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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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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 (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.

Residual plots from a simple simulated Poisson regression model. Left plot: raw residuals vs fitted mean show increasing spread with mean. There is diagonal "banding" in the residuals because the data are discrete. Right plot: Pearson residuals show what looks like constant spread as mean changes, and the diagonal bands are now curved.

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  • $\begingroup$ Could you clarify why you write that we are dividing by the square root of the variance when you are actually dividing by the square root of the expected value? I know the variance equals the mean for a poisson distribution, but it is a constant for a particular distribution, so what variance are we talking about here? $\endgroup$ – kdarras Feb 12 '20 at 14:58
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    $\begingroup$ 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). $\endgroup$ – Glen_b Feb 13 '20 at 3:36

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