Description

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).

Usage

Arguments

Details

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 in object$details. The sampling variance of D-hat 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 = 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.

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(theta-hat)) reported by 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 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.

Value

Dataframe with one row for each derived parameter (‘esa’, ‘D’) and columns as below

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’.

Warning

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.

Note

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

References

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.

See Also

predict.secr, print.secr, secr.fit, empirical.varD par.derived

Examples