Poisson surrogate for a conditional multinomial model (${\mu }_{sl}=e^{{\gamma }_s+x_s^T{\beta }_l}$)

Independent Poisson distributions with a separate nuisance intercept in every stratum become multinomial distributions after conditioning on their totals. Profiling the intercept gives $e^{{\hat{\gamma }}_s}=N_s/{\sum }_l{e}^{x_s^T{\beta }_l}$. Substitution produces the same parameter-dependent log-likelihood as the conditional multinomial model. Both approaches fit ${\hat{\mu }}_{sl}=N_s{\hat{p}}_{sl}$. This exact equivalence differs from approximating a single binomial event count by a rare-event Poisson distribution.