chat: Overdispersion of Activity Centres

Description

Activity centres may be clumped (overdispersed) relative to a Poisson distribution, the model used in secr.fit (Borchers and Efford 2008). This causes the sampling variance of density estimates to be understated. The problem is addressed with a variance inflation factor, essentially a measure of overdispersion.

Functions described here compute the expected number of individuals detected at each detector (Enk) and use that to compute Fletcher's $\hat c$ estimate of overdispersion for use as a variance inflation factor.

adjustVarD adjusts the SE and confidence limits of density estimates using Fletcher's $\hat c$.

See Cooch and White (2022) for an introduction to measurement of overdispersion in capture--recapture. The focus here is on overdispersion of activity centres relative to a Poisson distribution, rather than on non-independence in the spatial detection process.

Usage

Enk(D, mask, traps, detectfn = 0, detectpar = list(g0 = 0.2,
    sigma = 25, z = 1), noccasions = NULL, binomN = NULL,
    userdist = NULL, ncores = NULL) 
chat.nk(object)
adjustVarD(object, chatmin = 1, alpha = 0.05)

Value

For Enk, a vector of expected counts, one for each detector in traps.

For chat.nk, a list comprising --

expected.nk: expected number at each detector
nk: observed number at each detector
stats: vector of summary statistics: mean(expected.nk), var(expected.nk), mean(nk), var(nk), nu (=df), X2/nu
chat: Fletcher's $\hat c$

For adjustVarD, a dataframe with one row for each session, based on predict.secr or derived.secr, with extra column `c-hat'.

Arguments

D: numeric density, either scalar or vector of length nrow(mask)
mask: single-session habitat mask
traps: traps object
detectfn: integer code for detection function q.v.
detectpar: a named list giving a value for each parameter of detection function
noccasions: number of sampling intervals (occasions)
binomN: integer code for discrete distribution (see secr.fit)
userdist: user-defined distance function or matrix (see userdist)
ncores: integer number of threads
object: fitted secr model (see Warnings for restrictions)
chatmin: minimum value of Fletcher's $\hat c$
alpha: alpha level for confidence intervals

Warning

These functions are experimental in secr 4.5.8, and do not work with polygon-like detectors. No allowance is made for modelled variation in detection parameters with respect to occasion, detector or animal.

Details

If traps has a usage attribute then noccasions is set accordingly; otherwise it must be provided.

The environment variable RCPP_PARALLEL_NUM_THREADS determines the number of parallel threads. It is set to the value of ncores, unless that is NULL (see setNumThreads).

The conventional lack-of-fit statistic 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 $s_k = (n_k - E(n_k)) / E(n_k)$.

No adjustment is made by adjustVarD when $\hat c$ is less than the minimum.

References

Cooch, E. and White, G. (eds) (2022) Program MARK: A Gentle Introduction. 22nd edition. Available online at http://www.phidot.org/software/mark/docs/book/.

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

Examples

Run this code

# \donttest{
  temptrap <- make.grid()
  msk <- make.mask(temptrap)
  ## expected number of individuals per detector. 
  Enk (D = 5, msk, temptrap, detectpar = list(g0 = 0.2, sigma = 25),
    noccasions = 5)

  chat.nk(secrdemo.0)    
# }

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