secr (version 2.0.0)

empirical.varD: Design-based Variance of Density

Description

Compute the variance of a density estimate from a fitted spatially explicit model, using the empirical spatial variance of the number observed in replicated sampling units.

Usage

empirical.varD(object, esa = NULL, se.esa = NULL)

Arguments

object

fitted secr model object or vector of number observed in each sampling unit

esa

scalar estimate of effective sampling area ($\hat{a}$)

se.esa

estimated standard error of effective sampling area ($\widehat{SE}(\hat{a})$)

Value

• A vector of values --
• D
  Horvitz-Thompson-like estimate of population density
• seD
  SE of density estimate
• CVD
  relative SE of density estimate
• CVn
  relative SE of number observed (across sampling units)
• CVa
  relative SE of effective sampling area

Details

The variance of a Horvitz-Thompson-like estimate of density may be estimated as the sum of two components, one due to uncertainty in the estimate of effective sampling area ($\hat{a}$) and the other due to spatial variance in the total number of animals $n$ observed on $J$ replicate sampling units ($n = \sum_{j=1}^{J}{n_j}$). We use a delta-method approximation that assumes independence of the components: $$\widehat{\mbox{var}}(\hat{D}) = \hat{D}^2 {\frac{\widehat{\mbox{var}}(n)}{n^2} + \frac{\widehat{\mbox{var}}(\hat{a})}{\hat{a}}}$$ where $\widehat{\mbox{var}}(n) = \frac{J}{J-1} \sum_{j=1}^{J}(n_j-n/J)^2$. The estimate of $\mbox{var}(\hat{a})$ is model-based while that of $\mbox{var}(n)$ is design-based. This formulation follows that of Buckland et al. (2001, p. 78) for conventional distance sampling. Given sufficient independent replicates, it is a robust way to allow for unmodelled spatial overdispersion. There is a complication in SECR owing to the fact that $\hat{a}$ is a derived quantity (actually an integral) rather than a model parameter. Its sampling variance $\mbox{var}(\hat{a})$ is estimated indirectly in secr by combining the asymptotic estimate of the covariance matrix of the fitted detection parameters $\theta$ with a numerical estimate of the gradient of $a(\theta)$ with respect to $\theta$. This calculation is performed in derived. Input may be either a single fitted model or a vector of counts (the $n_j$) along with $\hat{a}$ and $\widehat{SE}(\hat{a})$. If object is a single fitted model then it must span multiple sessions and each session is interpreted as a replicate sample. This form does not allow zero counts; if some units have zero counts use the second style of input.

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.

See Also

derived, esa

Examples

## 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)
empirical.varD(nj, temp['esa',1], temp['esa',2])

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