# Residuals in poisson regression

Zuur 2013 Beginners Guide to GLM & GLMM suggests validating a Poisson regression by plotting Pearsons residuals against fitted values. Zuur states we shouldn't see the residuals fanning out as fitted values increase, like attached (hand drawn) plot.

But I thought a key characteristic of the Poisson distribution is that variance increases as mean increases. So shouldn't we expect to see increasing variation in the residuals as fitted values increase?

As a result, ordinary raw residuals ($$r_i=y_i-\hat\mu_i$$) should have a spread that increases with fitted values (though not in proportion).
However, Pearson residuals are residuals divided by the square root of the variance according to the model ($$r^P_i=\frac{y_i-\hat\mu_i}{\sqrt{\hat\mu_i}}$$ for a Poisson model). This means that if the model is correct, the Pearson residuals should have constant spread.
• The conditional distribution of the response may be different at each combination of predictors. Hence the use of the subscript on the mean; $\mu_i$ is the population mean (and thereby also population variance) for observation $i$, given its predictor values (the values of its IVs). Feb 13, 2020 at 3:36