Mixed ANOVA#

The Mixed ANOVA is the go-to method when your design includes both a between-subjects factor (e.g., group membership) and a within-subjects factor (e.g., repeated measurements over time). Also known as a split-plot design, it gives you three key results: the main effect of each factor and their interaction.

When to Use#

You have two or more groups (between-subjects factor) measured at multiple time points (within-subjects factor)
Your study uses a classic pre-post design with a control group or a longitudinal comparison across groups
You want to know not just whether groups differ or scores change over time, but whether the pattern of change differs between groups (interaction)
Your dependent variable is continuous (interval or ratio scale)
You have complete data — with missing values, linear mixed models (LMM) are a better choice

Assumptions#

Normality in each cell (Group x Time point)
Sphericity of the within-subjects factor (Mauchly's test)
Homogeneity of variances across groups (Levene's test)
Homogeneity of covariance matrices (Box's M test)

Tip: If sphericity is violated, correct the degrees of freedom using Greenhouse-Geisser (conservative) or Huynh-Feldt (liberal). If Box's M is significant, interpret between-subjects effects with caution.

Formula#

The Mixed ANOVA computes three separate F-tests:

Main effect of the between-subjects factor:

$F_{\text{between}} = \frac{MS_{\text{between}}}{MS_{\text{error(between)}}}$

Main effect of the within-subjects factor:

$F_{\text{within}} = \frac{MS_{\text{within}}}{MS_{\text{error(within)}}}$

Interaction:

$F_{\text{interaction}} = \frac{MS_{\text{interaction}}}{MS_{\text{error(within)}}}$

Each F-value is compared against the F-distribution with degrees of freedom determined by the number of groups and measurement occasions.

Example#

Practical Example: Pain Treatment Over Time

A clinic compares a new drug to a placebo. 40 patients are randomly assigned to two groups (between-subjects factor: drug vs. placebo). Each patient rates their pain on a 0–10 scale at three time points: baseline, 2 weeks, and 4 weeks (within-subjects factor: time).

Results:

Main effect of group: $F(1, 38) = 8.42$ , $p = .006$ — the drug group has overall lower pain scores
Main effect of time: $F(2, 76) = 24.15$ , $p < .001$ — pain changes over time
Group x Time interaction: $F(2, 76) = 11.87$ , $p < .001$ — the key finding: pain reduction is greater in the drug group than in the placebo group

Effect Size#

Partial eta-squared ( $\eta_p^2$ ) is reported for each effect:

$\eta_p^2 = \frac{SS_{\text{effect}}}{SS_{\text{effect}} + SS_{\text{error}}}$

$\eta_p^2$	Interpretation
0.01	small effect
0.06	medium effect
0.14	large effect

Tip: Report $\eta_p^2$ separately for each of the three effects. The interaction effect is often the most interesting — it reveals whether groups develop differently over time.