AICcmodavg (version 2.00)

mb.gof.test: Compute MacKenzie and Bailey Goodness-of-fit Test for Single Season Occupancy Models

Description

These functions compute the MacKenzie and Bailey (2004) goodness-of-fit test for single season occupancy models based on Pearson's chi-square.

Usage

mb.chisq(mod, print.table = TRUE)
mb.gof.test(mod, nsim = 5, plot.hist = TRUE)

Arguments

mod

the model for which a goodness-of-fit test is required.

print.table

logical. Specifies if the detailed table of observed and expected values is to be included in the output.

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

• mb.chisq returns the following components:
• chisq.table
  the table of observed and expected values for each detection history and its chi-square component (if print.table = TRUE).
• chi.square
  the Pearson chi-square statistic.
• mb.gof.test returns the following components:
• chisq.table
  the table of observed and expected values for each detection history and its chi-square component (if print.table = TRUE).
• chi.square
  the Pearson chi-square statistic.
• t.star
  the bootstrapped chi-square test statistics (i.e., obtained for each of the simulated data sets).
• p.value
  the 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.est
  the estimate of the overdispersion parameter, c-hat, computed as the observed test statistic divided by the mean of the simulated test statistics.
• nsim
  the 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

MacKenzie and Bailey (2004) and MacKenzie et al. (2006) suggest using the Pearson chi-square to assess the fit of single season occupancy models. Given low expected frequencies, the chi-square statistic will deviate from the theoretical distribution and it is recommended to use a parametric bootstrap approach to obtain P-values with the parboot function of the unmarked package. mb.chisq computes the table of observed and expected values based on the detection histories and single season occupancy model used. mb.gof.test calls internally mb.chisq and parboot to generate simulated data sets based on the model and compute the MacKenzie and Bailey test statistic. Missing values are accomodated by creating cohorts for each pattern of missing values.

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.

Note that values of c-hat > 1 indicate overdispersion (variance > mean), but that 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.

MacKenzie, D. I., Nichols, J. D., Royle, J. A., Pollock, K. H., Bailey, L. L., Hines, J. E. (2006) Occupancy estimation and modeling: inferring patterns and dynamics of species occurrence. Academic Press: New York.

See Also

AICc, c_hat, evidence, modavg, importance, modavgPred, Nmix.gof.test, occu, parboot

Examples

##single-season occupancy model example modified from ?occu
if(require(unmarked)) {
##single season
data(frogs)
pferUMF <- unmarkedFrameOccu(pfer.bin)
## add some fake covariates for illustration
siteCovs(pferUMF) <- data.frame(sitevar1 = rnorm(numSites(pferUMF)),
                                sitevar2 = rnorm(numSites(pferUMF))) 
     
## observation covariates are in site-major, observation-minor order
obsCovs(pferUMF) <- data.frame(obsvar1 = rnorm(numSites(pferUMF) *
                                 obsNum(pferUMF))) 

##run model
fm1 <- occu(~ obsvar1 ~ sitevar1, pferUMF)

##compute observed chi-square
obs <- mb.chisq(fm1)
obs
##round to 4 digits after decimal point
print(obs, digits.vals = 4)
}

\dontrun{
##compute observed chi-square, assess significance, and estimate c-hat
obs.boot <- mb.gof.test(fm1, 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)
}


\dontrun{
##data with missing values
mat1 <- matrix(c(0, 0, 0), nrow = 120, ncol = 3, byrow = TRUE)
mat2 <- matrix(c(0, 0, 1), nrow = 23, ncol = 3, byrow = TRUE)
mat3 <- matrix(c(1, NA, NA), nrow = 42, ncol = 3, byrow = TRUE)
mat4 <- matrix(c(0, 1, NA), nrow = 33, ncol = 3, byrow = TRUE)
y.mat <- rbind(mat1, mat2, mat3, mat4)
y.sim.data <- unmarkedFrameOccu(y = y.mat)
m1 <- occu(~ 1 ~ 1, data = y.sim.data)

mb.gof.test(m1, nsim = 3)
##note that more bootstrap samples are recommended
##(e.g., 1000, 5000, or 10 000) 
}

detach(package:unmarked)