AICcmodavg (version 1.0)

c_hat: Compute Estimate of Dispersion for Poisson and Binomial GLM's

Description

This function computes an estimate of c-hat for binomial or Poisson GLM's based on Pearson's chi-square divided by the residual degrees of freedom.

Usage

c_hat(mod)

Arguments

mod

the model for which a c-hat estimate is required.

Value

• 'c_hat' returns the estimated c-hat value

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, McCullagh and Nelder (1989) recommend using Pearson's chi-square divided by the residual degrees of freedom, which performs better. The latter is the method implemented by 'c_hat'.

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 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, on can create 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 (and distribution) should be investigated.

Note that it is only possible to estimate c-hat for binomial models with trials > 1 (i.e., success/trial or cbind(success, failure) syntax) or with Poisson GLM'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.

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.

See Also

AICc, confset, evidence, modavg, importance, modavgpred

Examples

#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)
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)