Two-Way Mixed ANOVA (1 between × 1 within)

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Tests effects of BOTH a between-subjects factor and a within-subjects factor on a continuous DV, including their interaction (the most common substantive question — does the group effect change over t

ime?). Three F-tests: Between main effect, Within main effect, Between×Within interaction. Auto-applies Greenhouse-Geisser / Huynh-Feldt sphericity corrections to within-involving F's when Mauchly rejects. Reports per-cell means, η²p effect sizes, and pairwise post-hoc with Bonferroni correction.

Worked example

Does an outcome change over time differently between two groups?

A mixed ANOVA crosses a between-subjects factor (group) with a within-subjects factor (time), testing the Group × Time interaction.

Result

A significant Group × Time interaction, F(2, 76) = 5.9, p = .004, showed the treatment group improved while the control did not.

How you'd report it (APA)

A mixed ANOVA revealed a significant Group × Time interaction, F(2, 76) = 5.9, p = .004.

When to use it

  • Group × Time design (treatment groups measured over time)
    Two therapy groups (CBT vs medication) measured on anxiety at baseline, 4 weeks, and 8 weeks.
  • Split-plot design (between × within)
    Male and female athletes (between) perform under 3 stress conditions (within: low, medium, high).

When NOT to — use instead

Hypotheses

Three null hypotheses: (1) H₀_B: no main effect of the between-subjects factor (group marginal means equal); (2) H₀_W: no main effect of the within-subjects factor (within-level marginal means equal); (3) H₀_BxW: no between × within interaction (the effect of the within factor is the same across between groups).
Hₐ_B, Hₐ_W, Hₐ_BxW: at least one component of each null is non-zero in the population.

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

Assumptions (and what to do if they fail)

No significant outliers in any of the between × 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 groups at each within-subjects level (Levene's test)medium

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

If violated: Unequal between-group 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, the within main effect and the between × within interaction F-tests have inflated Type I error.

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