Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 46 / 1 / d / Solution

For pair $i$, use conditionally independent counts $Y_{i0}\sim \mathrm{Pois}({\lambda }_i)$ and $Y_{i1}\sim \mathrm{Pois}({\lambda }_i{e}^{\beta })$, 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 factorsThis 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 ${\hat{\lambda }}_i=N_i/(1+e^{\beta })$, and substitution leaves precisely the same $\beta$-dependent factor as the conditional likelihood. Hence 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.