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?

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1 Answer 1


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
    Commented Feb 12, 2020 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
    Commented Feb 13, 2020 at 3:36

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