derived {secr} | R Documentation

## Derived Parameters of Fitted SECR Model

### 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

    derived(object, sessnum = NULL, groups = NULL, alpha = 0.05,
        se.esa = FALSE, se.D = TRUE, loginterval = FALSE,
        distribution = NULL)
    esa(object, sessnum = 1, beta = NULL, real = NULL)

### Arguments

`object` | `secr` object output from `secr.fit`
`sessnum` | index of session in object$capthist for which output required
`groups` | indices defining 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(D-hat)
`loginterval` | logical for whether to obtain interval for log(D)
`distribution` | character string for distribution of the number of individuals detected
`beta` | vector of fitted parameters on transformed (link) scale
`real` | vector of 'real' parameters

### 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 theta.

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).

The effective sampling area 'esa' reported by `derived` is equal to the mean of the a_i (theta-hat).

A 100(1–alpha)% asymptotic confidence interval is reported for density. By default, this is symmetrical about the estimate because the variance is computed on the natural scale. You may also choose an asymmetric (lognormal) interval.

`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 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.

### Value

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
varcomp1 | variance due to variation in n (Huggins' s^2)
varcomp2 | variance due to uncertainty in estimates of detection parameters

For a multi-session or multi-group analysis the value is a list with one component for each session and group.

### Author(s)

Murray Efford murray.efford@otago.ac.nz

### References

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.

### See Also

`predict.secr`, `print.secr`, `secr.fit`

### Examples

    ## extract derived parameters from a model fitted previously
    ## by maximizing the conditional likelihood
    data(secrdemo)
    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))

[Package secr version 1.2.11 Index]