---
tag: STHTW16717
kind: concept
name: Poisson regression
msc: [62J12]
specializes: [K6C48KMRKC]
sources:
  - url: https://en.wikipedia.org/wiki/Poisson_regression
same_as:
  wikidata: Q954529
  wikipedia: Poisson regression
---

## Summary

Poisson regression models a count response as [[Poisson distributed|0XB9GQTN9G]] with a mean whose logarithm is linear in the regressors.
It is the [[generalized linear model|K6C48KMRKC]] for Poisson responses with the canonical log link, and it is fitted by [[maximum likelihood|0FREPFKNS7]].
Each coefficient acts multiplicatively on the expected count, and an offset handles different exposure times or population sizes.
It is the basic model for counts of events and for log-linear models of [[contingency tables|PANSMHE0SW]].

## Explanations

### L1 human

Counts such as the number of accidents at a crossing per year cannot be negative, and their spread grows with their size.
Poisson regression says that the expected count is $e^{x^T \beta}$, which is always positive.
A coefficient of $0.1$ for a regressor means that one more unit of it multiplies the expected count by $e^{0.1} \approx 1.105$, an increase of about $10.5\%$.
A Poisson variable has variance equal to its mean, and when real counts vary more than that, analysts switch to a negative binomial or a quasi-Poisson model.