empirical.varD:

Description

Compute Horvitz-Thompson-like estimate of population density from a previously fitted spatial detection model, and estimate its sampling variance using the empirical spatial variance of the number observed in replicate sampling units. Wrapper functions are provided for several different scenarios, but all ultimately call derived.nj. The function derived also computes Horvitz-Thompson-like estimates, but it assumes a Poisson or binomial distribution of total number when computing the sampling variance.

Usage

derived.nj ( nj, esa, se.esa, method = c("SRS", "local", "poisson",
    "binomial"), xy = NULL, alpha = 0.05, loginterval = TRUE, area =
    NULL )
derived.mash ( object, sessnum = NULL,  method = c("SRS", "local"),
    alpha = 0.05, loginterval = TRUE)
derived.cluster ( object, sessnum = NULL,  method = c("SRS", "local"),
    alpha = 0.05, loginterval = TRUE)
derived.session ( object,  method = c("SRS", "local"), xy = NULL,
    alpha = 0.05, loginterval = TRUE )
derived.external ( object, sessnum = NULL, nj, cluster, buffer = 100,
    mask = NULL, noccasions = NULL,  method = c("SRS", "local"), xy = NULL,
    alpha = 0.05, loginterval = TRUE)

Arguments

object

fitted secr model

vector of number observed in each sampling unit (cluster)

esa

scalar estimate of effective sampling area (

\hat{a}

)

se.esa

estimated standard error of effective sampling area (

\hat{S E} (\hat{a})

)

method

character string `SRS' or `local'

dataframe of x- and y- coordinates (method = "local" only)

alpha

alpha level for confidence intervals

loginterval

logical for whether to base interval on log(N)

area

area of region for method = "binomial" (hectares)

sessnum

index of session in object$capthist for which output required

cluster

`traps' object for a single cluster

buffer

width of buffer in metres (ignored if mask provided)

mask

mask object for a single cluster of detectors

noccasions

number of occasions (for nj)

Value

A dataframe with one row and the columns --

estimate

Horvitz-Thompson-like estimate of population density

SE.estimate

SE of density estimate

lcl

lower 100(1--alpha)% confidence limit

ucl

upper 100(1--alpha)% confidence limit

CVn

relative SE of number observed (across sampling units)

CVa

relative SE of effective sampling area

CVD

relative SE of density estimate

Details

derived.nj accepts a vector of counts (nj), along with $\hat{a}$ and $\hat{S E} (\hat{a})$ . The argument esa may include both $\hat{a}$ and $\hat{S E} (\hat{a})$ ) - any form will do if it can be coerced to a vector of length 2. In the special case that nj is of length 1, or method takes the values `poisson' or `binomial', the variance is computed using a theoretical variance rather than an empirical estimate. The value of method corresponds to `distribution' in derived, and defaults to `poisson'. For method = 'binomial' you must specify area (see Examples).

derived.cluster accepts a model fitted to data from clustered detectors; each cluster is interpreted as a replicate sample. It is assumed that the sets of individuals sampled by different clusters do not intersect, and that all clusters have the same geometry (spacing, detector number etc.).

derived.mash accepts a model fitted to clustered data that have been `mashed' for fast processing (see mash); each cluster is a replicate sample: the function uses the vector of cluster frequencies ( $n_{j}$ ) stored as an attribute of the mashed capthist by mash.

derived.external combines detection parameter estimates from a fitted model with a vector of frequencies nj from replicate sampling units configured as in cluster. Detectors in cluster are assumed to match those in the fitted model with respect to type and efficiency, but sampling duration (noccasions), spacing etc. may differ. The mask should match cluster; if mask is missing, one will be constructed using the buffer argument and defaults from make.mask.

derived.session accepts a single fitted model that must span multiple sessions; each session is interpreted as a replicate sample.

Spatial variance may be calculated assuming simple random sampling (method = "SRS") or using the neighbourhood variance estimator recommended by Stevens and Olsen (2003) for generalized random tessellation stratified (GRTS) samples and implemented in package spsurvey (method = "local"). For `local' variance estimates, the centre of each replicate must be provided in xy, except where centres may be inferred from the data. The options method = "poisson" and method = "binomial" use theoretical (model-based) variance rather than the empirical spatial variance.

References

Buckland, S. T., Anderson, D. R., Burnham, K. P., Laake, J. L., Borchers, D. L. and Thomas, L. (2001) Introduction to Distance Sampling: Estimating Abundance of Biological Populations. Oxford University Press, Oxford.

Stevens, D. L. Jr and Olsen, A. R. (2003) Variance estimation for spatially balanced samples of environmental resources. Environmetrics 14, 593--610.

