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A Poisson distribution has probability mass $e^{-\mu }{\mu }^y/y!=\exp \{y\theta -e^{\theta }-\log (y!)\}$, where $\theta =\log \mu$. It is an exponential family with $b(\theta )=e^{\theta }$, mean $\mu$ and variance $\mu$. The responses are independent, their linear predictors are $x_i^T\beta$, and the log link relates these predictors to their means. These are precisely the components of a generalized linear model; the log link is its canonical link function and the dispersion is one.

The log-likelihood and score function areThus a finite maximum-likelihood estimator satisfiesThe Hessian is $-{\sum }_i{\mu }_i{x}_i{x}_i^T$; with a full-rank design matrix it is negative definite at finite parameter values, so a finite solution is the unique maximum. A finite maximum need not exist for every possible response/design combination; the displayed equations describe it when it does exist. If the first component of every $x_i$ is one, the first score equation gives ${\sum }_i{y}_i={\sum }_i{\hat{\mu }}_i$.

The saturated model fits each mean as $y_i$, with the limiting mean zero allowed for a zero count. Subtracting the fitted log-likelihood from the saturated log-likelihood gives the Poisson deviancewhere $0\log (0/{\hat{\mu }}_i)=0$. With an intercept, the sum of the second terms is zero, but retaining them is necessary without that score equation.

Take low pH as a reference level. For group $j$, the model with interaction haswhere $b_i$ is biomass. The additive model sets every ${\delta }_j$ to zero. The interaction model estimates six parameters and the additive model four. In the additive model the log-mean curves are parallel straight lines, while the mean curves are positive exponentials with fixed ratios $e^{{\gamma }_j-{\gamma }_k}$. In the interaction model the log-mean slopes differ, and the mean ratio is $e^{{\gamma }_j-{\gamma }_k+({\delta }_j-{\delta }_k)b}$. The following sketches illustrate common versus factor-specific slopes in Poisson regression; their coefficients are illustrative, not estimates from the data.

The additive model is nested in the interaction model with two restrictions. The likelihood-ratio test therefore useswhose p-value is $e^{-8}=0.0003355$. There is strong evidence that the biomass slope on the log-mean scale depends on pH. The interaction-model residual deviance $83.2$ is close to its $90-6=84$ residual degrees of freedom, giving no obvious indication of overdispersion from this comparison alone. The supplied deviances contain no slope estimates: they do not establish which pH group has more species, whether each slope is positive or negative, or whether any actual curves cross. Those conclusions require the fitted coefficients and their uncertainty.