i,t] ← survived[i,t] + gamma[t]*r[i,t] |
z[i,t] ~ dbin (muz[i,t],1) |
muy[i,t] ← z[i,t]*p[t] |
y[i,t] ~ dbin (muy[i,t],J[t]) |
} |
} |
Definition of a basic Jolly-Seber type model with unknown super-population size, N_{super}, using PX-DA
for (t in 1:T){ |
phi[t] ~ dunif (0,1) |
p[t] ~ dunif (0,1) |
g[t] ~ dunif (0,1) |
N[i] ← sum (z[1:M,i]) |
} |
for (i in 1:M){ |
z[i,1] ~ dbin (g[1],1) |
mu[i] ← z[i,1]*p[1] |
y[i,1] ~ dbin (mu[i],J[1]) |
r[i,1] ← 1 |
for (t in 2:T){ |
survived[i,t] ← phi[t]*z[i,t-1] |
r[i,t]<- r[i,(t-1)] * (1-z[i,t-1]) |
muz[i,t] ← survived[i,t] + g[t]*r[i,t] |
z[i,t] ~ dbin (muz[i,t],1) |
muy[i,t] ← z[i,t]*p[t] |
y[i,t] ~ dbin (muy[i,t],J[t]) |
} |
} |
for (i in 1:M){ # Compute Nsuper |
Nind[i] ← sum(z[i,1:T]) |
Nalive[i] ← 1-equals(Nind[i],0) |
} |
Nsuper ← sum(Nalive[1:M]) |
Example: Microtus data
for (i in 1:m){ |
eta[i] ~ dnorm(0,tauphi) |
for(t in 2:T){ |
... |
logit(phi[i,t]) ← phiyr[t]+eta[i] # yr + individual |
survived[i,t] ← phi[i,t]*z[i,t-1] |
... |
} |
} |
Estimates of open-population model parameters for Microtus data under models with (left half of table) and without individual heterogeneity on ϕ
| Heterogeneity (survival) model | Ordinary JS model (SA) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Mean | SD | 2.5% | 50% | 97.5% | Mean | SD | 2.5% | 50% | 97.5% | |
N_{1} | 56.37 | 0.62 | 56.00 | 56.00 | 58.00 | 56.35 | 0.62 | 56.00 | 56.00 | 58.00 |
N_{2} | 75.81 | 1.88 | 73.00 | 76.00 | 80.00 | 77.97 | 2.21 | 74.00 | 78.00 | 83.00 |
N_{3} | 55.47 | 1.33 | 54.00 | 55.00 | 59.00 | 56.98 | 1.82 | 54.00 | 57.00 | 61.00 |
N_{4} | 60.08 | 1.12 | 59.00 | 60.00 | 63.00 | 61.10 | 1.48 | 59.00 | 61.00 | 64.00 |
N_{5} | 51.34 | 0.60 | 51.00 | 51.00 | 53.00 | 51.84 | 0.90 | 51.00 | 52.00 | 54.00 |
N_{6} | 78.99 | 1.47 | 77.00 | 79.00 | 82.00 | 80.03 | 1.72 | 77.00 | 80.00 | 84.00 |
N_{super} | 173.72 | 1.73 | 171.00 | 174.00 | 178.00 | 173.44 | 1.63 | 171.00 | 173.00 | 177.00 |
p_{1} | 0.64 | 0.03 | 0.58 | 0.64 | 0.70 | 0.64 | 0.03 | 0.58 | 0.64 | 0.70 |
p_{2} | 0.44 | 0.03 | 0.39 | 0.44 | 0.50 | 0.44 | 0.03 | 0.38 | 0.44 | 0.49 |
p_{3} | 0.44 | 0.03 | 0.38 | 0.44 | 0.50 | 0.44 | 0.03 | 0.37 | 0.43 | 0.50 |
p_{4} | 0.51 | 0.03 | 0.45 | 0.51 | 0.57 | 0.51 | 0.03 | 0.45 | 0.51 | 0.57 |
p_{5} | 0.57 | 0.03 | 0.51 | 0.57 | 0.63 | 0.57 | 0.03 | 0.51 | 0.57 | 0.63 |
p_{6} | 0.53 | 0.03 | 0.48 | 0.53 | 0.58 | 0.53 | 0.03 | 0.48 | 0.53 | 0.58 |
ϕ_{1} | 0.96 | 0.03 | 0.87 | 0.96 | 0.99 | 0.85 | 0.05 | 0.73 | 0.85 | 0.94 |
ϕ_{2} | 0.50 | 0.10 | 0.30 | 0.50 | 0.70 | 0.54 | 0.06 | 0.42 | 0.54 | 0.65 |
ϕ_{3} | 0.56 | 0.12 | 0.31 | 0.57 | 0.77 | 0.70 | 0.06 | 0.57 | 0.70 | 0.82 |
ϕ_{4} | 0.33 | 0.12 | 0.12 | 0.33 | 0.57 | 0.57 | 0.06 | 0.44 | 0.57 | 0.69 |
ϕ_{5} | 0.79 | 0.11 | 0.53 | 0.81 | 0.96 | 0.86 | 0.05 | 0.75 | 0.87 | 0.95 |
σ | 2.57 | 0.71 | 1.41 | 2.49 | 4.38 | |||||
γ_{1} | 0.15 | 0.02 | 0.12 | 0.15 | 0.19 | 0.21 | 0.03 | 0.16 | 0.21 | 0.26 |
γ_{2} | 0.10 | 0.02 | 0.06 | 0.10 | 0.13 | 0.14 | 0.02 | 0.10 | 0.14 | 0.19 |
γ_{3} | 0.06 | 0.02 | 0.03 | 0.06 | 0.09 | 0.09 | 0.02 | 0.05 | 0.09 | 0.13 |
γ_{4} | 0.08 | 0.02 | 0.05 | 0.08 | 0.12 | 0.13 | 0.03 | 0.08 | 0.13 | 0.18 |
γ_{5} | 0.07 | 0.02 | 0.04 | 0.07 | 0.11 | 0.12 | 0.03 | 0.07 | 0.12 | 0.18 |
γ_{6} | 0.15 | 0.02 | 0.11 | 0.15 | 0.20 | 0.27 | 0.04 | 0.19 | 0.26 | 0.35 |
Relationship to models of occupancy dynamics
Jolly-Seber models under PX-DA are multi-state models
