 Conditional multinomial sufficient statistics

For independent multinomial distributions with fixed stratum totals $N_s$ and baseline logits $\log (p_{sl}/p_{sL})=x_s^T{\beta }_l$, the sufficient statistics are the displayed response-specific design-weighted counts, $l<L$. Expanding the log-likelihood gives ${\sum }_{l<L}{\beta }_l^T{T}_l-{\sum }_s{N}_s\log (1+{\sum }_{l<L}{e}^{x_s^T{\beta }_l})$, plus a data-only term. The Fisher-Neyman factorization theorem therefore proves sufficiency; with an unrestricted identifiable canonical parameter, a likelihood-ratio argument proves minimal sufficiency. Factor indicators give the corresponding response/covariate margins.