Examples

Run this code


## The `ovensong' data are pooled from 75 replicate positions of a
## 4-microphone array. The array positions are coded as the first 4
## digits of each sound identifier. The sound data are initially in the
## object `signalCH'. We first impose a 52.5 dB signal threshold as in
## Dawson & Efford (2009, J. Appl. Ecol. 46:1201--1209). The vector nj
## includes 33 positions at which no ovenbird was heard. The first and
## second columns of `temp' hold the estimated effective sampling area
## and its standard error.

signalCH.525 <- subset(signalCH, cutval = 52.5)
nonzero.counts <- table(substring(rownames(signalCH.525),1,4))
nj <- c(nonzero.counts, rep(0, 75 - length(nonzero.counts)))
temp <- derived(ovensong.model.1, se.esa = TRUE)
derived.nj(nj, temp["esa",1:2])

## The result is very close to that reported by Dawson & Efford
## from a 2-D Poisson model fitted by maximizing the full likelihood.

## If nj vector has length 1, a theoretical variance is used...
msk <- ovensong.model.1$mask
A <- nrow(msk) * attr(msk, "area")
derived.nj (sum(nj), temp["esa",1:2], method = "poisson")
derived.nj (sum(nj), temp["esa",1:2], method = "binomial", area = A)

## Not run: ------------------------------------
# 
# ## Set up an array of small (4 x 4) grids,
# ## simulate a Poisson-distributed population,
# ## sample from it, plot, and fit a model.
# ## mash() condenses clusters to a single cluster
# 
# testregion <- data.frame(x = c(0,2000,2000,0),
#     y = c(0,0,2000,2000))
# t4 <- make.grid(nx = 4, ny = 4, spacing = 40)
# t4.16 <- make.systematic (n = 16, cluster = t4,
#     region = testregion)
# popn1 <- sim.popn (D = 5, core = testregion,
#     buffer = 0)
# capt1 <- sim.capthist(t4.16, popn = popn1)
# fit1 <- secr.fit(mash(capt1), CL = TRUE, trace = FALSE)
# 
# ## Visualize sampling
# tempmask <- make.mask(t4.16, spacing = 10, type =
#     "clusterbuffer")
# plot(tempmask)
# plot(t4.16, add = TRUE)
# plot(capt1, add = TRUE)
# 
# ## Compare model-based and empirical variances.
# ## Here the answers are similar because the data
# ## were simulated from a Poisson distribution,
# ## as assumed by \code{derived}
# 
# derived(fit1)
# derived.mash(fit1)
# 
# ## Now simulate a patchy distribution; note the
# ## larger (and more credible) SE from derived.mash().
# 
# popn2 <- sim.popn (D = 5, core = testregion, buffer = 0,
#     model2D = "hills", details = list(hills = c(-2,3)))
# capt2 <- sim.capthist(t4.16, popn = popn2)
# fit2 <- secr.fit(mash(capt2), CL = TRUE, trace = FALSE)
# derived(fit2)
# derived.mash(fit2)
# 
# ## The detection model we have fitted may be extrapolated to
# ## a more fine-grained systematic sample of points, with
# ## detectors operated on a single occasion at each...
# ## Total effort 400 x 1 = 400 detector-occasions, compared
# ## to 256 x 5 = 1280 detector-occasions for initial survey.
# 
# t1 <- make.grid(nx = 1, ny = 1)
# t1.100 <- make.systematic (cluster = t1, spacing = 100,
#     region = testregion)
# capt2a <- sim.capthist(t1.100, popn = popn2, noccasions = 1)
# ## one way to get number of animals per point
# nj <- attr(mash(capt2a), "n.mash")
# derived.external (fit2, nj = nj, cluster = t1, buffer = 100,
#     noccasions = 1)
# 
# ## Review plots
# library(MASS)
# base.plot <- function() {
#     eqscplot( testregion, axes = FALSE, xlab = "",
#         ylab = "", type = "n")
#     polygon(testregion)
# }
# par(mfrow = c(1,3), xpd = TRUE, xaxs = "i", yaxs = "i")
# base.plot()
# plot(popn2, add = TRUE, col = "blue")
# mtext(side=3, line=0.5, "Population", cex=0.8, col="black")
# base.plot()
# plot (capt2a, add = TRUE,title = "Extensive survey")
# base.plot()
# plot(capt2, add = TRUE, title = "Intensive survey")
# par(mfrow = c(1,1), xpd = FALSE, xaxs = "r", yaxs = "r")  ## defaults
## ---------------------------------------------

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