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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?

enter image description here

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up vote 7 down vote accepted

The distinction is clear as soon as you understand what a Pearson residual is.

You are correct that for a Poisson model, variance increases as mean increases.

As a result, ordinary raw residuals ($r_i=y_i-\hat\mu_i$) should have a spread that increases with fitted values.

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.

enter image description here

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