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 can cause the sampling variance of density estimates to be understated. One solution currently under investigation is to apply a variance inflation factor, a measure of overdispersion, based on the number of individuals detected at each detector (Bischof et al. 2020).

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

Enk uses exact formulae for 'multi', 'proximity' and 'count' detector types. Other types may be simulated by setting a positive value for 'nrepl', which should be large (e.g., nrepl = 10000).

adjustVarD adjusts the SE and confidence limits of density estimates using Fletcher's $\hat c$. The implementation is limited to simple detection models (see Warnings).

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

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

Value

For nk, a vector of observed counts, one for each detector in traps(capthist).

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

For chat.nk, usually 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: $\hat c$ (Fletcher or Wedderburn depending on `type')

There are two variations --

If `verbose = FALSE' then only the numeric value of $\hat c$ is returned (a vector of 2 values if `type = "both"').

If chat.nk is called with `nsim > 0' then the output is a list comprising --

type: from input
nsim: from input
sim.chat: vector of simulated $\hat c$
chat: $\hat c$ (Fletcher or Wedderburn depending on `type')
p: probability of observing $\hat c$ as large as this (from rank of chat among sim.chat)

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

Arguments

capthist: secr capthist object
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
nrepl: integer number of replicates for E(nk) by simulation (optional)
object: fitted secr model or dataframe (see Warnings for restrictions)
nsim: integer number of c-hat values to simulate (optional)
...: other arguments passed to Fletcher.chat (verbose, type)
chatmin: minimum value of Fletcher's $\hat c$
alpha: alpha level for confidence intervals
chat: numeric chat (optional)

Warning

These functions are experimental in secr 4.6, and do not work with polygon-like and single-catch detectors. No allowance is made for modelled variation in detection parameters with respect to occasion, detector or animal; this includes mixture models (e.g., g0~h2).

Versions before 4.5.11 did not correctly compute expected counts for multi-catch detectors.

Furthermore, we doubt that the adjustment actually solves the problem of overdispersion (Efford and Fletcher unpubl.).

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).

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 $s_k = (n_k - E(n_k)) / E(n_k)$.

chat.nk may be used to simulate $\hat c$ values under the given model (set nsim > 0). The ... argument may include 'ncores = x' (x>1) to specify parallel processing of simulations - the speed up is large on unix-like machines for which the cluster type of makeCluster is "FORK" rather than "PSOCK". If 'ncores' is not provided then the value returned by setNumThreads() is used.

No adjustment is made by adjustVarD when $\hat c$ is less than the minimum. adjustVarD by default computes Fletcher's `chat' using chat.nk, but a value may be provided.

If chat has been computed separately and provided in the argument of that name, adjustVarD also accepts a single dataframe as the argument `object'; the dataframe should have row `D' and columns `link', `estimate', `SE.estimate' as in the output from predict.secr.

References

Bischof, R., P. Dupont, C. Milleret, J. Chipperfield, and J. A. Royle. 2020. Consequences of ignoring group association in spatial capture--recapture analysis. Wildlife Biology wlb.00649. DOI 10.2981/wlb.00649

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.

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

Examples

Run this code

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

# }

# useful plotting function for simulated chat (nsim>0)
plotchat <- function(chat, head = '', breaks = seq(0.5,2.5,0.05)) {
    hist(chat$sim.chat, xlim = range(breaks), main = head, xlab = 'c-hat',
        breaks = breaks, cex.main = 1, yaxs = 'i')
    abline(v = chat$chat, lwd = 1.5, col = 'blue')
}

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