Compute MacKenzie and Bailey Goodness-of-fit Test for Single Season and Dynamic Occupancy Models

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Description

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

Usage

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mb.chisq(mod, print.table = TRUE, ...)

## S3 method for class 'unmarkedFitOccu'
mb.chisq(mod, print.table = TRUE, ...)

## S3 method for class 'unmarkedFitColExt'
mb.chisq(mod, print.table = TRUE, ...)

mb.gof.test(mod, nsim = 5, plot.hist = TRUE, report = NULL, ...)

## S3 method for class 'unmarkedFitOccu'
mb.gof.test(mod, nsim = 5, plot.hist = TRUE,
                     report = NULL, ...)

## S3 method for class 'unmarkedFitColExt'
mb.gof.test(mod, nsim = 5, plot.hist = TRUE,
                     report = NULL, plot.seasons = FALSE, ...)

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. For dynamic occupancy models, this produces a histogram of the sum of the season-specific chi-squares for each bootstrap sample.

report

If NULL, the test statistic for each iteration is not printed in the terminal. Otherwise, an integer indicating the number of values of the test statistic that should be printed on the same line. For example, if report = 3, the values of the test statistic for three iterations are reported on each line.

plot.seasons

logical. For dynamic occupancy models, specifies that a histogram of the bootstrapped test statistic for each primary period (season) is to be included in the output.

...

additional arguments passed to the function.

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 (MacKenzie et al. 2002). 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.

This test is extended to dynamic occupancy models of MacKenzie et al. (2003) by using the occupancy estimates for each season obtained from the model. These estimates are then used to compute the predicted and observed frequencies separately within each season. The chi-squares are then summed to be used as the test statistic for the dynamic occupancy model.

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.

Value

mb.chisq returns the following components for single-season occupancy models:

chisq.table

the table of observed and expected values for each detection history and its chi-square component (if print.table = TRUE). Note that the table only shows the observed detection histories. Unobserved detection histories are not shown, but are included in the computation of the test statistic.

chi.square

the Pearson chi-square statistic. This test statistic should be compared against a bootstrap distribution instead of the theoretical chi-square distribution because low expected frequencies invalidate the chi-square assumption.

model.type

the model type, either single-season or dynamic.

mb.chisq returns the following additional components for dynamic occupancy models:

tables

a list containing the season-specific chi-square tables (if print.table = TRUE).

all.chisq

an element containing the season-specific chi-squares.

n.seasons

the number of primary periods (seasons).

mb.gof.test returns the following components for single-season occupancy models:

chisq.table

the table of observed and expected values for each detection history and its chi-square component.

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.

mb.gof.test returns the following additional components for dynamic occupancy models:

chisq.table

a list including the table of observed and expected values for each detection history and its chi-square component for each primary period (season).

chi.square

the chi-square test statistic, as the sum of the chi-squares across the primary periods.

p.value

a list of the P-values for each of the primary periods, computed separately as the proportion of the simulated test statistics greater than or equal to the observed test statistic.

p.global

the P-value of the chi-square test statistic for the dynamic occupancy model. This P-value is computed as the proportion of the simulated sums of chi-squares greater than or equal to the observed sum of chi-squares across the primary periods.

Author(s)

Marc J. Mazerolle

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., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

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, colext, evidence, modavg, importance, modavgPred, Nmix.gof.test, occu, parboot

Examples

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##single-season occupancy model example modified from ?occu
## Not run: 
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)


##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)


##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)

## End(Not run)

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