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 derivednj. 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

Arguments

Details

derivednj accepts a vector of counts (nj), along with a-hat and SE(a-hat). The argument esa may be a scalar or (if se.esa is NULL) a 2-column matrix with a_j-hat and SE(a_j-hat) for each replicate j (row). 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).

If independent.esa is TRUE then independence is assumed among cluster-specific estimates of esa, and esa variances are summed. The default is a weighted sum leading to higher overall variance.

derivedCluster 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.).

derivedMash 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.

derivedExternal 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.

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

Spatial variance is calculated by one of these methods

The weighted options R2 and R3 substitute a_j-hat for line length l_k in the corresponding formulae of Fewster et al. (2009, Eq 3,5). Density is estimated by \hat D = n/A where A = ∑ a_j. The variance of A is estimated as the sum of the cluster-specific variances, assuming independence among clusters. Fewster et al. (2009) found that an alternative estimator for line transects derived by Thompson (2002) performed better when there were strong density gradients correlated with line length (R2 in Fewster et al. 2009, Eq 3).

The neighborhood variance estimator is implemented in package spsurvey and was originally proposed for generalized random tessellation stratified (GRTS) samples. For ‘local’ variance estimates, the centre of each replicate must be provided in xy, except where centres may be inferred from the data. It is unclear whether ‘local’ can or should be used when clusters vary in size.

derivedSystematic implements the 'boxlet' variance estimator of Fewster (2011) for systematic designs using clustered detectors (an alternative to derivedCluster and derivedSessions). The method is experimental in secr 3.2.0 and may change. The ‘design’ argument is a list with components corresponding to arguments of make.systematic, (n and origin are ignored if provided):

If region is omitted from design then an attempt will be made to retrieve it from the mask attribute of object (this works if the call to make.mask used keep.poly = TRUE).

Value

Dataframe with one row for each derived parameter (‘esa’, ‘D’) and columns as below

Note

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 (a-hat) and the other due to spatial variance in the total number of animals n observed on J replicate sampling units (sum(n_j)). We use a delta-method approximation that assumes independence of the components:

var(D-hat) = D-hat^2 (var(n)/n^2 + var(a-hat)/a-hat^2)

where var(n) = J/(J-1).sum((n_j - n/J)^2). The estimate of var(a-hat) is model-based while that of 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 a-hat is a derived quantity (actually an integral) rather than a model parameter. Its sampling variance var(a-hat) 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.

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.

Fewster, R. M. (2011) Variance estimation for systematic designs in spatial surveys. Biometrics 67, 1518–1531.

Fewster, R. M., Buckland, S. T., Burnham, K. P., Borchers, D. L., Jupp, P. E., Laake, J. L. and Thomas, L. (2009) Estimating the encounter rate variance in distance sampling. Biometrics 65, 225–236.

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

Thompson, S. K. (2002) Sampling. 2nd edition. Wiley, New York.

See Also

derived, esa

Examples