Poisson trick

For each group $g$, let independent cell counts have Poisson means ${\mu }_{gj}$. Conditional on their total $n_g$, the cells have a multinomial distribution with probabilitiesIncluding a free group main effect in the Poisson log-linear model makes its profiled likelihood for the remaining parameters proportional to the corresponding multinomial likelihood. The two fits therefore give the same fitted proportions and likelihood-ratio comparisons.