# Compute MacKenzie and Bailey Goodness-of-fit Test for Single Season and Dynamic 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 and extend it to dynamic (multiple season) occupancy models.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
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, ...)
## S3 method for class 'unmarkedFitOccu'
mb.gof.test(mod, nsim = 5, plot.hist = TRUE,
...)
## S3 method for class 'unmarkedFitColExt'
mb.gof.test(mod, nsim = 5, plot.hist = TRUE,
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. |

`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 |

`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 |

`mb.chisq`

returns the following additional components for dynamic
occupancy models:

`tables` |
a list containing the season-specific chi-square tables
(if |

`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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | ```
##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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