unmarked (version 0.9-1)

distsamp: Fit the hierarchical distance sampling model of Royle et al. (2004)

Description

Fit the hierarchical distance sampling model of Royle et al. (2004) to line or point transect data recorded in discrete distance intervals.

Usage

distsamp(formula, data, keyfun=c("halfnorm", "exp",
    "hazard", "uniform"), output=c("density", "abund"),
    unitsOut=c("ha", "kmsq"), starts, method="BFGS",
    control=list(), se=TRUE, rel.tol=1e-4)

Arguments

formula

Double right-hand formula describing detection covariates followed by abundance covariates. ~1 ~1 would be a null model.

data

object of class unmarkedFrameDS, containing response matrix, covariates, distance interval cut points, survey type ("line" or "point"), transect lengths (for survey = "line"), and units ("m" or "km") for cut points and transect lengths. Se

keyfun

One of the following detection functions: "halfnorm", "hazard", "exp", or "uniform." See details.

output

Model either "density" or "abund"

unitsOut

Units of density. Either "ha" or "kmsq" for hectares and square kilometers, respectively.

starts

Vector of starting values for parameters.

method

Optimization method used by optim.

control

Other arguments passed to optim.

se

logical specifying whether or not to compute standard errors.

rel.tol

relative accuracy for the integration of the detection function. See integrate. You might try adjusting this if you get an error message related to the integral.

Value

• unmarkedFitDS object (child class of unmarkedFit-class) describing the model fit.

Details

Unlike conventional distance sampling, which uses the 'conditional on detection' likelihood formulation, this model is based upon the unconditional likelihood and thus allows for modeling both abundance and detection function parameters. The latent transect-level abundance distribution $f(N | \mathbf{\theta})$ is currently assumed to be Poisson with mean $\lambda$. The detection process is modeled as multinomial: $y_{ij} \sim Multinomial(N_i, pi_{ij})$, where $pi_ij$ is the multinomial cell probability for transect i in distance class j. These are computed based upon a detection function $g(x | \mathbf{\sigma})$, such as the half-normal, negative exponential, or hazard rate. Parameters $\lambda$ and $\sigma$ can be vectors affected by transect-specific covariates using the log link.

References

Royle, J. A., D. K. Dawson, and S. Bates (2004) Modeling abundance effects in distance sampling. Ecology 85, pp. 1591-1597.

See Also

unmarkedFrameDS, unmarkedFit-class fitList, formatDistData, parboot, sight2perpdist, detFuns. Also look at vignette("distsamp").

Examples

## Line transect examples

data(linetran)

ltUMF <- with(linetran, {
   unmarkedFrameDS(y = cbind(dc1, dc2, dc3, dc4),
   siteCovs = data.frame(Length, area, habitat),
   dist.breaks = c(0, 5, 10, 15, 20),
   tlength = linetran$Length * 1000, survey = "line", unitsIn = "m")
   })

ltUMF
summary(ltUMF)
hist(ltUMF)

# Half-normal detection function. Density output (log scale). No covariates.
(fm1 <- distsamp(~ 1 ~ 1, ltUMF))

# Some methods to use on fitted model
summary(fm1)
backTransform(fm1, type="state")       # animals / ha
exp(coef(fm1, type="state", altNames=TRUE))     # same
backTransform(fm1, type="det")         # half-normal SD
hist(fm1, xlab="Distance (m)")	# Only works when there are no detection covars

# Effective strip half-width
(eshw <- integrate(gxhn, 0, 20, sigma=10.9)$value)

# Detection probability
eshw / 20 # 20 is strip-width


# Halfnormal. Covariates affecting both density and and detection.
(fm2 <- distsamp(~area + habitat ~ habitat, ltUMF))

# Hazard-rate detection function.
(fm3 <- distsamp(~ 1 ~ 1, ltUMF, keyfun="hazard"))

# Plot detection function.
fmhz.shape <- exp(coef(fm3, type="det"))
fmhz.scale <- exp(coef(fm3, type="scale"))
plot(function(x) gxhaz(x, shape=fmhz.shape, scale=fmhz.scale), 0, 25,
	xlab="Distance (m)", ylab="Detection probability")

## Point transect example

data(pointtran)

ptUMF <- with(pointtran, {
	unmarkedFrameDS(y = cbind(dc1, dc2, dc3, dc4, dc5),
	siteCovs = data.frame(area, habitat),
	dist.breaks = seq(0, 25, by=5), survey = "point", unitsIn = "m")
	})

# Half-normal.
(fmp1 <- distsamp(~ 1 ~ 1, ptUMF))
hist(fmp1, ylim=c(0, 0.07), xlab="Distance (m)")

# effective radius
sig <- exp(coef(fmp1, type="det"))
ea <- 2*pi * integrate(grhn, 0, 25, sigma=sig)$value # effective area
sqrt(ea / pi) # effective radius

# detection probability
ea / (pi*25^2)