I have what I believe are an interesting QQ plot of the residuals of some sports data fitted using Poisson regression. My model is actually a live model to predict the number of remaining events in the game (basketball, soccer etc) from a set of explanatory variables. It looks also very much like the data is Poisson distributed, with equal mean and variance.

Residuals are as we know $r_i=Y_i-\mu_i$, where $Y_i$ are the observations, and $\mu_i$ the fitted mean of $Y_i$ ($Y_n$ may have different explanatory variables than, for example, $Y_{n-1}$).

Here is the first residual QQ plot of fitting my model to soccer match data (whole 90 minutes): Residual QQ plot number 1

Interpreting QQ plots if I want to check for normality of some data is not too hard. It is easy to see that the residuals have too thick tails. But assuming the data is without a doubt Poisson distributed, something must therefore be wrong with $\mu_i$. Looking at the lower tail, I cannot reach any other conclusion than that the values of $\mu_i$ generally are too big - such that $r_i$ becomes too small. But also, looking at the upper tail, I cannot reach any other conclusion than that $\mu_i$ is too small - such that $r_i$ becomes too big. Is this a contradiction, or merely just a reflection that $\mu_i$ is not fitted good enough for extreme cases?

  • 2
    $\begingroup$ Note that the data are assumed to be Poisson, but the QQ plot is for normality. Beware of over-interpreting the appearance of the extreme tail. $\endgroup$
    – Glen_b
    Commented Jun 11, 2014 at 9:23
  • $\begingroup$ Correct, but the residuals are normal, are they not? $\endgroup$
    – Erosennin
    Commented Jun 11, 2014 at 9:43
  • 4
    $\begingroup$ No, they are not. $\endgroup$
    – Glen_b
    Commented Jun 11, 2014 at 10:13
  • 3
    $\begingroup$ The individual y's are Poisson. $\hat y's$ are some complicated function of the set of $y$'s and the $x$s. The difference is not normal. The point of the residual plot is if the Poisson means aren't too small, the Poissons will be approximately normal. But the observations with small Poisson means won't be very well approximated by normals - and neither will the residuals. So you can't necessarily expect the QQ plot to match well everywhere, especially in the lower tail. $\endgroup$
    – Glen_b
    Commented Jun 11, 2014 at 11:22
  • 1
    $\begingroup$ Poisson regression explicitly assumes Poisson variation of the responses, so why would Normality even be a consideration? Why is it of any interest at all? $\endgroup$
    – whuber
    Commented Jun 11, 2014 at 15:51