= Common versus factor-specific slopes in Poisson regression
For a continuous predictor $x$ and a categorical <factor>, an additive <Poisson regression> with <log link> has $\log\mu_j(x)=a_j+b x$. The log-mean curves are parallel and $\mu_j(x)/\mu_k(x)=e^{a_j-a_k}$ is constant in $x$. Adding a predictor-by-factor <interaction> gives $\log\mu_j(x)=a_j+b_jx$, so this ratio becomes $e^{a_j-a_k+(b_j-b_k)x}$. With $J$ factor levels, the common-slope restriction has $J-1$ independent constraints. A <likelihood-ratio test> compares the two <deviances> with an asymptotic <chi-squared distribution> on $J-1$ <degrees of freedom>. Evidence against common slopes establishes varying effects on the log scale; it does not by itself determine their signs or the ordering of the mean curves.
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