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So correct me if i'm wrong. The usual link function for poisson regression is log, so that you're performing regression on log(y)~x1+x2+x3+x4+...

The variable y is typically a count, meaning it is restricted to integers from 0 to positive infinity. The input variables x1...xn are not restricted to the positive integers.

So how does the regression proceed when y = 0? Is log(0) merely ignored?

Also, to be clear, this question is not about zero-inflated poisson regression (which distinguishes between different kinds of zeros).

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2 Answers 2

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The Poisson model is $$y = \exp \left(\alpha + \beta \cdot x + \varepsilon \right).$$

The way you get an outcome of zero is when the index $\alpha + \beta \cdot x + \varepsilon$ is large and negative. The coefficients do not come from a regression of logged outcome on the covariates, but from maximization of the log likelihood. You can also use this model on non-integer outcomes, though that is more controversial.

You can learn more about this model from this blog post, including the zeros issue and a comparison to logged outcome regression.

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    $\begingroup$ I don't think that is correct (what is $\epsilon$ in the above equation?) I believe $y \sim Poisson(\lambda=\alpha + \beta x)$ is a better way to communicate the idea. $\endgroup$ Jan 10, 2018 at 2:47
  • $\begingroup$ The most correct way is to add an expectation operator to y. @MatthewDrury $\endgroup$
    – SmallChess
    Jan 10, 2018 at 3:23
  • $\begingroup$ Right, like $E[y \mid x] = exp(\alpha + \beta x)$. But that doesn't work out correctly the way you have it written, with the random component $\epsilon$ inside the exponential. $\endgroup$ Jan 10, 2018 at 3:31
  • $\begingroup$ That's the DGP, not the conditional expectation. You can always rewrite that to make the error enter multiplicatively if that is easier to digest. $\endgroup$
    – dimitriy
    Jan 10, 2018 at 3:41
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    $\begingroup$ +1 I was sceptical, and have seen William Gould's post on using Poisson models before, but upon re-reading it have to agree that what you write here is an accurate account of his approach. $\endgroup$
    – whuber
    Jan 10, 2018 at 5:16
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The poisson regression model is that the logarithm of the expected values can be modeled by a linear combination of predictors. The expected values of y are not 0, even though there may be 0 counts in the real data. The expectation is a positive real number.

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