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

The mean ratio is $R=(\lambda e^{\beta })/\lambda =e^{\beta }$, so its logarithm is exactly $\beta$; the baseline coefficient $\alpha$ cancels. Under the stated asymptotic normal approximation, the Wald confidence interval for $\beta$ is $0.69\pm 1.96(0.15)=(0.396,0.984)$. Exponentiation is monotone and therefore preserves the interval's coverage event. HenceThe standard error of $\alpha$ is not needed for this confidence interval: the Poisson regression already supplies the standard error of the contrast $\beta$. It would be incorrect to combine the two reported standard errors as if the mean ratio depended on both coefficients independently.

Let $Y_0$ and $Y_1$ be the aggregate counts in the control and treated groups. Assuming independent physicians, their Poisson distributions have means $12\lambda$ and $12\lambda e^{\beta }$. Condition on $E=\{Y_0+Y_1=m\}$. By independent Poisson conditioning,The PDF's full table gives $Y_0=67$, $Y_1=133$ and $m=200$. Thus the Two-group Poisson ratio conditional likelihood isConditioning the individual observations instead gives a multinomial distribution with cell probabilities $(1-p)/12$ in the first group and $p/12$ in the second; its parameter-dependent likelihood is the same. No factor involves $\lambda$. The conditional estimates are $\hat{p}=133/200$ and $\hat{\beta }=\log (133/67)\simeq 0.6857$.

The solid Poisson regression profile and dashed negative binomial regression profile peak at essentially the same $\beta \simeq 0.69$. The dashed profile is slightly wider and has heavier shoulders: it gives larger relative likelihood to values farther from the common maximizer. Thus the estimated mean ratio is stable, while the negative binomial model expresses slightly greater uncertainty.

The Poisson distribution constrains variance to equal the mean. A common negative binomial parametrization instead has $\mathrm{Var}(Y)=\mu +{\mu }^2/k$, with additional shape $k>0$; it permits overdispersion due, for example, to unexplained between-physician heterogeneity. Profiling this extra parameter lets the data support greater count variability and usually reduces information about the mean contrast. In this two-group model, the fitted group means remain the sample means for fixed $k$, which explains the nearly identical maximizers. The small separation of the profiles indicates a modest effect here, not overwhelming evidence for strong overdispersion. The comparison concerns the original PDF figure; no figure is reproduced.

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.