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\name{gpcount}
\alias{gpcount}
\title{
Generalized binomial N-mixture model for repeated count data
}
\description{
Fit the model of Chandler et al. (2011) to repeated count data collected
using the robust design. This model allows for inference about
population size, availability, and detection probability.
}
\usage{
gpcount(lambdaformula, phiformula, pformula, data,
mixture = c("P", "NB", "ZIP"), K, starts, method = "BFGS", se = TRUE,
engine = c("C", "R"), threads=1, ...)
}
\arguments{
\item{lambdaformula}{
Right-hand sided formula describing covariates of abundance.
}
\item{phiformula}{
Right-hand sided formula describing availability covariates
}
\item{pformula}{
Right-hand sided formula for detection probability covariates
}
\item{data}{
An object of class unmarkedFrameGPC
}
\item{mixture}{
Either "P", "NB", or "ZIP" for Poisson, negative binomial, or
zero-inflated Poisson distributions
}
\item{K}{
The maximum possible value of M, the super-population size.
}
\item{starts}{
Starting values
}
\item{method}{
Optimization method used by \code{\link{optim}}
}
\item{se}{
Logical. Should standard errors be calculated?
}
\item{engine}{
Either "C" or "R" for the C++ or R versions of the likelihood. The C++
code is faster, but harder to debug.
}
\item{threads}{Set the number of threads to use for optimization in C++, if
OpenMP is available on your system. Increasing the number of threads
may speed up optimization in some cases by running the likelihood
calculation in parallel. If \code{threads=1} (the default), OpenMP is disabled.
}
\item{\dots}{
Additional arguments to \code{\link{optim}}, such as lower and upper
bounds
}
}
\details{
The latent transect-level super-population abundance distribution
\eqn{f(M | \mathbf{\theta})}{f(M | theta)} can be set as either a
Poisson, negative binomial, or zero-inflated Poisson random variable, depending on the
setting of the \code{mixture} argument. The expected value of
\eqn{M_i} is \eqn{\lambda_i}{lambda_i}. If \eqn{M_i \sim NB}{M_i ~ NB},
then an additional parameter, \eqn{\alpha}{alpha}, describes
dispersion (lower \eqn{\alpha}{alpha} implies higher variance). If
\eqn{M_i \sim ZIP}{M_i ~ ZIP}, then an additional zero-inflation parameter
\eqn{\psi}{psi} is estimated.
The number of individuals available for detection at time j
is a modeled as binomial:
\eqn{N_{ij} \sim Binomial(M_i, \mathbf{\phi_{ij}})}{N(i,j) ~
Binomial(M(i), phi(i,j))}.
The detection process is also modeled as binomial:
\eqn{y_{ikj} \sim Binomial(N_{ij}, p_{ikj})}{y(i,k,j) ~
Binomial(N(i,t), p(i,k,j))}.
Parameters \eqn{\lambda}{lambda}, \eqn{\phi}{phi} and \eqn{p}{p} can be
modeled as linear functions of covariates using the log, logit and logit
links respectively.
}
\value{
An object of class unmarkedFitGPC
}
\references{
Royle, J. A. 2004. N-Mixture models for estimating population size from
spatially replicated counts. \emph{Biometrics} 60:108--105.
Chandler, R. B., J. A. Royle, and D. I. King. 2011. Inference about
density and temporary emigration in unmarked populations. Ecology
92:1429-1435.
}
\author{
Richard Chandler \email{rbchan@uga.edu}
}
\note{
In the case where availability for detection is due to random temporary
emigration, population density at time j, D(i,j), can be estimated by
N(i,j)/plotArea.
This model is also applicable to sampling designs in which the local
population size is closed during the J repeated counts, and availability
is related to factors such as the probability of vocalizing. In this
case, density can be estimated by M(i)/plotArea.
If availability is a function of both temporary emigration and other
processess such as song rate, then density cannot be directly estimated,
but inference about the super-population size, M(i), is possible.
Three types of covariates can be supplied, site-level,
site-by-year-level, and observation-level. These must be formatted
correctly when organizing the data with \code{\link{unmarkedFrameGPC}}
}
\seealso{
\code{\link{gmultmix}}, \code{\link{gdistsamp}},
\code{\link{unmarkedFrameGPC}}
}
\examples{
set.seed(54)
nSites <- 20
nVisits <- 4
nReps <- 3
lambda <- 5
phi <- 0.7
p <- 0.5
M <- rpois(nSites, lambda) # super-population size
N <- matrix(NA, nSites, nVisits)
y <- array(NA, c(nSites, nReps, nVisits))
for(i in 1:nVisits) {
N[,i] <- rbinom(nSites, M, phi) # population available during vist j
}
colMeans(N)
for(i in 1:nSites) {
for(j in 1:nVisits) {
y[i,,j] <- rbinom(nReps, N[i,j], p)
}
}
ym <- matrix(y, nSites)
ym[1,] <- NA
ym[2, 1:nReps] <- NA
ym[3, (nReps+1):(nReps+nReps)] <- NA
umf <- unmarkedFrameGPC(y=ym, numPrimary=nVisits)
\dontrun{
fmu <- gpcount(~1, ~1, ~1, umf, K=40, control=list(trace=TRUE, REPORT=1))
backTransform(fmu, type="lambda")
backTransform(fmu, type="phi")
backTransform(fmu, type="det")
}
}
|
