Three-Way Mixed ANOVA BWW (1 between × 2 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 one between-subjects factor (e.g., Treatment) and two within-subjects factors (e.g., Time × Condition).

Each participant is in ONE between-cell AND measured at every combination of the two within-factors. Tests SEVEN F-tests — three main effects, three two-way interactions, and the three-way interaction — with the correct split-plot error terms (subjects-within-groups for between effects, within-subject residuals for within-involving effects). Auto-applies Greenhouse-Geisser / Huynh-Feldt sphericity corrections per within-factor.

Worked example

One between-subjects factor and two within factors — how do they combine?

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

Result

A significant three-way interaction, F(2, 76) = 4.1, p = .021, showed the two within-subject effects combined differently across groups.

How you'd report it (APA)

A three-way mixed (BWW) ANOVA showed a significant three-way interaction, F(2, 76) = 4.1, p = .021.

When to use it

  • Treatment × Time × Condition (longitudinal experimental)
    60 participants are randomised to (Drug A / Drug B / Placebo) (between).
  • Group × Session × Phase (clinical / educational longitudinal)
    40 students in (Intervention vs Control) (between) complete tests at (Sessions: 1 / 2 / 3) × (Phases: Pre-task / Mid-task / Post-task) (within × within).

When NOT to — use instead

Hypotheses

Seven null hypotheses — three main effects (H₀_A: between; H₀_W1: within1; H₀_W2: within2), three two-way interactions (H₀_AW1, H₀_AW2, H₀_W1W2), and the three-way interaction (H₀_AW1W2). Each H₀ states that the corresponding variance component is zero in the population.
Hₐ for each effect: at least one component is non-zero.

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

Assumptions (and what to do if they fail)

No significant outliers in any of the between × within1 × within2 cellsmedium

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

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

Residuals are approximately normally distributedmedium

Check: Shapiro-Wilk on residuals.

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

Homogeneity of variances between groups at each within1 × within2 cell (Levene's test)medium

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

If violated: Unequal between-group variances inflate F-test Type I error.

Sphericity — checked separately for each within-subjects effect family (within1, within2, within1 × within2 interaction contrasts)medium

Check: Mauchly's test per within family; non-significant → sphericity met for that family.

If violated: Violated sphericity inflates the F-test's Type I error rate for the affected family.

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

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

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