Three-Way Mixed ANOVA BBW (2 between × 1 within)

Coming soon

Repeat / Within

This test is implemented and is currently going through StatMinds’ production verification: every statistic is independently checked against a trusted reference (scipy / R), locked with regression tests, and the screen is exercised across assumption-met/violated and significant/non-significant scenarios before it opens up.

See what’s live now

Split-plot factorial ANOVA with two between-subjects factors (e.g., Sex × Treatment) and one within-subjects factor (e.g., Time).

Each participant is in ONE cell of the between-subjects grid AND measured at ALL levels of the within-factor. Tests SEVEN F-tests: three main effects, three two-way interactions, and the three-way interaction — all with the correct split-plot error terms (subjects-within-cells for between effects, within-subjects residual for within-involving effects). Auto-applies Greenhouse-Geisser / Huynh-Feldt corrections when Mauchly sphericity is violated.

Worked example

Two between-subjects factors and one within factor — how do they combine?

A three-way mixed ANOVA (between-between-within) tests two grouping factors and one repeated factor plus their interactions.

Result

A significant three-way interaction, F(2, 84) = 3.7, p = .029, showed the repeated-measure change depended on both grouping factors.

How you'd report it (APA)

A three-way mixed (BBW) ANOVA showed a significant three-way interaction, F(2, 84) = 3.7, p = .029.

When to use it

  • Treatment × demographic × time (longitudinal RCT)
    A 12-week RCT randomises 80 patients to CBT or Medication (between-1) and stratifies by Sex (M/F, between-2).
  • Intervention × severity × session (educational / clinical)
    An education researcher tests three reading-tutoring interventions (CBT-based / Phonics-based / Control, between-1) across two student-severity tiers (Mild Difficulty / Severe Diff

When NOT to — use instead

Hypotheses

Seven null hypotheses — three main effects (H₀_A, H₀_B, H₀_W: between1, between2, within), three two-way interactions (H₀_AB: A × B, H₀_AW: A × within, H₀_BW: B × within), and the three-way interaction (H₀_ABW: A × B × within). Each H₀ states that the corresponding variance component is zero in the population.
Hₐ for each of the seven effects: at least one component of that null is non-zero in the population.

Parameter tested: set of cell means {μ_{ijk}} and their marginal / interaction decomposition

Assumptions (and what to do if they fail)

No significant outliers in any of the between1 × between2 × within cellsmedium

Check: Inspect per-cell boxplots; flag points beyond the whiskers.

If violated: Outliers in a cell can inflate within-cell variance and distort the F-tests.

Residuals (deviations from cell means) are approximately normally distributedmedium

Check: Shapiro-Wilk on residuals.

If violated: Severely non-normal residuals threaten the F-test's Type I error rate.

Homogeneity of variances between the between-subjects cells at each within-level (Levene's test)medium

Check: Levene's p > .05 at every within-level → homogeneity met.

If violated: Unequal between-cell variances at some within-levels inflate F-test Type I error.

Sphericity of within-subjects effects — variances of pairwise differences between within-levels are equalmedium

Check: Mauchly's test on the within-factor; non-significant → sphericity met.

If violated: When sphericity is violated, every within-involving F-test has inflated Type I error.

Ready to run a Three-Way Mixed ANOVA BBW (2 between × 1 within) on your own data?

Guided setup, automatic assumption checks, effect sizes, figures and an APA write-up.

Run this test →