Poisson Regression

GLM for count data (0, 1, 2, …) — models the expected value via log link, assuming variance ≈ mean.

Also known as: Poisson GLM, Poisson model

Poisson regression is the GLM for count data such as "defects per batch", "complaints per week", or "failures per unit". It uses the log link so that the predicted rate always stays positive.

ln E [Y ∣ X] = β_{0} + β_{1} X_{1} + \dots + β_{p} X_{p}

Coefficients are read as rate ratios: $exp (β_{j})$ = factor by which the expected count changes for $+ 1$ in $X_{j}$
Assumption: $Var (Y) = E [Y]$ — the core condition of the Poisson distribution
Offset — for unequal observation duration/size: $ln Exposure$ as a fixed offset in the model (e.g. "defects per 1000 parts")

Used in

Regression (Attributive)

In the Algorithm Lab

Sources

Cameron, A. C. & Trivedi, P. K. — Regression Analysis of Count Data, 2nd Edition, Cambridge University Press
Hilbe, J. M. — Modeling Count Data, Cambridge University Press