---
title: "Multiple linear regression (ordinary least squares)"
method_id: SM.REL.REG.MUL.OLS
family: Relationship
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/multiple_linear_regression.md
source: Clarus method library
---

# Multiple linear regression (ordinary least squares)

> 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:** multiple regression; OLS regression; linear model; multivariable linear regression.

## Hypotheses

- **Null (H0):** beta_j = 0 for predictor j (the predictor has no linear association with {dependent} after adjusting for the other predictors); jointly, all slopes equal 0.
- **Alternative (H1):** beta_j != 0 for at least one predictor (the predictor shifts {dependent} after adjustment); jointly, at least one slope is nonzero.

## When to use it

Use this method when your goal is to:
- predict continuous outcome from multiple predictors
- quantify unique association of each predictor
- model linear relationship controlling for covariates

## Data it expects

- **dependent:** 1 continuous
- **independent:** >= 2 predictors (continuous and/or dummy-coded categorical)
- **pairing:** independent (cross-sectional)

## Assumptions Clarus checks

- **linearity** (severity: critical; on violation: warn_only)
- **no multicollinearity** (gate: vif; 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** (Wherry (1931); adjusted R-squared penalizes added predictors)

## Honest limitations

- Coefficients are PARTIAL effects: each is adjusted for the other predictors and changes if predictors are added or removed.
- A high R-squared with observational data is association, not proof of cause; only random assignment licenses a causal claim.
- Multicollinearity inflates standard errors and can flip coefficient signs without affecting overall fit — inspect the VIF.
- Adjusted R-squared and f2 describe in-sample fit; out-of-sample prediction will usually be weaker.
- Influential cases (high Cook's distance) can dominate the fit — check and report them.
- Results generalize only to the sampled population and the observed predictor range; do not extrapolate beyond it.

## Primary sources

- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum. (f2 effect size, pp. 410-414).
- Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd 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
- Green, S. B. (1991). How many subjects does it take to do a regression analysis? Multivariate Behavioral Research, 26(3), 499-510. https://doi.org/10.1207/s15327906mbr2603_7
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). Sage. (Ch.8, regression diagnostics).

## 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, "Multiple linear regression (ordinary least squares)", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/multiple_linear_regression.md

**APA**

Clarus. (2026). Multiple linear regression (ordinary least squares) (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/multiple_linear_regression.md

**BibTeX**

```bibtex
@misc{clarus-multiple-linear-regression,
  author       = {Clarus},
  title        = {Multiple linear regression (ordinary least squares)},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/multiple_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.MUL.OLS · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library