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

distsamp(formula, data, keyfun=c("halfnorm", "exp",
"hazard", "uniform"), output=c("density", "abund"),
unitsOut=c("ha", "kmsq"), starts, method="BFGS", se=TRUE,
engine=c("C", "R", "TMB"), rel.tol=0.001, ...)

## Arguments

formula Double right-hand formula describing detection covariates followed by abundance covariates. ~1 ~1 would be a null model. 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. See example for set up. One of the following detection functions: "halfnorm", "hazard", "exp", or "uniform." See details. Model either "density" or "abund" Units of density. Either "ha" or "kmsq" for hectares and square kilometers, respectively. Vector of starting values for parameters. Optimization method used by optim. logical specifying whether or not to compute standard errors. Use code written in C++ or R Requested relative accuracy of the integral, see integrate Additional arguments to optim, such as lower and upper bounds

## Details

Unlike conventional distance sampling, which uses the 'conditional on detection' likelihood formulation, this model is based upon the unconditional likelihood and allows for modeling both abundance and detection function parameters.

The latent transect-level abundance distribution $$f(N | \mathbf{\theta})$$ assumed to be Poisson with mean $$\lambda$$ (but see gdistsamp for alternatives).

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.

## Value

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

## Note

You cannot use obsCovs.

## Author

Richard Chandler rbchan@uga.edu

## References

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

Sillett, S. and Chandler, R.B. and Royle, J.A. and Kery, M. and Morrison, S.A. In Press. Hierarchical distance sampling models to estimate population size and habitat-specific abundance of an island endemic. Ecological Applications

## See also

unmarkedFrameDS, unmarkedFit-class fitList, formatDistData, parboot, sight2perpdist, detFuns, gdistsamp, ranef. 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 #> Data frame representation of unmarkedFrame object. #> y.1 y.2 y.3 y.4 Length area habitat #> 1 3 6 5 0 5 5.953128 A #> 2 8 5 6 4 7 4.523539 A #> 3 0 0 0 0 3 5.812511 A #> 4 2 1 1 1 4 5.929066 A #> 5 0 2 0 0 2 5.039786 A #> 6 4 1 2 1 6 3.872571 A #> 7 3 3 2 0 4 3.890749 B #> 8 1 2 0 0 3 6.246779 B #> 9 2 1 0 1 1 6.813926 B #> 10 6 0 3 2 4 4.323164 B #> 11 5 1 2 0 4 7.059182 B #> 12 8 5 2 2 5 4.752804 Bsummary(ltUMF) #> unmarkedFrameDS Object #> #> line-transect survey design #> Distance class cutpoints (m): 0 5 10 15 20 #> #> 12 sites #> Maximum number of distance classes per site: 4 #> Mean number of distance classes per site: 4 #> Sites with at least one detection: 11 #> #> Tabulation of y observations: #> 0 1 2 3 4 5 6 8 #> 14 9 10 4 2 4 3 2 #> #> Site-level covariates: #> Length area habitat #> Min. :1 Min. :3.873 A:6 #> 1st Qu.:3 1st Qu.:4.473 B:6 #> Median :4 Median :5.426 #> Mean :4 Mean :5.351 #> 3rd Qu.:5 3rd Qu.:6.027 #> Max. :7 Max. :7.059 hist(ltUMF) # Half-normal detection function. Density output (log scale). No covariates. (fm1 <- distsamp(~ 1 ~ 1, ltUMF)) #> #> Call: #> distsamp(formula = ~1 ~ 1, data = ltUMF) #> #> Density: #> Estimate SE z P(>|z|) #> -0.171 0.134 -1.28 0.201 #> #> Detection: #> Estimate SE z P(>|z|) #> 2.39 0.127 18.7 2.46e-78 #> #> AIC: 164.7524 # Some methods to use on fitted model summary(fm1) #> #> Call: #> distsamp(formula = ~1 ~ 1, data = ltUMF) #> #> Density (log-scale): #> Estimate SE z P(>|z|) #> -0.171 0.134 -1.28 0.201 #> #> Detection (log-scale): #> Estimate SE z P(>|z|) #> 2.39 0.127 18.7 2.46e-78 #> #> AIC: 164.7524 #> Number of sites: 12 #> optim convergence code: 0 #> optim iterations: 31 #> Bootstrap iterations: 0 #> #> Survey design: line-transect #> Detection function: halfnorm #> UnitsIn: m #> UnitsOut: ha #> backTransform(fm1, type="state") # animals / ha #> Backtransformed linear combination(s) of Density estimate(s) #> #> Estimate SE LinComb (Intercept) #> 0.843 0.113 -0.171 1 #> #> Transformation: exp exp(coef(fm1, type="state", altNames=TRUE)) # same #> lam(Int) #> 0.8427749 backTransform(fm1, type="det") # half-normal SD #> Backtransformed linear combination(s) of Detection estimate(s) #> #> Estimate SE LinComb (Intercept) #> 10.9 1.38 2.39 1 #> #> Transformation: exp hist(fm1, xlab="Distance (m)") # Only works when there are no det covars # Empirical Bayes estimates of posterior distribution for N_i plot(ranef(fm1, K=50)) # Effective strip half-width (eshw <- integrate(gxhn, 0, 20, sigma=10.9)$value)
#> [1] 12.7523
# Detection probability
eshw / 20 # 20 is strip-width
#> [1] 0.6376152

