---
title: "Mixed (split-plot) ANOVA"
method_id: SM.DIF.PAR.MIX.MANOVA
family: Difference
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/mixed_anova.md
source: Clarus method library
---

# Mixed (split-plot) 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:** split-plot ANOVA; between-within ANOVA; mixed-design ANOVA; mixed factorial ANOVA; two-way ANOVA with one repeated measure.

## Hypotheses

- **Null (H0):** no between main effect (all between-group marginal means equal) AND no within main effect (all within-condition marginal means equal) AND no interaction (between x within cell means are additive)
- **Alternative (H1):** at least one of the three effects is non-null: a between main effect, a within main effect, or a between x within interaction

## When to use it

Use this method when your goal is to:
- compare means across one between and one within factor

## Data it expects

- **dependent:** 1 continuous
- **independent:** 1 nominal between-subjects factor + 1 nominal within-subjects (repeated) factor
- **pairing:** mixed (split-plot): subjects nested in the between factor, measured under every level of the within factor

## Assumptions Clarus checks

- **normality** (gate: shapiro_wilk; severity: warning; on violation: warn_only)
- **sphericity** (gate: mauchly; severity: warning; on violation: correct)
- **variance homogeneity** (gate: levene; severity: warning; on violation: warn_only)
- **independence** (severity: warning; on violation: warn_only)

## Effect size reported

- **partial eta squared** (Cohen (1988); Lakens (2013))

## Honest limitations

- Interpret the interaction first: if it is significant, the main effects can be misleading because the effect of one factor depends on the level of the other.
- When the repeated factor has 3 or more levels and sphericity is violated, rely on the Greenhouse-Geisser-corrected p-values, not the uncorrected ones.
- A significant between-subjects effect is not causal unless participants were randomly assigned to the between-subjects groups.
- Large n can pair statistical significance with a trivial effect — read the partial eta-squared for each effect.
- A subject missing any repeated measurement is dropped entirely; substantial dropout can bias the results and limits generalization.
- A-priori power and minimum-detectable-effect are not computed for this design (no standard closed-form formula); a non-significant effect may reflect limited power rather than a true null.
- Results generalize only to the sampled population.

## Primary sources

- Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd.
- Mauchly, J. W. (1940). Significance test for sphericity of a normal n-variate distribution. The Annals of Mathematical Statistics, 11(2), 204-209. https://doi.org/10.1214/aoms/1177731915
- Greenhouse, S. W., & Geisser, S. (1959). On methods in the analysis of profile data. Psychometrika, 24(2), 95-112. https://doi.org/10.1007/BF02289823
- Levene, H. (1960). Robust tests for equality of variances. In Contributions to Probability and Statistics (pp. 278-292). Stanford University Press.
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Olejnik, S., & Algina, J. (2003). Generalized eta and omega squared statistics: Measures of effect size for some common research designs. Psychological Methods, 8(4), 434-447. https://doi.org/10.1037/1082-989X.8.4.434
- Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. 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: A Model Comparison Perspective (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, "Mixed (split-plot) ANOVA", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/mixed_anova.md

**APA**

Clarus. (2026). Mixed (split-plot) ANOVA (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/mixed_anova.md

**BibTeX**

```bibtex
@misc{clarus-mixed-anova,
  author       = {Clarus},
  title        = {Mixed (split-plot) ANOVA},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/mixed_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.MIX.MANOVA · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library