= Solution
The solid <Poisson regression> profile and dashed <negative binomial regression> profile peak at essentially the same $\beta\simeq0.69$. The dashed profile is slightly wider and has heavier shoulders: it gives larger relative <likelihood> to values farther from the common maximizer. Thus \b[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 $\operatorname{Var}(Y)=\mu+\mu^2/k$, 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.
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