# weighted glm model selection

Can AIC values between different weighed models be compared to select the best model (ie the model with the lowest weighted AIC)?

For example, if my response variable is the 'Average Sales Per Person' where this equals to (Sum of Sales / Number of People), so naturally I would like to assign more weight to groups with higher number of people.

I run the following two models:

Gamma

glm(formula = AvrSales ~ x + y, family = Gamma(link = "identity"),weights=NumPeople)

Null deviance: 692476  on 2181  degrees of freedom
Residual deviance: 574155  on 2171  degrees of freedom

AIC: 28802851


Normal

glm(formula = AvrSales ~ x + y, family = gaussian(link = "identity"),
weights = NumPeople)

Null deviance: 5.1948e+10  on 2181  degrees of freedom
Residual deviance: 4.4921e+10  on 2171  degrees of freedom

AIC: 33158


Using AIC as a model selection criteria, would the Normal model then be selected since it has a lower AIC? Or are both models bad at explaining the response since the Residual Deviance is so high?

If I remove the weights, then the opposite conclusion can be reached.

Gamma (with out weights)

glm(formula = AvrSales ~ x + y, family = Gamma(link = "identity"))

Null deviance: 10903  on 2181  degrees of freedom
Residual deviance: 10685  on 2171  degrees of freedom

AIC: 25738


Normal (with out weights)

glm(formula = AvrSales ~ x + y, family = gaussian(link = "identity"))

Null deviance: 847754154  on 2181  degrees of freedom
Residual deviance: 824882849  on 2171  degrees of freedom

AIC: 34239


What makes sense here? If I must use weights inside my model, how is then possible to compare different models subjectively? Any advice, or references to literature would be greatly appreciated.