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.

gpcount(lambdaformula, phiformula, pformula, data,
mixture = c("P", "NB"), K, starts, method = "BFGS", se = TRUE,
engine = c("C", "R"), threads=1, ...)

## Arguments

lambdaformula Right-hand sided formula describing covariates of abundance. Right-hand sided formula describing availability covariates Right-hand sided formula for detection probability covariates An object of class unmarkedFrameGPC Either "P" or "NB" for Poisson and negative binomial distributions The maximum possible value of M, the super-population size. Starting values Optimization method used by optim Logical. Should standard errors be calculated? Either "C" or "R" for the C++ or R versions of the likelihood. The C++ code is faster, but harder to debug. 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 threads=1 (the default), OpenMP is disabled. Additional arguments to optim, such as lower and upper bounds

## Details

The latent transect-level super-population abundance distribution $$f(M | \mathbf{\theta})$$ can be set as either a Poisson or a negative binomial random variable, depending on the setting of the mixture argument. The expected value of $$M_i$$ is $$\lambda_i$$. If $$M_i \sim NB$$, then an additional parameter, $$\alpha$$, describes dispersion (lower $$\alpha$$ implies higher variance).

The number of individuals available for detection at time j is a modeled as binomial: $$N_{ij} \sim Binomial(M_i, \mathbf{\phi_{ij}})$$.

The detection process is also modeled as binomial: $$y_{ikj} \sim Binomial(N_{ij}, p_{ikj})$$.

Parameters $$\lambda$$, $$\phi$$ and $$p$$ can be modeled as linear functions of covariates using the log, logit and logit links respectively.

## Value

An object of class unmarkedFitGPC

Royle, J. A. 2004. N-Mixture models for estimating population size from spatially replicated counts. 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 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 unmarkedFrameGPC

gmultmix, gdistsamp, 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)
#>  4.35 3.80 3.70 4.10
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)

if (FALSE) {
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")
}