# Halfnormal. Covariates affecting both density and and detection.
(fm2 <- distsamp(~area + habitat ~ habitat, ltUMF))
#>
#> Call:
#> distsamp(formula = ~area + habitat ~ habitat, data = ltUMF)
#>
#> Density:
#>             Estimate    SE     z P(>|z|)
#> (Intercept)   -0.376 0.191 -1.97  0.0490
#> habitatB       0.439 0.266  1.65  0.0992
#>
#> Detection:
#>             Estimate     SE     z  P(>|z|)
#> (Intercept)    3.091 0.5111  6.05 1.47e-09
#> area          -0.110 0.0884 -1.24 2.14e-01
#> habitatB      -0.271 0.2711 -1.00 3.17e-01
#>
#> AIC: 166.456
# Hazard-rate detection function.
(fm3 <- distsamp(~ 1 ~ 1, ltUMF, keyfun="hazard"))
#>
#> Call:
#> distsamp(formula = ~1 ~ 1, data = ltUMF, keyfun = "hazard")
#>
#> Density:
#>  Estimate    SE      z P(>|z|)
#>    -0.119 0.219 -0.546   0.585
#>
#> Detection:
#>  Estimate    SE    z  P(>|z|)
#>      2.13 0.433 4.91 9.34e-07
#>
#> Hazard-rate(scale):
#>  Estimate    SE    z P(>|z|)
#>     0.315 0.543 0.58   0.562
#>
#> AIC: 167.0218
# 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 examples

# Analysis of the Island Scrub-jay data.
# See Sillett et al. (In press)

data(issj)
str(issj)
#> 'data.frame':	307 obs. of  8 variables:
#>  $issj[0-100] : int 0 0 0 0 0 0 0 0 0 0 ... #>$ issj(100-200]: int  0 0 0 0 0 0 0 0 0 0 ...
#>  $issj(200-300]: int 2 0 0 0 0 0 0 0 0 0 ... #>$ x            : num  234870 237083 235732 237605 234239 ...
#>  $y : num 3767154 3766804 3766717 3766719 3766570 ... #>$ elevation    : num  51.4 156.9 144.8 184.3 111.4 ...
#>  $forest : num 0.022054 0.006731 0.016182 0.257626 0.000716 ... #>$ chaparral    : num  0.242 0.466 0.769 0.206 0 ...
jayumf <- unmarkedFrameDS(y=as.matrix(issj[,1:3]),
siteCovs=data.frame(scale(issj[,c("elevation","forest","chaparral")])),
dist.breaks=c(0,100,200,300), unitsIn="m", survey="point")

(fm1jay <- distsamp(~chaparral ~chaparral, jayumf))
#>
#> Call:
#> distsamp(formula = ~chaparral ~ chaparral, data = jayumf)
#>
#> Density:
#>             Estimate    SE      z  P(>|z|)
#> (Intercept)   -2.799 0.160 -17.55 6.02e-69
#> chaparral      0.912 0.145   6.31 2.82e-10
#>
#> Detection:
#>             Estimate     SE     z  P(>|z|)
#> (Intercept)    4.729 0.0840 56.27 0.000000
#> chaparral     -0.249 0.0739 -3.36 0.000772
#>
#> AIC: 969.9142

if (FALSE) {

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)

}