I have area, period, popsum, and nsum, as well as log of popsum, which is used as the offset term for the regression. So I put:

proc genmod data=want;

class period area;

model nsum= area period area*period / dist=NB link=log offset=logpop type1;

run;

I have 51 observations (17 years for each of the 3 study areas) and for area 1 nsum is around 3,300 for each year, area 2 nsum is around 14,000, and for area 3 nsum is around 10,000. The log says 'fitting saturated model' and I get p=1.000 for the interaction term under LR Statistics for Type 1 Analysis. In the output with the actual estimates, period*area shows p around 0.96-0.99 for each year and areas 1/2 (with area 3 as the reference group).

I tried doing Poisson instead of NB and it still says the model is saturated, but gives a p value for the interaction term instead of p=1.000, so not totally sure which way I should be going (and if I should be including type 1 or type 3 analysis). Thanks!