unmarked (version 0.11-0)

gmultmix: Generalized multinomial N-mixture model

Description

A three level hierarchical model for designs involving repeated counts that yield multinomial outcomes. Possible data collection methods include repeated removal sampling and double observer sampling. The three model parameters are abundance, availability, and detection probability.

Usage

gmultmix(lambdaformula, phiformula, pformula, data,
mixture = c("P", "NB"), K, starts, method = "BFGS", se = TRUE, ...)

Arguments

lambdaformula

Righthand side (RHS) formula describing abundance covariates

phiformula

RHS formula describing availability covariates

pformula

RHS formula describing detection covariates

data

An object of class unmarkedFrameGMM

mixture

Either "P" or "NB" for Poisson and Negative Binomial mixing distributions.

K

The upper bound of integration

starts

Starting values

method

Optimization method used by optim

se

Logical. Should standard errors be calculated?

...

Additional arguments to optim, such as lower and upper bounds

Value

An object of class unmarkedFitGMM.

Details

The latent transect-level super-population abundance distribution $f(M | theta)$ can be set as either a Poisson or a negative binomial random variable, depending on the setting of the mixture argument. mixture = "P" or mixture = "NB" select the Poisson or negative binomial distribution respectively. The mean of $M_i$ is $lambda_i$. If $M_i ~ NB$, 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: $N(i,j) ~ Binomial(M(i), phi(i,j))$.

The detection process is modeled as multinomial: $y(i,1:J,t) ~ Multinomial(N(i,t), pi(i,1,t), pi(i,2,t), ..., pi(i,J,t))$, where $pi(ijt)$ is the multinomial cell probability for plot i at time t on occasion j.

Cell probabilities are computed via a user-defined function related to the sampling design. Alternatively, the default functions removalPiFun or doublePiFun can be used for equal-interval removal sampling or double observer sampling. Note that the function for computing cell probabilites is specified when setting up the data using unmarkedFrameGMM.

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

References

Royle, J. A. (2004) Generalized estimators of avian abundance from count survey data. Animal Biodiversity and Conservation 27, pp. 375--386.

Chandler, R. B., J. A. Royle, and D. I. King. 2011. Inference about density and temporary emigration in unmarked populations. Ecology 92:1429-1435.

See Also

unmarkedFrameGMM for setting up the data and metadata. multinomPois for surveys where no secondary sampling periods were used. Example functions to calculate multinomial cell probabilities are described piFuns

Examples

# Simulate data using the multinomial-Poisson model with a
# repeated constant-interval removal design.

n <- 100  # number of sites
T <- 4    # number of primary periods
J <- 3    # number of secondary periods

lam <- 3
phi <- 0.5
p <- 0.3

#set.seed(26)
y <- array(NA, c(n, T, J))
M <- rpois(n, lam)          # Local population size
N <- matrix(NA, n, T)       # Individuals available for detection

for(i in 1:n) {
    N[i,] <- rbinom(T, M[i], phi)
    y[i,,1] <- rbinom(T, N[i,], p)    # Observe some
    Nleft1 <- N[i,] - y[i,,1]         # Remove them
    y[i,,2] <- rbinom(T, Nleft1, p)   # ...
    Nleft2 <- Nleft1 - y[i,,2]
    y[i,,3] <- rbinom(T, Nleft2, p)
    }

y.ijt <- cbind(y[,1,], y[,2,], y[,3,], y[,4,])


umf1 <- unmarkedFrameGMM(y=y.ijt, numPrimary=T, type="removal")

(m1 <- gmultmix(~1, ~1, ~1, data=umf1, K=30))

backTransform(m1, type="lambda")        # Individuals per plot
backTransform(m1, type="phi")           # Probability of being avilable
(p <- backTransform(m1, type="det"))    # Probability of detection
p <- coef(p)

# Multinomial cell probabilities under removal design
c(p, (1-p) * p, (1-p)^2 * p)

# Or more generally:
head(getP(m1))

# Empirical Bayes estimates of super-population size
re <- ranef(m1)
plot(re, layout=c(5,5), xlim=c(-1,20), subset=site%in%1:25)