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Appendix S1: Bayesian methods for parameter estimation
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Bayesian methods have become popular for analyzing mark-recapture data. Since
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the early studies (see Clark et al. 2005 for a literature review), methods have been widely
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applied and customized for specific applications (e.g. Royle & Dorazio 2008). The trends
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for methodological development have included using hierarchical structure to model
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heterogeneity, dealing efficiently with missing data, and using covariates to model
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parameters of interest. We use Bayesian methods for estimating parameters for multistate
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mark-recapture data (Calvert et al. 2009, Clark et al. 2005). Other authors have used
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Bayesian methods to estimate movement in open population models (Dupuis & Schwarz
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2007), examine age-dependence in survival (Zheng et al. 2007), model metapopulation
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dynamics (Royle & Kery 2007), model individual effects in survival models (Royle
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2008), and estimate population size using data augmentation for both closed and open
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populations (Royle & Dorazio 2010).
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We developed a software program in MATLAB that implements a Bayesian
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technique for estimating population vital rates from mark-recapture data with multiple
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life stages. The Bayesian framework for mark-recapture data allows the probabilistic
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estimation of vital rate parameters (survival, transition and recapture probabilities).
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Instead of point estimates, the Bayesian approach gives the solution as a distribution of
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different possible vital rates that explain the observed data. These distributions are often
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analytically intractable, but there are numerical methods to approximate the distributions
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by producing random samples from them. We used a Gibbs sampling algorithm for
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estimating population vital rates, described in detail here.
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We estimated separate survival, transition and recapture probabilities for each
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year and each life stage. This model is developed for structured populations, where the
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population can be stratified by life stage at each time step (Dupuis 1995). We used the
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following notation:
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xkt observed life stage for individual k at year t
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zkt true life stage for individual k at year t
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Sit survival probability from year t to year t+1 for life stage i
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Rit recapture probability in year t for life stage i
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Pijt transition probability from life stage j at year t to life stage i at year t+1
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There are four life stages i, represented by the following numbers: 1=juvenile,
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2=subadult, 3=male, and 4=female.
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Raw mark-recapture data were entered directly into the model, where two
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matrices summarized the capture history for a given individual (Dupuis 1995). Each entry
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in matrix x was the observed life stage of an individual. If the individual was not
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observed, we set xkt = 0. Following Clark et al. (2005), we defined the matrix z, which
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contained the estimated life stage of an individual (which includes both the observed and
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unobserved life stages). The first matrix consists of raw data, and life stage values for the
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other matrix are unknown state parameters, estimated as part of the Bayesian analysis.
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From a previous analysis, we determined that capture probabilities varied by both
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life stage and year (McCaffery & Maxell 2010). We used the secondary sampling
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sessions within each summer to estimate the recapture probability for that year. Thus, the
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likelihood for observing x for a given year t, with recapture probabilities R was
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P(x*t | R*t )   Rit oit (1  Rit ) uit
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(1)
i1

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where the indicator variables okt and ukt indicate the number of observed and unobserved
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individuals at year t and life stage i, respectively (counted from all secondary capture
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sessions). Survival and transition probabilities were estimated between primary capture
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sessions. The likelihood for life stages z for a given year t, with survival and transition
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probabilities S and P was
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P(z*t | S*t )   Sit Sit (1  Sit ) d it
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(2)
i1
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
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P(z*t | P**t )    Pijt #(i j )t
(3)
i1 j 1
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where indicator variables sit, dit, and #(ij)t indicate the number of survivals, deaths and
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
transitions at year t, respectively (counted between primary sessions).
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For estimating the true life stages, we followed Clark et al. (2005) and
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conditioned the life stage at time t on life stages at the previous and next time point
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(surviving previous time, stage transition, not capturing, surviving next time and stage
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transition):
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P(zkt  j | zk,t 1  m,zk,t 1  l,R,S,P)  Sm,t 1Pjm,t 1 (1  R jt )S jt Pljt
(4)
The true life stages z are implicitly connected to the observed life stages x: the
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value of x affects the neighboring z values through equation (4), as such affecting the
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survival and transition probabilities S and P. However, the recapture probabilities R are
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calculated directly from the data and are therefore not affected by z.
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Our prior assumptions followed the robust design framework (Pollock 1982):
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since secondary capture sessions were very close to each other (1 day), we assumed that
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survival probability was one between the secondary sessions and no transitions were
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possible between them. These priors were implicit in our computations, since we
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estimated survival and transition only between primary capture sessions. For all survival
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and transition estimates among years, we used uniform priors, spanning all possible
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probabilities (0 to 1 interval). Thus, from our ten-year capture histories, we estimated
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posterior probability distributions for annual survival, recapture, and transition
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probabilities.
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The total likelihood for observing data x and having true life stages z, given the
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vital rate parameters, was obtained by taking a product of equations (1-3) over years. The
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posterior distributions for the vital rates can be written down analytically (beta
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distribution for recapture and survival and Dirichlet distribution for transition). Thus,
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given our observed data x and an estimate for life stages z, we sampled new values for R,
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S, and P from these known distributions. In contrast, for some given values of the vital
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rates, we calculated the probabilities for different zit using equation (4) and sampled new
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values from the resulting multinomial distribution. Alternating these conditional
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samplings, we arrived at the Gibbs sampler (Gelman et al. 2004), which can be
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summarized by the following four steps:
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1. Set the initial values for R, S, P and z and iteration counter i=1
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2. Sample R, S and P from their conditional posteriors, given x and z, using
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equations (1-3)
3. Loop over individuals and time points where xkt = 0, sample a new life stage
zkt using equation (4), given previously sampled R, S, and P
4. Set i=i+1 and go to step 2, until a desired number of iterations is obtained
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We determined that the Gibbs sampler had converged by checking that the chains had
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mixed well. Additionally, we observed that the likelihood stabilized after an initial period
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of increase, which indicated convergence. We produced 5000 samples from the posterior.
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In addition to the above survival and stage transitions, we estimated two
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additional terms, breeding probability and survival probability from egg to one year. For
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the breeding probability, we estimated of the number of females for each year. These
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were calculated by dividing the number of observed females (counted from the field data)
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by the recapture probability estimate. To get the estimate for breeding probability as a
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distribution, this calculation was performed by numerical integration over the distribution
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of recapture rates given by the Bayesian mark-recapture analysis, i.e., repeating
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calculations over the distribution of estimates generated by the Gibbs sampler. We then
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divided the number of observed egg masses by the number of females to determine the
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proportion of females breeding each year.
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Similarly, for survival probabilities from egg to one year, we needed estimates of
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the number of one-year-old frogs. These were calculated from a separate data set, which
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contains capture histories for one-year-old individuals (identified as frogs 34 mm and
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smaller). The above Gibbs sampling algorithm was used for estimating the recapture
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probability for this age class (in this estimation, we only had one life stage), and the
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number of observed individuals was then divided by different recapture probability
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estimates to get a distribution of the number of one-year-old frogs. To estimate survival
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from egg to one year, we then divided the number of one-year-old frogs by the number of
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eggs laid in the basin the previous spring.
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Literature Cited
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