Overdispersed poisson or negative binomial regression I am performing a GLM on count data (insurance claims) and I wish to compare Overdispersed Poisson Regression (ODP) against Negative Binomial regression. 
would know whether there is a practical index (AIC, logLik) that in standard R could support me in fitting which one to use. I am selecting significant predictors with backward deletion (using anova(fittedModel, test="Chisq") type III tests). 
Therefore it is not assumed the final model within each distribution family to have the same predictor sets.
 A: Instead of overdispersed (or quasi-)poisson regression you can use the NB1 distribution, which has the same linear variance function as ODP and a full-fledged likelihood function instead of the quasilikelihood of ODP. NB1 is implemented in the gamlss package as family=NBII, whereas regular Negative Binomial can be called through family=NBI. All credit for this part of the answer goes to @Achim Zeileis for helping me with a similar question here: Why is the Quasipoisson in glm not treated as a special case of Negative Binomial? ,  see his post for more info regarding NB/NB1 (and the confusing naming conventions).
Regarding ANOVA, I have not been able to find a built-in method for gamlss objects, but it is not hard to write your own implementation of the chi-squared test statistic. An example:
set.seed(123)
data = rNBII(100,mu = 6,sigma=0.5)  #generate NB1 data with mean mu=5 and variance (1+sigma)*mu = 9
h0 = gamlss(data~1,family=PO) #null model: poisson
h1 = gamlss(data~1,family=NBII) #alternative model: NB1/ODP

df = h1$df.fit - h0$df.fit
deviance = as.numeric(-2*logLik(h0) + 2*logLik(h1))
p.value = pchisq(deviance,df,lower.tail=F)
> p.value
[1] 0.01429169 #reject the null model at > 95% confidence

A: I am not sure exactly what is meant by "in standard R" but if you are open to downloading packages, I believe the pscl package's vuong function may do what you want. It implements a model comparison test that is designed specifically to compare non-nested models; it can compare nested ones as well, but there are more familiar ones that can serve that purpose. Like most other model comparison tests, it is based on comparing the likelihoods of the two models. The Vuong test involves some correction for parsimony and such as well. 
A decent summary is available at Wikipedia. 
Here's the original citation:
Vuong, Q.H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica. 57(2). 307–333.
