Fit the MacKenzie et al. (2002) Occupancy Model

This function fits the single season occupancy model of MacKenzie et al (2002).

occu(formula, data, knownOcc=numeric(0), linkPsi=c("logit", "cloglog"),
            starts, method="BFGS", se=TRUE, engine=c("C", "R", "TMB"),
            threads = 1, ...)

Arguments

formula	Double right-hand side formula describing covariates of detection and occupancy in that order.
data	An unmarkedFrameOccu object
knownOcc	Vector of sites that are known to be occupied. These should be supplied as row numbers of the y matrix, eg, c(3,8) if sites 3 and 8 were known to be occupied a priori.
linkPsi	Link function for the occupancy model. Options are "logit" for the standard occupancy model or "cloglog" for the complimentary log-log link, which relates occupancy to site-level abundance. See details.
starts	Vector of parameter starting values.
method	Optimization method used by optim.
se	Logical specifying whether or not to compute standard errors.
engine	Code to use for optimization. Either "C" for fast C++ code, "R" for native R code, or "TMB" for Template Model Builder. "TMB" is used automatically if your formula contains random effects.
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 threads=1 (the default), OpenMP is disabled.
...	Additional arguments to optim, such as lower and upper bounds

Details

See unmarkedFrame and unmarkedFrameOccu for a description of how to supply data to the data argument.

occu fits the standard occupancy model based on zero-inflated binomial models (MacKenzie et al. 2006, Royle and Dorazio 2008). The occupancy state process (\(z_i\)) of site \(i\) is modeled as

$$z_i \sim Bernoulli(\psi_i)$$

The observation process is modeled as

$$y_{ij}|z_i \sim Bernoulli(z_i p_{ij})$$

By default, covariates of \(\psi_i\) and \(p_{ij}\) are modeled using the logit link according to the formula argument. The formula is a double right-hand sided formula like ~ detform ~ occform where detform is a formula for the detection process and occform is a formula for the partially observed occupancy state. See formula for details on constructing model formulae in R.

When linkPsi = "cloglog", the complimentary log-log link function is used for \(psi\) instead of the logit link. The cloglog link relates occupancy probability to the intensity parameter of an underlying Poisson process (Kery and Royle 2016). Thus, if abundance at a site is can be modeled as \(N_i ~ Poisson(\lambda_i)\), where \(log(\lambda_i) = \alpha + \beta*x\), then presence/absence data at the site can be modeled as \(Z_i ~ Binomial(\psi_i)\) where \(cloglog(\psi_i) = \alpha + \beta*x\).

Value

unmarkedFitOccu object describing the model fit.

References

Kery, Marc, and J. Andrew Royle. 2016. Applied Hierarchical Modeling in Ecology, Volume 1. Academic Press.

MacKenzie, D. I., J. D. Nichols, G. B. Lachman, S. Droege, J. Andrew Royle, and C. A. Langtimm. 2002. Estimating Site Occupancy Rates When Detection Probabilities Are Less Than One. Ecology 83: 2248-2255.

MacKenzie, D. I. et al. 2006. Occupancy Estimation and Modeling. Amsterdam: Academic Press.

Royle, J. A. and R. Dorazio. 2008. Hierarchical Modeling and Inference in Ecology. Academic Press.

Author

Ian Fiske

See also

unmarked, unmarkedFrameOccu, modSel, parboot

Examples

data(frogs)
pferUMF <- unmarkedFrameOccu(pfer.bin)
plot(pferUMF, panels=4)
# add some fake covariates for illustration
siteCovs(pferUMF) <- data.frame(sitevar1 = rnorm(numSites(pferUMF)))

# observation covariates are in site-major, observation-minor order
obsCovs(pferUMF) <- data.frame(obsvar1 = rnorm(numSites(pferUMF) * obsNum(pferUMF)))

(fm <- occu(~ obsvar1 ~ 1, pferUMF))
#> 
#> Call:
#> occu(formula = ~obsvar1 ~ 1, data = pferUMF)
#> 
#> Occupancy:
#>  Estimate   SE     z P(>|z|)
#>      8.05 20.4 0.394   0.694
#> 
#> Detection:
#>             Estimate    SE       z  P(>|z|)
#> (Intercept)   -1.929 0.164 -11.730 8.90e-32
#> obsvar1       -0.152 0.177  -0.864 3.88e-01
#> 
#> AIC: 262.5825 

confint(fm, type='det', method = 'normal')
#>                 0.025     0.975
#> p(Int)     -2.2507318 -1.606287
#> p(obsvar1) -0.4984393  0.193446
confint(fm, type='det', method = 'profile')
#> Profiling parameter 1 of 2 ... done.
#> Profiling parameter 2 of 2 ... done.
#>                 0.025      0.975
#> p(Int)     -2.2650988 -1.5736092
#> p(obsvar1) -0.5010324  0.1933641

# estimate detection effect at obsvars=0.5
(lc <- linearComb(fm['det'],c(1,0.5)))
#> Linear combination(s) of Detection estimate(s)
#> 
#>  Estimate    SE (Intercept) obsvar1
#>        -2 0.194           1     0.5
#> 

# transform this to probability (0 to 1) scale and get confidence limits
(btlc <- backTransform(lc))
#> Backtransformed linear combination(s) of Detection estimate(s)
#> 
#>  Estimate     SE LinComb (Intercept) obsvar1
#>     0.119 0.0203      -2           1     0.5
#> 
#> Transformation: logistic 
confint(btlc, level = 0.9)
#>        0.05      0.95
#>  0.08915656 0.1563634

# Empirical Bayes estimates of proportion of sites occupied
re <- ranef(fm)
sum(bup(re, stat="mode"))
#> [1] 130