Profile likelihood for a common Poisson rate ratio with unequal exposures (${\hat{\lambda }}_i(r)=t_i/(p_{i1}+r{p}_{i2})$)

For independent counts with means $p_{i1}{\lambda }_i$ and $p_{i2}{\lambda }_{i}r$, let $t_i=y_{i1}+y_{i2}$. Maximizing the Poisson regression likelihood in each baseline rate gives the displayed estimate. The remaining log-rate-ratio score isEquivalently, conditional on $t_i$, the second count has a binomial distribution with success probability $r{p}_{i2}/(p_{i1}+r{p}_{i2})$. Each positive-total term increases strictly with $r$; if both treatment totals are positive, the score has a unique positive root. This reduces a many-parameter fit to a scalar equation.