 Paired Poisson conditional likelihood

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.