Under PX-DA, “individuals” in the augmented dataset belong to one of three types: (1) alive; (2) previously alive but now dead; and (3) have not previously been alive. This is easily described using conventional “multi-state” models with 3 states, and where state 2 is an absorbing state. Clearly, this formulation is equivalent to a multi-state occupancy model as well and provides an alternative and direct implementation in the BUGS language (but one which we omit here).
Summary and conclusions
We have shown that PX-DA provides a flexible tool for analyzing closed- and open-population models with individual effects, such as random effects or covariate effects. The utility of PX-DA has been established in many different contexts, and the method has been used to solve problems of extraordinary complexity, for which solutions were unimaginable only a few years ago. For example, PX-DA effectively renders spatial capture–recapture models as ordinary individual-effects models, and PX-DA has been proposed for analysis of both closed (Royle and Young 2008; Royle et al. 2009) and open (Gardner et al. 2010b) spatial capture–recapture models. PX-DA has also been used in the formulation and analysis of metacommunity models based on species-level occurrence models (Dorazio et al. 2006; Kéry and Royle 2009) and to extend those models for open metacommunities (Kéry et al. 2009; Dorazio et al. 2010).
Our use of PX-DA in capture–recapture problems is not without limitations. For example, N is removed as a formal parameter of the model by marginalizing over a specific prior—a binomial mixture with mixing distribution given by the uniform prior assumed for ψ (Eq. 2). This prior implies only vague knowledge of N, which may not always be the case. However, in these instances, binomial mixtures can still be used to specify a flexible class of priors for N by assuming different mixing distributions for ψ. As an example, the beta-binomial prior for N arises by assuming a beta mixing distribution for ψ. Other types of hierarchical priors for N are also possible. We have extended the principles of PX-DA to accommodate variability in N among sub-populations (Converse and Royle 2010); however, it is clear that additional research is needed in the formulation of priors for use in PX-DA.
The essential concept underlying PX-DA is that excess “observations” are added to a dataset and then the new (augmented) dataset is analyzed using a new model, expanded to accommodate the augmented data. In the context of capture–recapture models, we add the “all zero” encounter histories which are not, in practice, observable. The model for this dataset is naturally a zero-inflated version of either a binomial or a multinomial base model. Thus, PX-DA yields a generally consistent formulation for the analysis of both closed- and open-population models of all types. In addition, in doing so, PX-DA unifies the inference framework across a huge range of models that in the classical literature are treated in a relatively diffuse manner as unrelated “black boxes” and named procedures. We have identified interesting parallels between PX-DA-based capture–recapture models for closed populations and “occupancy models” and between open-population models and “multi-state” occupancy models. These parallels suggest the potential for convergence of many software platforms that have been developed for the analysis of ecological data. This convergence has already been achieved by Bayesian analysis using software such as WinBUGS, OpenBUGS, and other MCMC engines.
Acknowledgments
We thank Beth Gardner and Elise Zipkin for reviewing drafts of this manuscript. We thank Ullas Karanth (camera-trapping data) and Jim Nichols (Microtus data) for making data from their research available for our use. Use of trade, product, or firm names does not imply endorsement by the U.S. Government.