= Profile likelihood for a common Poisson rate ratio with unequal exposures
{title2=$\widehat\lambda_i(r)=t_i/(p_{i1}+rp_{i2})$}
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 is
$$
\sum_i\left[y_{i2}-\frac{t_i r p_{i2}}{p_{i1}+rp_{i2}}\right]=0.
$$
Equivalently, conditional on $t_i$, the second count has a <binomial distribution> with success <probability> $rp_{i2}/(p_{i1}+rp_{i2})$. 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.
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