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Two-Way Within-Subjects (Repeated Measures) ANOVA

Two-way repeated measures ANOVA combines elements of two-factor between-subjects design and single-factor within-subjects design. That is, the variability is computed for the main effects of Factor A, the main effect of Factor B, and the interaction (AB), but the design requires three error terms, one for each effect.

Two-Way Repeated Measures ANOVA Output

Multiple Comparisons/Post-Hoc Tests

Supported Worksheet Layout

Two-Way Repeated Measures ANOVA Output

The default output includes the two-way ANOVA table. The additional output depends on selected options, and can include interaction plots, the results for sphericity test and correction, and table reports for selected multiple comparisons tests.

The Default Output: One-Way ANOVA Table

This table displays:

The source of variability (i) attributable to Factor A, (ii) attributable to Factor B, (iii) attributable to AB interaction, and (iv) attributable to error terms for each effect (i.e., A x Subjects, B x Subjects, AB x Subjects)
The sum of the squared deviations from the mean (Sum of Squares); the degrees of freedom (df); the Mean Square for each of the variability components
The two recommended F statistics: F⁺ (which is the preferred test statistic when omnibus sphericity is tenable), the F^x statistics (which can be used when omnibus sphericity is not tenable, but local sphericity condition is satisfied)
The p⁺ value corresponding to F⁺, and the p^x value corresponding to F^x
The partial omega squared test result (testing strength of association)

Interaction Plots

Interaction plots are a graphical display of the effect of one factor at each level of the other factor. With no or insignificant interaction, the lines are approximately parallel. The more the lines diverge from being parallel, the stronger the interaction.

The error bars displayed on the interaction plots can represent:

Standard error of mean
Standard Deviation
Confidence Interval (including options of 90.0%, 95.0%, 97.5%, 99.0%)

Sphericity Evaluation (Locally Best Invariant Test)

Aabel uses the Locally Best Invariant test to evaluate whether or not the sphericity is tenable.

Greenhouse & Geisser and Huynd & Feldt Corrections

For sphericity corrections, Aabel provides Greenhouse & Geisser and Huynd & Feldt methods.

Multiple Comparisons/Post-Hoc Tests

Simple comparisons (also know as pair-wise comparisons) accompanying two-way within-subjects ANOVA include:

Tukey's HSD test (see the example above)
Tukey B test on ordered means
Fisher's LSD test
The Newman-Keuls (Neuman-Keuls) test on ordered means
Tukey Kramer test
Scheffé test
Bonferroni-Dunn test

The example below is the Fisher's LSD test result for the two-way within-subjects ANOVA example illustrated above. For more information regarding multiple comparisons, click here.

Supported Worksheet Layout

This ANOVA design requires (i) two categorical columns for storing levels of the design within-subjects factors (i.e., Factor A with p >= 2 levels, and Factor B with q >=2 levels), and (ii) a numeric column for storing the experimental scores (response values) from all of the pq experimental conditions (see the right-hand side image below).

In this layout, a subject occurs in multiple worksheet rows: hence, it is required to have a data column (numeric or categorical) with the subject ID.

Below, is an example of the data layout used for two-way repeated measures ANOVA: