What relationship does a negative binomial regression have to heteroskedasticity and if one still needs to check and/or correct for it how would this be done?
Heteroskedasticity is relevant with ordinary linear regression, where there is an assumption that variance is constant (do not depend on the mean), known as homoskedasticity. But with alternative regression models, like poisson regression or negative binomial regression, there is no assumption of constant variance! In its place this models do assume some other mean-variance relationship, in the case of poisson regression that the variance=mean. For negative binomial regression, there o exist several variants with different mean-variance relationships. So in place of investigating violation of homoskedasticity, you should investigate violation of the assumed mean-variance relationship. You could start with a plot of residuals versus predicted values.