Fletcher.chat {secr} | R Documentation |
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.