= Poisson trick
{c}
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 probabilities
$$
\pi_{gj}=\frac{\mu_{gj}}{\sum_k\mu_{gk}}.
$$
Including 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.
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