= Paired Poisson conditional likelihood
{title2=$Y_{i1}\mid N_i\sim\operatorname{Bin}(N_i,e^\beta/(1+e^\beta))$}
For conditionally independent paired counts with means $\lambda_i$ and $\lambda_i e^\beta$, condition on every pair total $N_i=Y_{i0}+Y_{i1}$. The joint <conditional likelihood> removes all pair-specific rates, leaving the displayed independent <binomial distributions>. A zero-total pair contributes no information. With unrestricted fixed baseline rates, this conditional likelihood has the same parameter-dependent part as the ordinary <profile likelihood>; a specified <Gamma–Poisson hierarchical model> can retain additional information by imposing structure on those rates.
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