Grouped-binomial logistic regression

For independent grouped counts $Y_i\sim \mathrm{Bin}(n_i,p_i)$, grouped-binomial logistic regression specifiesFitting the proportions $Y_i/n_i$ with binomial weights $n_i$ gives the same likelihood.

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 $1/w_1$.

For each treatment year let $C$ be cycles, $S$ singleton deliveries, $T$ twin deliveries and $H$ deliveries of three or more babies. A singleton-per-cycle model uses $S\sim \mathrm{Bin}(C,p)$. A multiple-delivery model conditional on a live delivery instead uses $T+H\sim \mathrm{Bin}(S+T+H,q)$. The latter denominator counts deliveries, not babies, and it excludes failed cycles. Confusing these denominators changes the statistical question.