Two-Way Mixed-Model ANOVA (Between-Within Design)

Two-way mixed model ANOVA is also known as Two-way Mixed factorial ANOVA and split-plot factorial design. This Design combines one independent sample factor (Factor A) and one correlated groups factor (Factor B), evaluating the main effects of the Factor A, Factor B, and the interaction (AB).

Two-Way Mixed-Model 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 corrections, and table reports for selected multiple comparisons tests.

The Default Output: Two-Way Mixed (Between-Within) ANOVA Table

This table displays:

• The source of variability: The between-subjects variability (i) attributable to Factor A, and (ii) attributable to subjects within group error (Subjects WG); The within-subjects variability (i) attributable to Factor B, (ii) attributable to AB interaction, and (iii) attributable to B x Subjects WG error
• 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 test statistic (F) and the p value
• 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 (in the example below, the main effect is due to the pq condition "Low Temperature" x "High Humidity").

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 (Haris W)

• In factorial ANOVA with a mixed design Factor B represents multi-sample repeated measures. Accordingly, evaluating the tenability of sphericity assumption requires a multi-sample sphericity test, for which a variety of approaches have been made.
• Aabel uses the Harris approach [Harris, P. (1984); Kirk, R.E. (1995)] for testing the multi-sample sphericity assumption for Factor B.

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 mixed between-within ANOVA design 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 Bonferroni-Dunn test result for the two-way mixed between-within ANOVA example illustrated above. For more information regarding multiple comparisons, click here.

Supported Worksheet Layout

For this ANOVA design, Aabel supports two different worksheet layouts.

• The layout I allows storing the design experimental scores (response data) from all of the pq levels of the design factors (Factor A with p >= 2 levels, and Factor B with q >=2 levels) in a single numeric column, and uses the pq levels of the design factors to split the response data accordingly. 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.
• The layout II requires storing the design experimental scores (response data) from each level of the within-subjects factor (Factor B) in a separate numeric column, and uses the p >= 2 levels of the between-subjects factor (Factor A) to split the response data accordingly.