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).
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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 = 1, ...) ## 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 = 1, ...) esa(object, sessnum = 1, beta = NULL, real = NULL, noccasions = NULL)
index of session in object$capthist for which output required
vector of covariate names to define group(s) (see Details)
alpha level for confidence intervals
logical for whether to calculate SE(mean(esa))
logical for whether to calculate SE(D-hat)
logical for whether to base interval on log(D)
character string for distribution of the number of individuals detected
integer number of cores available for parallel processing
vector of fitted parameters on transformed (link) scale
vector of ‘real’ parameters
integer number of sampling occasions (see Details)
other arguments (not used)
The derived estimate of density is a Horvitz-Thompson-like estimate:
D-hat = sum( a_i (theta-hat)^--1)
where (theta-hat) is the estimate of effective sampling area for animal i with detection parameter vector θ.
A non-null value of the argument
distribution overrides the value
object$details. The sampling variance of D-hat
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
groups should appear in the covariates dataframe of
object$capthist (or each of its components, in the case of a
esa is used by
derived to compute individual-specific
effective sampling areas:
a_i = integral p.(X; z_i, theta_i) dX
where p.(X) is the probability an individual at X is detected at least once and the z_i are optional individual covariates. Integration is over the area A of the habitat mask.
noccasions may be used to vary the number of
sampling occasions; it works only when detection parameters are constant
across individuals and across time.
The effective sampling area ‘esa’ (a(theta-hat))
derived is equal to the harmonic mean of the
a_i (theta-hat) (arithmetic
mean prior to version 1.5). The sampling variance of
a(theta-hat) is estimated by
var(a(theta)) = G-hat^T V-hat G-hat,
where V-hat is the asymptotic estimate of the variance-covariance matrix of the beta detection parameters (theta) and G-hat is a numerical estimate of the gradient of a(theta) with respect to theta, evaluated at theta-hat.
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
real, with the former taking precedence. If
neither is provided then the fitted values in
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
se.D as desired.
ncores > 1 the parallel package is used to create
processes on multiple cores (see Parallel for more).
derived across multiple models in parallel see
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’.
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 D-hat due to variation in n, i.e., Huggins' s^2), and ‘varcomp2’ (the variance in D-hat due to uncertainty in estimates of detection parameters).
These quantities are related to CVn and CVa as follows:
CVn = sqrt(varcomp1) / D-hat
CVa = sqrt(varcomp2) / D-hat
Borchers, D. L. and Efford, M. G. (2007) Supplements to Biometrics paper. Available online at http://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.
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## 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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