---
title: "Simple linear regression (OLS)"
method_id: SM.REL.REG.SIM.OLS
family: Relationship
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/simple_linear_regression.md
source: Clarus method library
---

# Simple linear regression (OLS)

> 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:** OLS regression (one predictor); bivariate linear regression; least-squares line.

## Hypotheses

- **Null (H0):** beta_1 = 0 (the slope of {dependent} on {independent} is zero)
- **Alternative (H1):** beta_1 != 0 (there is a nonzero linear slope of {dependent} on {independent})

## When to use it

Use this method when your goal is to:
- quantify linear relationship
- predict continuous outcome from one predictor
- estimate slope per unit change

## Data it expects

- **dependent:** 1 continuous
- **independent:** 1 continuous
- **pairing:** paired_observations_same_unit

## Assumptions Clarus checks

- **linearity** (severity: warning; on violation: warn_only)
- **residual normality** (gate: shapiro_wilk; severity: warning; on violation: warn_only)
- **homoscedasticity** (gate: breusch_pagan; severity: warning; on violation: correct)
- **independence** (gate: durbin_watson; severity: warning; on violation: warn_only)

## Effect size reported

- **r2 f2** (Cohen (1988))

## Honest limitations

- A regression slope describes association and prediction, not causation, unless the predictor was randomly assigned in a controlled experiment.
- R2 measures variance explained in THIS sample and is optimistically biased; read adjusted R2 for small n.
- Predictions outside the observed range of {independent} (extrapolation) are not supported by the data.
- A single high-leverage or influential point can dominate the slope; inspect the residual and influence plots.
- Large n can make a trivially small slope statistically significant - read f2 and the slope's practical size.
- Results generalize only to the population from which the sample was drawn.

## Primary sources

- Galton, F. (1886). Regression towards mediocrity in hereditary stature. Journal of the Anthropological Institute, 15, 246-263. https://doi.org/10.2307/2841583
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Breusch, T. S., & Pagan, A. R. (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica, 47(5), 1287-1294. https://doi.org/10.2307/1911963
- Durbin, J., & Watson, G. S. (1951). Testing for serial correlation in least squares regression II. Biometrika, 38(1/2), 159-177. https://doi.org/10.2307/2332325
- Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality. Biometrika, 52(3/4), 591-611. https://doi.org/10.2307/2333709
- Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd ed.). Routledge.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). Sage.

## 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, "Simple linear regression (OLS)", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/simple_linear_regression.md

**APA**

Clarus. (2026). Simple linear regression (OLS) (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/simple_linear_regression.md

**BibTeX**

```bibtex
@misc{clarus-simple-linear-regression,
  author       = {Clarus},
  title        = {Simple linear regression (OLS)},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/simple_linear_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.SIM.OLS · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library