Three-Way RM-ANOVA (3 within factors)

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

Fully within-subjects 3-factor ANOVA: same subjects measured at every combination of three crossed within-subjects factors.

Tests SEVEN F-tests — three main effects, three two-way interactions, and the three-way interaction — each with a subjects-within-cells error term. Auto-applies Greenhouse-Geisser / Huynh-Feldt sphericity corrections per within-factor when Mauchly rejects.

Worked example

Three fully within-subject factors — how do they interact?

A three-way repeated-measures ANOVA tests three within-subject factors and all their interactions, with per-effect sphericity checks.

Result

A significant three-way interaction, F(2, 46) = 4.9, p = .012, was found.

How you'd report it (APA)

A three-way repeated-measures ANOVA showed a significant three-way interaction, F(2, 46) = 4.9, p = .012.

When to use it

Stimulus × Difficulty × Session (cognitive / perceptual)
24 subjects perform a visual-search task across every (Stimulus: face / object) × (Difficulty: easy / hard) × (Session: 1 / 2 / 3) cell.
Drug × Dose × Time (pharmacology / dose-response)
16 lab rats receive every combination of (Drug: A / B / C) × (Dose: low / high) × (Time: 30min / 60min / 90min) post-administration on counterbalanced sessions.

When NOT to — use instead

Any factor between-subjects
Mixed designs (any combination of B/W) need the appropriate mixed ANOVA, not the fully-within version. → Three-Way Mixed ANOVA BBW (2 between \u00d7 1 within)
Two within-factors only
Use the two-way RM ANOVA (3 F-tests) — three-way is overkill for a 2-factor design. → Two-Way RM-ANOVA (2 within factors)
Missing within-cells (any subject misses any cell)
RM-ANOVA requires complete cases. → Linear Mixed Effects (LMM) \u2014 clustered / repeated data (ICC, BLUPs)
Ordinal / non-normal DV with small n
Three-way RM-ANOVA inherits ANOVA's normality assumption; small-n + non-normal needs a permutation or robust alternative. → Permutation ANOVA (distribution-free)

Hypotheses

Seven null hypotheses: 3 main effects (H₀_A, H₀_B, H₀_C), 3 two-way interactions (H₀_AB, H₀_AC, H₀_BC), and 1 three-way interaction (H₀_ABC).

Each alternative: at least one component of the corresponding 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 factor1 × factor2 × factor3 cellmedium

Check: Inspect per-cell boxplots.

If violated: Outliers distort within-cell variance.

Residuals are approximately normally distributedmedium

Check: Shapiro-Wilk on residuals.

If violated: Severe non-normality inflates Type I error.

Sphericity checked per within-subjects effect — Greenhouse-Geisser / Huynh-Feldt correction auto-applied when violatedmedium

Check: Mauchly per effect; non-significant → met.

If violated: Inflates the F-test's Type I error rate for the affected effect.

Ready to run a Three-Way RM-ANOVA (3 within factors) on your own data?

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

Run this test →

🔎 Worked example

✓ When to use it

✗ When NOT to — use instead

Hypotheses

Assumptions (and what to do if they fail)

Worked example

When to use it

When NOT to — use instead