= Conditional multinomial sufficient statistics
{title2=$T_l=\sum_s n_{sl}x_s$}
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 $\sum_{l<L}\beta_l^TT_l-\sum_sN_s\log(1+\sum_{l<L}e^{x_s^T\beta_l})$, 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.
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