Must the response variable be gamma distributed to appropriately use a gamma-log model?

Question

I'm responsible for challenging a gamma model with log link. The developer claims that an assumption of the gamma-log generalized linear model is that the response variable, in this case average balances is distributed gamma. He then uses distributional tests (like Anderson-Darling) to test the empirical distribution and support his claim.

I'm a little unsure whether this is actually the assumption for generalized linear model and whether this method is appropriate to support his claim. Most of the assumptions I've been able to find about the GLM are around the error term and the selection of the link function. Other generalized linear models, like a logistic regression have an empirical binomial distribution, but use the logit because it theoretically maps the probabilities between 0 and 1.

Furthermore, the Anderson-Darling test seems inappropriate as a method to select distributions for the generalized linear model.

Answer 1

The developer claims that an assumption of the gamma-log generalized linear model is that the response variable, in this case average balances is distributed gamma. He then uses distributional tests (like Anderson-Darling) to test the empirical distribution and support his claim.

It sounds like you're saying he's testing the unconditional distribution of the response variable.

I'm a little unsure whether this is actually the assumption for generalized linear model

It isn't. The assumption is that the conditional distribution (i.e. at each fixed value of the predictors) is gamma; the unconditional distribution would then be a mixture of gammas, which might look distinctly non-gamma; similarly, a distinctly non-gamma conditional distribution may look gamma when treated unconditionally -- it depends on the arrangement of the predictor variables.

and whether this method is appropriate to support his claim. [...] Furthermore, the Anderson-Darling test seems inappropriate as a method to select distributions for the generalized linear model.

Failure to reject the goodness of fit test wouldn't be appropriate for establishing the suitability of the gamma even if the problematic issue of testing the unconditional distribution was avoided. It would at best establish that you couldn't rule it out (which would just indicate that the sample size was too small). The problem with goodness of fit testing is that we shouldn't believe that the distribution is exactly gamma in any case -- the question of whether it's close enough that our inference isn't unduly affected (a question of effect size in some sense) is not answered by a hypothesis test.