Mixed ANOVA | Licklider

Mixed ANOVA tests for differences in a continuous outcome when the design includes both a between-subjects factor (for example, treatment group) and a within-subjects factor (for example, time point). Each subject belongs to exactly one between-subjects group and is measured under all within-subject conditions.

When Licklider selects this test

Licklider selects mixed ANOVA when all of the following are true:

The study design is explicitly identified as mixed
The data includes a between-subjects grouping column, a within-subjects condition column, and a subject identifier column
There are at least two levels in both the between and within factors
The design is balanced: every subject appears in every within-subject condition exactly once
Each subject belongs to exactly one between-subjects group

If the design is not explicitly identified as mixed, Licklider stays on the standard group-comparison decision path and selects one-way ANOVA, repeated measures ANOVA, or their non-parametric alternatives as appropriate.

Manual override to mixed ANOVA is not available in this release. The mixed path is auto-selected only when the design metadata and required columns are both present.

How it differs from repeated measures ANOVA

Repeated measures ANOVA handles one within-subjects factor: all subjects pass through all conditions, and there is no between-subject grouping. Mixed ANOVA adds one between-subjects factor, allowing questions such as whether two treatment groups change differently over time.

Design	Between factor	Within factor	Test
Independent groups only	Yes	No	One-way ANOVA
Repeated measures only	No	Yes	Repeated measures ANOVA
Mixed	Yes	Yes	Mixed ANOVA

Sphericity and corrections

Like repeated measures ANOVA, mixed ANOVA assumes sphericity for the within-subjects factor. Licklider tests this with Mauchly's test and applies the same correction rule used elsewhere in the product:

Greenhouse-Geisser correction when epsilon < 0.75
Huynh-Feldt correction when epsilon ≥ 0.75
No correction when sphericity is met

These corrections apply to the within-subjects main effect and the interaction effect. The between-subjects main effect does not use a sphericity correction because it is based on independent groups.

Reading the output

When Licklider runs mixed ANOVA, the figure view's Stats panel shows a dedicated Mixed ANOVA (1 between x 1 within) card. The card includes a three-row effects table:

Effect	Source	F	df (num, den)	p	p (corr)	eta_p^2	Sphericity	Sig
between	Between-subjects group column	F statistic	numerator, denominator df	uncorrected p	n/a	partial eta-squared	none	yes / no
within	Within-subjects condition column	F statistic	numerator, denominator df	uncorrected p	corrected p when needed	partial eta-squared	correction + epsilon	yes / no
interaction	Between x within	F statistic	numerator, denominator df	uncorrected p	corrected p when needed	partial eta-squared	correction + epsilon	yes / no

The same card also reports:

Field	What it means
Mauchly W	Mauchly's test statistic
Mauchly p	p-value for the sphericity test
Sphericity	met, violated, or n/a
Recommended	none, greenhouse_geisser, or huynh_feldt
Follow-up analysis	whether simple-effects follow-up is available
Interaction interpretation	whether the main effects can be interpreted safely

Effect size interpretation (partial eta-squared):

Value	Interpretation
< 0.01	Negligible
0.01 - 0.06	Small
0.06 - 0.14	Medium
≥ 0.14	Large

Interaction effects

When the interaction between the between-subjects and within-subjects factors is significant, the Stats panel keeps the omnibus result but adds an interpretation guard. The current product message is that main-effect interpretation is incomplete because simple-effect follow-up is not implemented in v1.

This means the omnibus mixed ANOVA result is still valid, but you should not treat the main effects alone as the full story when the interaction is significant.

Normality and automatic routing

Mixed ANOVA is selected from the declared design, not from table structure alone. Once the mixed path is selected, Licklider does not auto-switch to a standard non-parametric mixed-design alternative when normality is weak, because no such fallback is currently implemented in the product.

Normality and other assumption information still appear in the broader analysis record and quality-check surfaces. Use those diagnostics to judge whether the omnibus result is robust enough for your study.

Data requirements

Mixed ANOVA requires:

A between-subjects grouping column
A within-subjects condition column
A subject identifier column
A value column
A balanced design: every subject appears in every within-subjects condition exactly once
Exactly one between-subjects group assignment per subject
At least two subjects in each between-subjects group

Licklider returns an error instead of a result when the same subject crosses between-group levels, when duplicate subject-condition rows are present, or when the design is incomplete or unbalanced.

Relationship to other tests

Scenario	Test selected
Mixed design (1 between x 1 within), balanced	Mixed ANOVA
Paired only, 3+ conditions, normal	Repeated measures ANOVA
Paired only, 3+ conditions, non-normal	Friedman
Independent only, 3+ groups, normal	One-way ANOVA
Independent only, 3+ groups, non-normal	Kruskal-Wallis
Paired, 2 conditions, normal	Paired t-test
Independent, 2 groups, normal	Welch's t-test

Current limitations

Only one between-subjects factor and one within-subjects factor are supported
Post hoc simple-effects comparisons are not yet available
Mixed-effects model fallback for unbalanced or missing data is not yet available
Generalized eta-squared is not yet reported; the current panel uses partial eta-squared
A standard non-parametric mixed-design fallback is not available
Manual override to mixed ANOVA is not supported in this release