= Solution
For pair $i$, use conditionally independent counts $Y_{i0}\sim\operatorname{Pois}(\lambda_i)$ and $Y_{i1}\sim\operatorname{Pois}(\lambda_i e^\beta)$, with a common effect $\beta$ and a separate positive baseline rate for each physician. Equivalently the linear predictor is $\alpha_i+\beta x$, with $\alpha_i=\log\lambda_i$. This <Poisson regression> controls each physician's stable baseline propensity.
Condition on all twelve pair totals $N_i=Y_{i0}+Y_{i1}$. The <paired Poisson conditional likelihood> has independent factors
$$
Y_{i1}\mid N_i\sim\operatorname{Bin}\left(N_i,\frac{e^\beta}{1+e^\beta}\right),\qquad
\boxed{L_c(\beta)\propto
\frac{e^{\beta\sum_iY_{i1}}}{(1+e^\beta)^{\sum_iN_i}}.}
$$
This eliminates every $\lambda_i$ exactly. It is more useful as a nuisance-elimination device than the unpaired calculation, which removes only one common baseline: many pair-specific rates are otherwise estimated from just two measurements each.
A possible concern is loss of information through conditioning on totals that are not ancillary for $\beta$, especially with sparse pairs; a zero-total pair contributes no conditional information. There is an important qualification in this exact fixed-baseline Poisson model. Maximizing over $\lambda_i$ gives $\widehat\lambda_i=N_i/(1+e^\beta)$, and substitution leaves precisely the same $\beta$-dependent factor as the <conditional likelihood>. Hence \b[there is no extra loss relative to ordinary profiling in this particular model]; one should not automatically assert an incidental-parameter bias in its estimate of $\beta$.
Additional baseline observations, a calibrated distribution of baseline rates, or informative hierarchical constraints can retain information in the pair totals. A <Gamma–Poisson hierarchical model> is one possible structured model, but merely giving it a completely free common <mean> does not automatically create extra information about $\beta$: that <mean> can absorb the factor $1+e^\beta$ in the totals. Such remedies trade exact nuisance elimination for additional data or assumptions. Finally, paired sampling alone does not establish conditional Poisson independence or equidispersion: remaining within-physician dependence or <overdispersion> should be checked, and an appropriate joint count model or physician-level robust <variance> used if needed. The common multiplicative-effect assumption is also part of the analysis.
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