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Introduction

When reading lecture slides from a lecture I missed, one slide seemed to suggest that when presented with count data (the response being a count of something) one should try to fit a usual linear regression and then study a histogram of residuals to determine whether the count data seem to follow a Poisson distribution. The slide seems to say that if the residual histogram looks like a Poisson distribution the Poisson regression should be used. An example histogram presented is:

enter image description here


My thoughts about the approach

I have not seen this approach elsewhere to determine if Poisson regression should be used and at first I was a bit doubtful; I thought that given the training data $\mathbf x$ the response is a random variable $Y \sim Pr(Y| x) = Poisson(\lambda)$ and the prediction $\hat y(x)$ is constant for a particular $x$ so that the residuals also become random variables but where the mean and variance are not equal:

$$E[Y - \hat y(x)] = E[Y] - E[\hat y(x)] = \lambda - \hat y(x) \neq \lambda, \quad \text{if } \hat y(x) \neq 0$$ while, $$Var(Y - \hat y(x)) = Var(Y) + Var(\hat y(x)) + 2 Cov(Y, \hat y(x)) = Var(Y) + 0 + 0 = \lambda \neq E[Y - \hat y(x)]$$ so that the residuals cannot be Poisson random variables since for a Poisson random variable the mean and variance are equal. Since the residuals do not follow a Poisson distribution the approach outlined on the slide seems doubtful. (I have also assumed that the histogram of residuals pertained to a specific $x$ and not all residuals).


Question

But I am not sure about the accuracy of my analysis or whether I may have misunderstood the slide (it barely contains any text). So therefore I wonder, is there any reason to study the histogram of residuals of a linear regression fit when determining whether to use Poisson regression?

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    $\begingroup$ +1. Unless all fitted values vary only a little, this approach is doomed to fail, because it yields a mixture of different distributions, none of which is Poisson! (How many sets of residuals have you ever seen that are all non-negative and integral? Probably none.) One would hope the slide was trying to recommend something altogether different, but perhaps did not do so clearly enough. $\endgroup$
    – whuber
    Jan 4, 2022 at 19:45
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    $\begingroup$ If some residuals are negative as in your example, how could their distribution resemble a Poisson distribution? $\endgroup$
    – EdM
    Jan 4, 2022 at 19:52
  • $\begingroup$ @EdM Well that certainly is a good point. I am not used to working with the Poisson distribution, it passed me by. $\endgroup$
    – Paradox
    Jan 4, 2022 at 19:57
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    $\begingroup$ Viewing regression as a model for the conditional distributions of Y given particular X, a better approach would be to look at histograms or bar charts of Y data within cohorts defined by ranges of X variables. I say "ranges" so you get enough data for such graphs to provide reasonable estimates of the distributions. $\endgroup$ Jan 4, 2022 at 20:52

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