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# Two-Way Between-Subjects ANOVA

Two-way between-subjects ANOVA is also known as between-subjects factorial ANOVA, completely randomized factorial ANOVA, two-way ANOVA, completely randomized factorial design with two treatments. It evaluates the effect of two independent variables (factors) on a response variable simultaneously. That is, it evaluates the variation among the differences between means for different levels of one factor over different levels of the other factor.

### Two-Way Between-Subjects ANOVA Output

The default output includes the two-way ANOVA table. The additional output depends on selected options, and can include the two-way mean table, interaction plots, the test results for homogeneity of variance, and table reports for selected multiple comparisons/post-hoc tests.

#### The Default Output: Two-Way ANOVA Table

- The source of variability: The between-groups variability (i) attributable to Factor
*A*, (ii) attributable to Factor*B*, and (iii) attributable to*AB*interaction; the within-groups variability (i.e., error or residual variability) - 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)

This table displays:

#### Testing for Homogeneity of Variance

- Hartley's Fmax test
- Bartlett's chi-square test

For evaluating the homogeneity of variance assumption, Aabel provides:

#### Interaction Plots

The interaction plot is a graphical display of the effect of one factor at each level of the other factor. 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%)

### Multiple Comparisons/Post-Hoc Tests

Simple comparisons (also know as pair-wise comparisons) accompanying two-way between-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

When you select a multiple comparison/post-hoc test accompanying a two-way between ANOVA, the test will be performed for any item of the
ANOVA table that has a *p* value less than or equal to the omnibus α.

The example below is the Newman-Keuls test result for the two-way between-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 between-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* independent groups.