---
title: "Two-way (factorial) ANOVA"
method_id: SM.DIF.PAR.FACT.ANOVA2
family: Difference
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/factorial_anova.md
source: Clarus method library
---

# Two-way (factorial) ANOVA

> A classical statistical method in the Clarus library (Difference 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:** factorial ANOVA; two-factor ANOVA; 2-way between-subjects ANOVA; A x B ANOVA.

## Hypotheses

- **Null (H0):** Main effect A: all marginal means of {factorA} are equal; Main effect B: all marginal means of {factorB} are equal; Interaction: all cell-mean deviations (interaction terms) are zero.
- **Alternative (H1):** At least one marginal mean of {factorA} differs; and/or at least one marginal mean of {factorB} differs; and/or at least one interaction term is non-zero (the factors interact).

## When to use it

Use this method when your goal is to:
- compare means across two crossed factors
- test interaction between two categorical factors
- compare more than two groups defined by two grouping variables

## Data it expects

- **dependent:** 1 continuous
- **independent:** 2 nominal/categorical factors (each >=2 levels), fully crossed
- **pairing:** independent (between-subjects)

## Assumptions Clarus checks

- **independence** (severity: critical; on violation: warn_only)
- **normality of residuals** (gate: shapiro_wilk; severity: warning; on violation: warn_only)
- **variance homogeneity** (gate: levene; severity: warning; on violation: warn_only)

## Effect size reported

- **partial eta squared** (Cohen (1973); Lakens (2013) — partial eta^2 = SS_effect / (SS_effect + SS_residual), computed separately for factorA, factorB and the A x B interaction.)

## Honest limitations

- When the interaction is significant, do not read the main effects at face value — the effect of one factor changes across levels of the other; report simple effects.
- Unequal cell sizes make the partition of variance ambiguous; results here use Type II sums of squares, and a different SS type can change the main-effect tests.
- A significant F localizes that something differs, not which specific cells — use the post-hoc/estimated-marginal-means comparisons.
- Large n can render a trivial effect statistically significant — read partial eta-squared, not just p.
- Differences are not causal unless participants were randomly assigned to the factor levels.
- Results generalize only to the sampled population and the observed factor levels.

## Primary sources

- Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver & Boyd.
- Tukey, J. W. (1949). Comparing individual means in the analysis of variance. Biometrics, 5(2), 99-114. https://doi.org/10.2307/3001913
- Cohen, J. (1973). Eta-squared and partial eta-squared in fixed factor ANOVA designs. Educational and Psychological Measurement, 33(1), 107-112. https://doi.org/10.1177/001316447303300111
- Olejnik, S., & Algina, J. (2003). Generalized eta and omega squared statistics. Psychological Methods, 8(4), 434-447. https://doi.org/10.1037/1082-989X.8.4.434
- Lakens, D. (2013). Calculating and reporting effect sizes. Frontiers in Psychology, 4, 863. https://doi.org/10.3389/fpsyg.2013.00863
- Maxwell, S. E., Delaney, H. D., & Kelley, K. (2018). Designing Experiments and Analyzing Data (3rd ed.). Routledge.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.), Ch.12-13.
- Levene, H. (1960). Robust tests for equality of variances. In Contributions to Probability and Statistics (pp. 278-292). Stanford University 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, "Two-way (factorial) ANOVA", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/factorial_anova.md

**APA**

Clarus. (2026). Two-way (factorial) ANOVA (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/factorial_anova.md

**BibTeX**

```bibtex
@misc{clarus-factorial-anova,
  author       = {Clarus},
  title        = {Two-way (factorial) ANOVA},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/factorial_anova.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.DIF.PAR.FACT.ANOVA2 · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library