 Solution

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.