# How does poisson regression handle zeros anyway?

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).

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.

• 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. Jan 10, 2018 at 2:47
• The most correct way is to add an expectation operator to y. @MatthewDrury Jan 10, 2018 at 3:23
• 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. Jan 10, 2018 at 3:31