Density and abundance estimates and variances

Description

Computes density and abundance estimates and variances based on Horvitz-Thompson-like estimator

Usage

dht(model, region.table, sample.table, obs.table = NULL, subset = NULL,
  se = TRUE, bootstrap = FALSE, options = list())

Arguments

Details

Density and abundance within the sampled region is computed based on a Horvitz-Thomspon-like estimator for groups and individuals (if a clustered population) and this is extrapolated to the entire survey region based on any defined regional stratification. The variance is based on replicate samples within any regional stratification. For clustered populations, E(s) and its standard error are also output.

Abundance is estimated with a Horvitz-Thompson-like estimator (Huggins 1989,1991; Borchers et al 1998; Borchers and Burnham 2004). The abundance in the sampled region is simply 1/p_1 + 1/p_2 + ... + 1/p_n where p_i is the estimated detection probability for the ith detection of n total observations. It is not strictly a Horvitz-Thompson estimator because the p_i are estimated and not known. For animals observed in tight clusters, that estimator gives the abundance of groups (group=TRUE in options) and the abundance of individuals is estimated as s_1/p_1 + s_2/p_2 + ... + s_n/p_n, where s_i is the size (e.g., number of animals in the group) of each observation(group=FALSE in options).

Extrapolation and estimation of abundance to the entire survey region is based on either a random sampling design or a stratified random sampling design. Replicate samples(lines)(sample.table are specified within regional strata region.table, if any. If there is no stratification, region.table should contain only a single record with the Area for the entire survey region. The sample.table is linked to the region.table with the Region.Label. The obs.table is linked to the sample.table with the Sample.Label and Region.Label. Abundance can be restricted to a subset (e.g., for a particular species) of the population by limiting the list the observations in obs.table to those in the desired subset. Alternatively, if Sample.Label and Region.Label are in the dataframe used to fit the model, then a subset argument can be given in place of the obs.table. To use the subset argument but include all of the observations, use subset=1==1 to avoid creating an obs.table.

In extrapolating to the entire survey region it is important that the unit measurements be consistent or converted for consistency. A conversion factor can be specified with the convert.units variable in the options list. The values of Area in region.table, must be made consistent with the units for Effort in sample.table and the units of distance in the dataframe that was analyzed. It is easiest to do if the units of Area is the square of the units of Effort and then it is only necessary to convert the units of distance to the units of Effort. For example, if Effort was entered in kilometers and Area in square kilometers and distance in meters then using options=list(convert.units=0.001) would convert meters to kilometers, density would be expressed in square kilometers which would then be consistent with units for Area. However, they can all be in different units as long as the appropriate composite value for convert.units is chosen. Abundance for a survey region can be expressed as: A*N/a where A is Area for the survey region, N is the abundance in the covered (sampled) region, and a is the area of the sampled region and is in units of Effort * distance. The sampled region a is multiplied by convert.units, so it should be chosen such that the result is in the same units of Area. For example, if Effort was entered in kilometers, Area in hectares (100m x 100m) and distance in meters, then using options=list(convert.units=10) will convert a to units of hectares (100 to convert meters to 100 meters for distance and .1 to convert km to 100m units).

If the argument se is set to TRUE, a standard error for density and abundance is computed and the coefficient of variation and log-normal confidence intervals are constructed using a Satterthwaite approximation for degrees of freedom (Buckland et al. 2001 pg 90). The function dht.se computes the variance and interval estimates. The variance has two components: 1) variation due to uncertanity from estimation of the detection function and 2) variation in abundance due to random sample selection. The first component is computed using a delta method estimate of variance (DeltaMethod (Huggins 1989, 1991, Borchers et al. 1998) in which the first derivatives of the abundance estimator with respect to the parameters in the detection function are computed numerically. The second component can be computed in one of three ways as set by the option varflag with values 0,1,2. A value of 0 is to use a binomial variance for the number of observations and it is only useful if the sampled region is the survey region and the objects are not clustered which will not occur very often. A value of 1 uses the standard variance for the encounter rate (Buckland et al. 2001 pg 78-79, although the actual encounter rate formula used by default is now estimator R2 from Fewster et al. (2009) - see varn for details). If the population is clustered the mean group size and standard error is also included. This variance estimator is not appropriate if size or a derivative of size is used in the any of the detection function models. In general if any covariates are used in the models, the default option 2 is preferable. It uses the variance estimator suggested by Innes et al (2002) which used the formula for the variance ecounter rate but replaces the number of observations per sample with the estimated abundance per sample. This latter variance is also given in Marques and Buckland (2004).

The argument options is a list of variable=value pairs that set options for the analysis. All but one of these has been described so far. The remaining variable pdelta should not need to be changed but was included for completeness. It controls the precision of the first derivative calculation for the delta method variance.

Value

list object of class dht with elements:

The list structure of clusters and individuals are the same:

dht options

Several options are available to control calculations and output:

ci.width

Confidence iterval width, expressed as a decimal between 0 and 1 (default 0.95, giving a 95% CI)

pdelta

delta value for computing numerical first derivatives (Default: 0.001)

varflag

0,1,2 (see Details) (Default: 2)

convert.units

multiplier for width to convert to units of length (Default: 1)

ervar

encounter rate variance type - see type argument to varn (Default: "R2")

Author(s)

Jeff Laake

References

Borchers, D.L., S.T. Buckland, P.W. Goedhart, E.D. Clarke, and S.L. Hedley. 1998. Horvitz-Thompson estimators for double-platform line transect surveys. Biometrics 54: 1221-1237.

Borchers, D.L. and K.P. Burnham. General formulation for distance sampling pp 10-11 In: Advanced Distance Sampling, eds. S.T. Buckland, D.R.Anderson, K.P. Burnham, J.L. Laake, D.L. Borchers, and L. Thomas. Oxford University Press.

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

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

Huggins, R.M. 1989. On the statistical analysis of capture experiments. Biometrika 76:133-140.

Huggins, R.M. 1991. Some practical aspects of a conditional likelihood approach to capture experiments. Biometrics 47: 725-732.

Innes, S. M.P. Heide-Jorgensen, J.L. Laake, K.L. Laidre, H.J. Cleator, P. Richard, and R.E.A. Stewart. 2002. Surveys of belugas and narwhals in the Canadian High Arctic in 1996. NAMMCO Scientific Publications 4: 169-190.

Marques, F.F.C. and S.T. Buckland. 2004. Covariate models for the detection function. In: Advanced Distance Sampling, eds. S.T. Buckland, D.R.Anderson, K.P. Burnham, J.L. Laake, D.L. Borchers, and L. Thomas. Oxford University Press.

See Also

print.dht dht.se

Results