Computes standard error, cv, and log-normal confidence intervals for abundance and density within each region (if any) and for the total of all the regions. It also produces the correlation matrix for regional and total estimates.
dht.se(
model,
region.table,
samples,
obs,
options,
numRegions,
estimate.table,
Nhat.by.sample
)List with 2 elements:
completed table with se, cv and confidence limits
correlation matrix of estimates
ddf model object
table of region values
table of samples(replicates)
table of observations
list of options that can be set (see dht)
number of regions
table of estimate values
estimated abundances by sample
Jeff Laake
The variance has two components:
variation due to uncertainty from estimation of the detection function parameters;
variation in abundance due to random sample selection.
The first component (model parameter uncertainty) is computed using a delta
method estimate of variance (huggins1989;nobracketsmrds; huggins1991;nobracketsmrds; borchers1998;nobracketsmrds) in
which the first derivatives of the abundance estimator with respect to the
parameters in the detection function are computed numerically (see
DeltaMethod).
The second component (encounter rate variance) can be computed in one of
several ways depending on the form taken for the encounter rate and the
estimator used. To begin with there three possible values for varflag
to calculate encounter rate:
0 uses a negative binomial variance for the number of
observations (equation 13 of borchers1998;nobracketsmrds).
This estimator is only
useful if the sampled region is the survey region and the objects are not
clustered; this situation will not occur very often;
1 uses the encounter rate \(n/L\) (objects observed per unit
transect) from buckland2001;textualmrds pg 78-79 (equation 3.78) for line
transects (see also fewster2009;nobracketsmrds estimator R2). This variance
estimator is not appropriate if size or a derivative of size
is used in the detection function;
2 is the default and uses the encounter rate estimator
\(\hat{N}/L\) (estimated abundance per unit transect) suggested by
innes2002;textualmrds and marques2004;textualmrds.
In general if any covariates are used in the models, the default
varflag=2 is preferable as the estimated abundance will take into
account variability due to covariate effects. If the population is clustered
the mean group size and standard error is also reported.
For options 1 and 2, it is then possible to choose one of the
estimator forms given in fewster2009;textualmrds. For line transects:
"R2", "R3", "R4", "S1", "S2",
"O1", "O2" or "O3" can be used by specifying ervar
in the list of options provided by the options argument
(default "R2"). For points, either the
"P2" or "P3" estimator can be selected (>=mrds 2.3.0
default "P2", <= mrds 2.2.9 default "P3"). See
varn and fewster2009;textualmrds
for further details on these estimators.
Exceptions to the above occur if there is only one sample in a stratum. In
this situation, varflag=0 continues to use a negative binomial
variance while the other options assume a Poisson variance (\(Var(x)=x\)),
where when varflag=1 x is number of detections in the covered region and
when varflag=2 x is the abundance in the covered region. It also assumes
a known variance so \(z=1.96\) is used for critical value. In all other cases
the degrees of freedom for the \(t\)-distribution assumed for the
log(abundance) or log(density) is based on the Satterthwaite approximation
(buckland2001;nobracketsmrds pg 90) for the degrees of freedom (df). The df are
weighted by the squared cv in combining the two sources of variation because
of the assumed log-normal distribution because the components are
multiplicative. For combining df for the sampling variance across regions
they are weighted by the variance because it is a sum across regions.
The coefficient of variation (CV) associated with the abundance estimates is calculated based on the following formula for the varflag options 1 and 2:
varflag=1
$$CV(\hat{N}) = \sqrt{\left(\frac{\sqrt{n}}{n}\right)^2+CV(\hat{p})^2}$$
varflag=2
$$CV(\hat{N}) = \sqrt{\left(\frac{\sqrt{\hat{N}}}{\hat{N}}\right)^2+CV(\hat{p})^2}$$ where n is the number of observations, \(\hat{N}\) is the estimated abundance and \(\hat{p}\) is the average probability of detection for an animal in the covered area.
A non-zero correlation between regional estimates can occur from using a common detection function across regions. This is reflected in the correlation matrix of the regional and total estimates which is given in the value list. It is only needed if subtotals of regional estimates are needed.
dht, print.dht