c_hat: Estimate Dispersion for Poisson and Binomial GLM's and GLMM's

Description

Functions to compute an estimate of c-hat for binomial or Poisson GLM's and GLMM's using different estimators of overdispersion.

Usage

c_hat(mod, method = "pearson", ...)
# S3 method for glm
c_hat(mod, method = "pearson", ...)
# S3 method for glmmTMB
c_hat(mod, method = "pearson", ...)
# S3 method for merMod
c_hat(mod, method = "pearson", ...)
# S3 method for vglm
c_hat(mod, method = "pearson", ...)

Value

c_hat returns an object of class c_hat with the estimated c-hat value and an attribute for the type of estimator used.

Arguments

mod: an object of class glm, glmmTMB, merMod, or vglm for which a c-hat estimate is required.
method: this argument defines the estimator used. The default "pearson" uses the Pearson chi-square divided by the residual degrees of freedom. Other methods include "deviance" consisting of the residual deviance divided by the residual degrees of freedom, "farrington" for the estimator suggested by Farrington (1996), and "fletcher" for the estimator suggested by Fletcher (2012).
...: additional arguments passed to the function.

Author

Marc J. Mazerolle

Details

Poisson and binomial GLM's do not have a parameter for the variance and it is usually held fixed to 1 (i.e., mean = variance). However, one must check whether this assumption is appropriate by estimating the overdispersion parameter (c-hat). Though one can obtain an estimate of c-hat by dividing the residual deviance by the residual degrees of freedom (i.e., method = "deviance"), McCullagh and Nelder (1989) and Venables and Ripley (2002) recommend using Pearson's chi-square divided by the residual degrees of freedom (method = "pearson"). An estimator based on Farrington (1996) is also implemented by the function using the argument method = "farrington". Recent work by Fletcher (2012) suggests that an alternative estimator performs better than the above-mentioned methods in the presence of sparse data and is now implemented with method = "fletcher". For GLMM's, only the Pearson chi-square estimator of overdispersion is currently implemented.

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 of the candidate model set and the same value of c-hat applied to the entire model set. Specifically, a global model is the most complex model which can be simplified to obtain all the other (nested) models of the set. 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 create 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 (and distribution) should be investigated.

Note that c_hat only supports the estimation of c-hat for binomial models with trials > 1 (i.e., success/trial or cbind(success, failure) syntax) or with Poisson GLM's or GLMM's.

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

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

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261--304.

Farrington, C. P. (1996) On assessing goodness of fit of generalized linear models to sparse data. Journal of the Royal Statistical Society B 58, 349--360.

Fletcher, D. J. (2012) Estimating overdispersion when fitting a generalized linear model to sparse data. Biometrika 99, 230--237.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169--180.

McCullagh, P., Nelder, J. A. (1989) Generalized Linear Models. Second edition. Chapman and Hall: New York.

Venables, W. N., Ripley, B. D. (2002) Modern Applied Statistics with S. Second edition. Springer: New York.

Examples

Run this code

#binomial glm example
set.seed(seed = 10)
resp <- rbinom(n = 60, size = 1, prob = 0.5)
set.seed(seed = 10)
treat <- as.factor(sample(c(rep(x = "m", times = 30), rep(x = "f",
                                           times = 30))))
age <- as.factor(c(rep("young", 20), rep("med", 20), rep("old", 20)))
#each invidual has its own response (n = 1)
mod1 <- glm(resp ~ treat + age, family = binomial)
if (FALSE) {
c_hat(mod1) #gives an error because model not appropriate for
##computation of c-hat
}

##computing table to summarize successes
table(resp, treat, age)
dat2 <- as.data.frame(table(resp, treat, age)) #not quite what we need
data2 <- data.frame(success = c(9, 4, 2, 3, 5, 2),
                    sex = c("f", "m", "f", "m", "f", "m"),
                    age = c("med", "med", "old", "old", "young",
                      "young"), total = c(13, 7, 10, 10, 7, 13))
data2$prop <- data2$success/data2$total
data2$fail <- data2$total - data2$success

##run model with success/total syntax using weights argument
mod2 <- glm(prop ~ sex + age, family = binomial, weights = total,
            data = data2)
c_hat(mod2)

##run model with other syntax cbind(success, fail)
mod3 <- glm(cbind(success, fail) ~ sex + age, family = binomial,
            data = data2) 
c_hat(mod3)

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