---
title: "Poisson regression"
method_id: SM.REL.REG.CNT.POISSON
family: Relationship
version: 1.0.0
date_modified: 2026-06-21
canonical: https://clarus.ofrencber.com/library/poisson_regression.md
source: Clarus method library
---

# Poisson regression

> A classical statistical method in the Clarus library (Relationship family).
> Clarus selects, assumption-checks and runs this method on your data with a
> deterministic rule engine and real numerical libraries (scipy / statsmodels);
> it never generates numbers, citations or results.

**Also known as:** Poisson regression; log-linear count regression; Poisson GLM; count regression.

## Hypotheses

- **Null (H0):** beta_j = 0 for predictor j (the predictor does not change the log-expected-count; rate ratio = 1). Overall H0: all slope coefficients are zero (tested by the likelihood-ratio / deviance test against the intercept-only model).
- **Alternative (H1):** beta_j != 0 (the predictor changes the log-expected-count; rate ratio != 1).

## When to use it

Use this method when your goal is to:
- model count outcome from predictors
- estimate rate ratios
- predict expected count or rate

## Data it expects

- **dependent:** 1 count (non-negative integers — events per observation/exposure)
- **independent:** >= 1 predictor (continuous and/or categorical)
- **pairing:** independent (cross-sectional)

## Assumptions Clarus checks

- **count outcome** (severity: critical; on violation: reroute)
- **no overdispersion** (gate: overdispersion_check; severity: warning; on violation: warn_only)
- **no multicollinearity** (gate: vif; severity: warning; on violation: warn_only)
- **log linearity** (severity: warning; on violation: warn_only)

## Effect size reported

- **rate ratio** (Nelder & Wedderburn (1972), §3.2; rate ratio = exp(beta), the multiplicative change in the expected count (rate) per one-unit increase in the predictor.)

## Honest limitations

- Coefficients are on the LOG scale and exp(coefficient) is a RATE RATIO (multiplicative change in the expected count), NOT an additive change in the count.
- Poisson assumes the variance equals the mean; real count data are often overdispersed (variance > mean), which understates the standard errors — check the dispersion index and prefer quasi-Poisson / negative binomial when it materially exceeds 1.
- If the counts are events per unit of exposure (time, area, trials), an offset (log exposure) is required to model rates; rate ratios without it are not comparable across exposures.
- Excess zeros beyond the Poisson expectation may indicate a zero-inflated process; a zero-inflated model may fit better.
- Cross-sectional / observational data permits no causal claim about the predictors.
- Results generalize only to the sampled population.

## Primary sources

- Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized Linear Models. Journal of the Royal Statistical Society, Series A (General), 135(3), 370-384. https://doi.org/10.2307/2344614
- McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). Chapman & Hall. (Ch. 6, log-linear models for counts.) https://doi.org/10.1007/978-1-4899-3242-6
- Cameron, A. C., & Trivedi, P. K. (2013). Regression Analysis of Count Data (2nd ed.). Cambridge University Press. https://doi.org/10.1017/CBO9781139013567
- McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior. In P. Zarembka (Ed.), Frontiers in Econometrics. Academic Press.

## How to cite this

To cite this Clarus method page (the page itself — for the method's own primary sources, see above):

**Plain text**

Clarus, "Poisson regression", version 1.0.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/poisson_regression.md

**APA**

Clarus. (2026). Poisson regression (Version 1.0.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/poisson_regression.md

**BibTeX**

```bibtex
@misc{clarus-poisson-regression,
  author       = {Clarus},
  title        = {Poisson regression},
  howpublished = {Clarus method library},
  version      = {1.0.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/poisson_regression.md}
}
```

_No DOI is minted for Clarus method pages yet; this citation uses the canonical URL, version and date. Cite the primary sources above for the method's scientific provenance._

---

METHOD SM.REL.REG.CNT.POISSON · VERSION 1.0.0 · UPDATED 2026-06-21 · SOURCE Clarus method library