Fletcher.chat: Estimate overdispersion

In secr: Spatially Explicit Capture-Recapture

Estimate overdispersion

Description

General function for estimating a variance inflation factor (\hat c) from observed counts.

Usage

Fletcher.chat (observed, expected, np, verbose = TRUE, 
    type = c('Fletcher', 'Wedderburn', 'both'), multinomial = FALSE)

Arguments

observed	integer vector of observed counts, or a list of such vectors
expected	numeric vector of expected counts
np	integer number of parameters estimated
verbose	logical; if TRUE returns extended output
type	character
multinomial	logical; if TRUE, one df is subtracted for the constraint

Details

Fletcher.chat applies the overdispersion formula of Fletcher (2012) or computes the conventional (Wedderburn 1974) variance inflation factor X^2/df. It is used by chat.nj.

A conventional variance inflation factor due to Wedderburn (1974) is \hat c_X = X^2/(K-p) where K is the number of detectors, p is the number of estimated parameters, and

X^2 = \sum_k (n_k - E (n_k))^2/ E(n_k).

Fletcher's \hat c is an improvement on \hat c_X that is less affected by small expected counts. It is defined by

\hat c = c_X / (1+ \bar s),

where \bar s = \sum_k s_k / K and s_k = (n_k - E(n_k)) / E(n_k).

The inputs ‘observed’ and ‘expected’ are vectors of counts (e.g., number of distinct individuals per detector); ‘observed’ may also be a list of such vectors, possibly simulated.

Value

Output depends on ‘verbose’, ‘observed’ and ‘type’:

– if ‘observed’ is a list of nk vectors (usually generated by simulation) then the output is a vector of (Fletcher or Wedderburn) \hat c values, one element for each component of ‘observed’, unless type = "both" when the output is a list of two such vectors. Argument ‘verbose’ is ignored.

– if ‘observed’ is a simple vector then ‘verbose’ output is a list comprising input values, various summary statistics, and the computed Fletcher overdispersion (‘chat’). The statistic ‘cX2’ is the conventional variance inflation factor of Wedderburn (1974) – X^2/df. For verbose = FALSE, a single estimate of \hat c is returned when type = "Fletcher" or type = "Wedderburn", otherwise a vector of the two estimates.

References

Fletcher, D. (2012) Estimating overdispersion when fitting a generalized linear model to sparse data. Biometrika 99, 230–237.

Wedderburn, R. W. M. (1974) Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika 61, 439–47.

See Also

chat.nj, adjustVarD

secr documentation built on Feb. 24, 2026, 9:07 a.m.