 Baseline logit estimator in a group-factor binomial model

For independent $Y_{ij}\sim \mathrm{Bin}(m,p_i)$ with $k$ replicates per group and $\mathrm{logit}(p_i)=\mu +{\alpha }_i$, ${\alpha }_1=0$, let $S_i={\sum }_j{Y}_{ij}$ and $N=km$. The likelihood equations give ${\hat{p}}_i=S_i/N$, $\hat{\mu }=\log (S_1/(N-S_1))$, and ${\hat{\alpha }}_i=\mathrm{logit}({\hat{p}}_i)-\hat{\mu }$ when $0<S_i<N$. Boundary totals give infinite logits. With $0<p_1<1$ fixed and $N\to \infty$, the central limit theorem and delta method giveThe other groups do not increase information about the baseline logit when they have free offsets. For two groups, the information matrix for $(\mu ,{\alpha }_2)$ is $\left(\begin{array}{cc}{w}_1+w_2& w_2\\ w_2& w_2\end{array}\right)$ with $w_i=N{p}_i(1-p_i)$, whose inverse has first diagonal entry $1/w_1$.