Nmix.gof.test: Compute Chi-square Goodness-of-fit Test for N-mixture Models

Description

These functions compute a goodness-of-fit test for N-mixture models based on Pearson's chi-square.

Usage

Nmix.chisq(mod)
Nmix.gof.test(mod, nsim = 5, plot.hist = TRUE)

Arguments

mod

the N-mixture model of 'unmarkedFitPCount' or 'unmarkedFitPCO' classes for which a goodness-of-fit test is required.

nsim

the number of bootstrapped samples.

plot.hist

logical. Specifies that a histogram of the bootstrapped test statistic is to be included in the output.

Value

'Nmix.chisq' returns a single value:
chi.squarethe Pearson chi-square statistic.
'Nmix.gof.test' returns the following components:
chi.squarethe Pearson chi-square statistic.
t.starthe bootstrapped chi-square test statistics (i.e., obtained for each of the simulated data sets).
p.valuethe P-value assessed from the parametric bootstrap, computed as the proportion of the simulated test statistics greater than or equal to the observed test statistic.
c.hat.estthe estimate of the overdispersion parameter, c-hat, computed as the observed test statistic divided by the mean of the simulated test statistics.
nsimthe number of bootstrap samples. The recommended number of samples varies with the data set, but should be on the order of 1000 or 5000, and in cases with a large number of visits, even 10 000 samples, namely to reduce the effect of unusually small values of the test statistics.

Details

The Pearson chi-square can be used to assess the fit of N-mixture models. Instead of relying on the theoretical distribution of the chi-square, a parametric bootstrap approach is implemented to obtain P-values with the 'parboot' function of the unmarked package. 'Nmix.chisq' computes the observed chi-square statistic based on the observed and expected values from the model. 'Nmix.gof.test' calls internally 'Nmix.chisq' and 'parboot' to generate simulated data sets based on the model and compute the chi-square test statistic.

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.

References

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

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.

Examples

Run this code

##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)
}
detach(package:unmarked)

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