Compute derived parameters of spatially explicit capture-recapture model. Density is a derived parameter when a model is fitted by maximizing the conditional likelihood. So also is the effective sampling area (in the sense of Borchers and Efford 2008).
derived(object, ...)# S3 method for secr
derived(object, sessnum = NULL, groups = NULL, alpha = 0.05,
se.esa = FALSE, se.D = TRUE, loginterval = TRUE, distribution = NULL,
ncores = NULL, bycluster = FALSE, ...)
# S3 method for secrlist
derived(object, sessnum = NULL, groups = NULL, alpha = 0.05,
se.esa = FALSE, se.D = TRUE, loginterval = TRUE, distribution = NULL,
ncores = NULL, bycluster = FALSE, ...)
# S3 method for secr
esa(object, sessnum = 1, beta = NULL, real = NULL, noccasions = NULL,
ncores = NULL, ...)
Dataframe with one row for each derived parameter (`esa', `D') and columns as below
| estimate | estimate of derived parameter |
| SE.estimate | standard error of the estimate |
| lcl | lower 100(1--alpha)% confidence limit |
| ucl | upper 100(1--alpha)% confidence limit |
| CVn | relative SE of number observed (Poisson or binomial assumption) |
| CVa | relative SE of effective sampling area |
| CVD | relative SE of density estimate |
For a multi-session or multi-group analysis the value is a list with one component for each session and group.
The result will also be a list if object is an `secrlist'.
secr object output from secr.fit, or an
object of class c("secrlist", "list")
index of session in object$capthist for which output required
vector of covariate names to define group(s) (see Details)
alpha level for confidence intervals
logical for whether to calculate SE(mean(esa))
logical for whether to calculate SE(D-hat)
logical for whether to base interval on log(D)
character string for distribution of the number of individuals detected
integer number of threads used for parallel processing
logical; if TRUE results are reported separately for each cluster of detectors
vector of fitted parameters on transformed (link) scale
vector of `real' parameters
integer number of sampling occasions (see Details)
other arguments passed to fdHess or ignored (esa)
derived() may be applied to detection models fitted by maximizing the full likelihood (CL = FALSE). However, models using g (groups) will not be handled correctly.
The derived estimate of density is a Horvitz-Thompson-like estimate: $$\hat{D} = \sum\limits _{i=1}^{n} {a_i (\hat{\theta})^{-1}}$$ where \(a_i (\hat{\theta})\) is the estimate of effective sampling area for animal \(i\) with detection parameter vector \(\theta\).
A non-null value of the argument distribution overrides the value
in object$details. The sampling variance of \(\hat{D}\)
from secr.fit by default is spatially unconditional
(distribution = "Poisson"). For sampling variance conditional on the population of the
habitat mask (and therefore dependent on the mask area), specify
distribution = "binomial". The equation for the conditional
variance includes a factor \((1 - a/A)\) that disappears in the
unconditional (Poisson) variance (Borchers and Efford 2007). Thus the
conditional variance is always less than the unconditional variance. The
unconditional variance may in turn be an overestimate or (more likely)
an underestimate if the true spatial variance is non-Poisson.
Derived parameters may be estimated for population subclasses (groups)
defined by the user with the groups argument. Each named factor
in groups should appear in the covariates dataframe of
object$capthist (or each of its components, in the case of a
multi-session dataset).
esa is used by derived to compute individual-specific
effective sampling areas: $$a_i (\hat{\theta}) = \int _A \:
p.(\mathbf{X};\mathbf{z}_i, \mathbf{\hat{\theta}}) \; \mathrm{d}
\mathbf{X}$$ where
\(p.(\mathbf{X})\) is the probability an individual at X is
detected at least once and the \(\mathbf{z}_i\) are optional
individual covariates. Integration is over the area \(A\) of the
habitat mask.
The argument noccasions may be used to vary the number of
sampling occasions; it works only when detection parameters are constant
across individuals and across time.
Setting ncores = NULL uses the existing value from the environment variable
RCPP_PARALLEL_NUM_THREADS (see setNumThreads).
The effective sampling area `esa' (\(a(\hat{\theta})\))
reported by derived is equal to the harmonic mean of the
\(a_i (\hat{\theta})\) (arithmetic
mean prior to version 1.5). The sampling variance of
\(a(\hat{\theta})\) is estimated by
$$\widehat{\mbox{var}}(a(\hat{\theta})) = \hat{G}_\theta^T
\hat{V}_\theta \hat{G}_\theta, $$
where \(\hat{V}\) is the asymptotic estimate of the
variance-covariance matrix of the beta detection parameters
(\(\theta\)) and \(\hat{G}\) is a numerical estimate
of the gradient of \(a(\theta)\) with respect to
\(\theta\), evaluated at \(\hat{\theta}\).
A 100(1--alpha)% asymptotic confidence interval is reported for density. By default, this is asymmetric about the estimate because the variance is computed by backtransforming from the log scale. You may also choose a symmetric interval (variance obtained on natural scale).
The vector of detection parameters for esa may be specified via
beta or real, with the former taking precedence. If
neither is provided then the fitted values in object$fit$par are
used. Specifying real parameter values bypasses the various
linear predictors. Strictly, the `real' parameters are for a naive
capture (animal not detected previously).
The computation of sampling variances is relatively slow and may be
suppressed with se.esa and se.D as desired.
For computing derived across multiple models in parallel see
par.derived.
From secr 5.0.0 the ... argument may be used to control the step size
(.relStep) used by fdHess when estimating gradients for SE(D) and SE(esa).
Borchers, D. L. and Efford, M. G. (2007) Supplements to Biometrics paper. Available online at https://www.otago.ac.nz/density/.
Borchers, D. L. and Efford, M. G. (2008) Spatially explicit maximum likelihood methods for capture--recapture studies. Biometrics, 64, 377--385.
Huggins, R. M. (1989) On the statistical analysis of capture experiments. Biometrika 76, 133--140.
predict.secr,
print.secr,
secr.fit,
empirical.varD
if (FALSE) {
## extract derived parameters from a model fitted previously
## by maximizing the conditional likelihood
derived (secrdemo.CL)
## what happens when sampling variance is conditional on mask N?
derived(secrdemo.CL, distribution = "binomial")
## fitted g0, sigma
esa(secrdemo.CL)
## force different g0, sigma
esa(secrdemo.CL, real = c(0.2, 25))
}
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