R: Generalized multinomial N-mixture model

gmultmix {unmarked} | R Documentation

## 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, engine=c("C","R"), ...)

### 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?
`engine` | Either "C" to use fast C++ code or "R" to use native R code during the optimization.
`...` | Additional arguments to optim, such as lower and upper bounds

### 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.

### Value

An object of class unmarkedFitGMM.

### 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`

### Author(s)

Richard Chandler rbchan@uga.edu and Andy Royle

### 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)

[Package unmarked version 1.0.0 Index]