derived  R Documentation 
Compute derived parameters of spatially explicit capturerecapture 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 class '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 class 'secrlist'
derived(object, sessnum = NULL, groups = NULL, alpha = 0.05,
se.esa = FALSE, se.D = TRUE, loginterval = TRUE, distribution = NULL,
ncores = NULL, bycluster = FALSE, ...)
esa(object, sessnum = 1, beta = NULL, real = NULL, noccasions = NULL,
ncores = NULL)
object 

sessnum 
index of session in object$capthist for which output required 
groups 
vector of covariate names to define group(s) (see Details) 
alpha 
alpha level for confidence intervals 
se.esa 
logical for whether to calculate SE(mean(esa)) 
se.D 
logical for whether to calculate SE(Dhat) 
loginterval 
logical for whether to base interval on log(D) 
distribution 
character string for distribution of the number of individuals detected 
ncores 
integer number of threads used for parallel processing 
bycluster 
logical; if TRUE results are reported separately for each cluster of detectors 
beta 
vector of fitted parameters on transformed (link) scale 
real 
vector of ‘real’ parameters 
noccasions 
integer number of sampling occasions (see Details) 
... 
other arguments (not used) 
The derived estimate of density is a HorvitzThompsonlike estimate:
\hat{D} = \sum\limits _{i=1}^{n} {a_i (\hat{\theta})^{1}}
where a_i (\hat{\theta})
is the estimate of effective sampling area for animal
i
with detection parameter vector \theta
.
A nonnull value of the argument distribution
overrides the value
in object$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 nonPoisson.
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$capthist (or each of its components, in the case of a
multisession dataset).
esa
is used by derived
to compute individualspecific
effective sampling areas:
a_i (\hat{\theta}) = \int _A \:
p.(\mathbf{X};\mathbf{z}_i, \mathbf{\hat{\theta}}) \; \mathrm{d}
\mathbf{X}
where
p.(\mathbf{X})
is the probability an individual at X is
detected at least once and the \mathbf{z}_i
are optional
individual covariates. Integration is over the area A
of the
habitat mask.
The argument noccasions
may be used to vary the number of
sampling occasions; it works only when detection parameters are constant
across individuals and across time.
Setting ncores = NULL
uses the existing value from the environment variable
RCPP_PARALLEL_NUM_THREADS (see setNumThreads
).
The effective sampling area ‘esa’ (a(\hat{\theta})
)
reported by derived
is equal to the harmonic mean of the
a_i (\hat{\theta})
(arithmetic
mean prior to version 1.5). The sampling variance of
a(\hat{\theta})
is estimated by
\widehat{\mbox{var}}(a(\hat{\theta})) = \hat{G}_\theta^T
\hat{V}_\theta \hat{G}_\theta,
where \hat{V}
is the asymptotic estimate of the
variancecovariance matrix of the beta detection parameters
(\theta
) and \hat{G}
is a numerical estimate
of the gradient of a(\theta)
with respect to
\theta
, evaluated at \hat{\theta}
.
A 100(1–alpha)% asymptotic confidence interval is reported for density. By default, this is asymmetric about the estimate because the variance is computed by backtransforming from the log scale. You may also choose a symmetric interval (variance obtained on natural scale).
The vector of detection parameters for esa
may be specified via
beta
or real
, with the former taking precedence. If
neither is provided then the fitted values in object$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
.
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(1alpha)% confidence limit 
ucl  upper 100(1alpha)% 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 multisession or multigroup 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’.
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.
Before version 2.1, the output table had columns for ‘varcomp1’ (the variance in \hat{D}
due to variation in n
, i.e.,
Huggins' s^2
), and ‘varcomp2’ (the variance in \hat{D}
due to uncertainty in estimates of detection parameters).
These quantities are related to CVn and CVa as follows:
\mbox{CVn} = \sqrt{ \mbox{varcomp1} } / \hat{D}
\mbox{CVa} = \sqrt{ \mbox{varcomp2} } / \hat{D}
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
par.derived
## Not run:
## 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))
## End(Not run)
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