Nmix.chisq(mod)
Nmix.gof.test(mod, nsim = 5, plot.hist = TRUE)It is also possible to obtain an estimate of the overdispersion parameter (c-hat) for the model at hand by dividing the observed chi-square statistic by the mean of the statistics obtained from simulation (MacKenzie and Bailey 2004, McKenny et al. 2006). This method of estimating c-hat is similar to the one implemented for capture-mark-recapture models in program MARK (White and Burnham 1999).
Note that values of c-hat > 1 indicate overdispersion (variance > mean). Values much higher than 1 (i.e., > 4) probably indicate lack-of-fit. In cases of moderate overdispersion, one usually multiplies the variance-covariance matrix of the estimates by c-hat. As a result, the SE's of the estimates are inflated (c-hat is also known as a variance inflation factor).
In model selection, c-hat should be estimated from the global model and the same value of c-hat applied to the entire model set. Specifically, a global model is the most complex model from which all the other models of the set are simpler versions (nested). When no single global model exists in the set of models considered, such as when sample size does not allow a complex model, one can estimate c-hat from 'subglobal' models. Here, 'subglobal' models denote models from which only a subset of the models of the candidate set can be derived. In such cases, one can use the smallest value of c-hat for model selection (Burnham and Anderson 2002).
Note that c-hat counts as an additional parameter estimated and should be added to K. All functions in package 'AICcmodavg' automatically add 1 when the 'c.hat' argument > 1 and apply the same value of c-hat for the entire model set. When c-hat > 1, functions compute quasi-likelihood information criteria (either QAICc or QAIC, depending on the value of the 'second.ord' argument) by scaling the log-likelihood of the model by c-hat. The value of c-hat can influence the ranking of the models: as c-hat increases, QAIC or QAICc will favor models with fewer parameters. As an additional check against this potential problem, one can generate several model selection tables by incrementing values of c-hat to assess the model selection uncertainty. If ranking changes little up to the c-hat value observed, one can be confident in making inference.
In cases of underdispersion (c-hat < 1), it is recommended to keep the value of c-hat to 1. However, note that values of c-hat << 1 can also indicate lack-of-fit and that an alternative model should be investigated.
MacKenzie, D. I., Bailey, L. L. (2004) Assessing the fit of site-occupancy models. Journal of Agricultural, Biological, and Environmental Statistics 9, 300--318.
McKenny, H. C., Keeton, W. S., Donovan, T. M. (2006). Effects of structural complexity enhancement on eastern red-backed salamander (Plethodon cinereus) populations in northern hardwood forests. Forest Ecology and Management 230, 186--196. White, G. C., Burnham, K. P. (1999). Program MARK: Survival estimation from populations of marked animals. Bird Study 46 (Supplement), 120--138.
AICc, c_hat, evidence,
modavg, importance,
mb.gof.test, modavgpred,
pcount, pcountOpen,
parboot##N-mixture model example modified from ?pcount
if(require(unmarked)) {
##single season
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
obsCovs = mallard.obs)
##run model
fm.mallard <- pcount(~ ivel+ date + I(date^2) ~ length + elev + forest,
mallardUMF, K=30)
##compute observed chi-square
obs <- Nmix.chisq(fm.mallard)
obs
##round to 4 digits after decimal point
print(obs, digits.vals = 4)
##compute observed chi-square, assess significance, and estimate c-hat
obs.boot <- Nmix.gof.test(fm.mallard, nsim = 3)
##note that more bootstrap samples are recommended
##(e.g., 1000, 5000, or 10 000)
obs.boot
print(obs.boot, digits.vals = 4, digits.chisq = 4)
}
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