# Negative binomial GLM: model checking in R

I'm using a negative binomial GLM in R, but I could not solve the following issues (using McCullagh and Nelder (1989), textbooks and Google):

1. Residual plots. McCullagh and Nelder (1989, p.398) recommend plotting the standardized deviance residuals against the fitted values transformed to the constant-information scale of the error distribution. What is the transformation for negative binomial errors? It is not provided by McCullagh and Nelder.

2. Checking the variance function. For a formal check (p.400), the variance function $(\mu + \mu^2/k)$ should be embedded in a suitable family. The improvement of the fit provides information on whether the variance function is adequate (this requires a quasi-likelihood curve; depending on an indexing parameter). Is there an R-package that supports this check?

3. Checking the link function. One test would be to add the estimated linear predictor (squared) as an extra covariate and to assess the fall in deviance. As I understand it, this would be implemented as follows:

data$linear_predictors <- model$linear.predictors
model.including_linear_predictors <- update(model, y ~ . + linear_predictors^2)
summary(model.including_linear_predictors)


But for my model, there is no change in deviance. Is my understanding/implementation correct? McCullagh and Nelder (1989, p.401) also mention a score-test. How would that be implemented in R?

The other formal check would be to embed the link function in a family indexed by a parameter $\lambda$ and test the prior value $\lambda_0$ in the usual way. How would that be implemented?

4. Checking the scales of the covariates. The check involves embedding the current scale x in a family $h(x;\theta)$ indexed by $\theta$; in a suitable grid, the $\theta$ that minimizes the deviance should be identified. Is this supported by an R package?

I know it's a lot - but I could not find answers to these questions on Google or in textbooks. I highly appreciate any help/suggestions!