= Baseline logit estimator in a group-factor binomial model
{title2=$\widehat\mu=\operatorname{logit}(S_1/(km))$}
For independent $Y_{ij}\sim\operatorname{Bin}(m,p_i)$ with $k$ replicates per group and $\operatorname{logit}(p_i)=\mu+\alpha_i$, $\alpha_1=0$, let $S_i=\sum_jY_{ij}$ and $N=km$. The likelihood equations give $\widehat p_i=S_i/N$, $\widehat\mu=\log(S_1/(N-S_1))$, and $\widehat\alpha_i=\operatorname{logit}(\widehat p_i)-\widehat\mu$ 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> give
$$
\sqrt N(\widehat\mu-\mu)\ \xrightarrow{d}\ N\left(0,\frac1{p_1(1-p_1)}\right).
$$
The 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 $\begin{pmatrix}w_1+w_2&w_2\\w_2&w_2\end{pmatrix}$ with $w_i=Np_i(1-p_i)$, whose inverse has first diagonal entry $1/w_1$.
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