ANOVA Methods and Multiple Comparisons

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One-Way ANOVA

This method allows comparing several independent groups/samples that may have a different mean for each group. One-way ANOVA is employed to evaluate whether or not there is a difference between at least two means in a set of data from k independent groups/samples (where k >=2).

Data from all samples/groups stored in one data column, and a grouping variable is used to split the data (see the left-hand side example data layout in the image below)
Each data column representing values obtained from a sample/group (see the right-hand side example data layout in the image below)

For multiple comparisons/post-hoc tests, see multiple comparisons built into the ANOVA methods.

Repeated Measures One-Way ANOVA

This method is an inferential test with two or more dependent samples. The samples represent repeated measurements on the same group of subjects and the design is called within-subjects or repeated measures ANOVA. Repeated measures one-way ANOVA is also referred to as the "single-factor within-subjects analysis of variance", "single-factor repeated-measures analysis of variance", or "randomized-blocks one-way analysis of variance".

The data layout for repeated measures ANOVA in Aabel is the same as that in the one-way ANOVA illustrated in the image above.
The output results include:
- The ANOVA table and a diamond plot
- Optional additional outputs including a mean table and results of sphericity tests and corrections

Two-Way ANOVA (Completely Randomized Factorial)

The two-way between subjects factorial ANOVA method is used to evaluate the effect of two independent categorical variables (i.e., factors) on a dependent variable simultaneously. The fixed effects design is the default, but Aabel also permits random effects models or mixed fixed/random models. If factor 1 has p levels and factor 2 has n levels, then there will be n*p groups.

The two-way ANOVA tests whether at least two of the levels of factor 1 represent populations with different mean values, and at least two of the levels of factor 2 represent populations with different mean values. In addition, it evaluates the variation among the differences between means for different levels of one factor over different levels of the other factor, i.e., whether there is a significant interaction between the factors.

To perform two-way factorial ANOVA, three columns of data are required. One column stores the numeric dependent variable, and two categorical columns with >= 2 levels contain the grouping values, i.e., factors.
For multiple comparisons/post-hoc tests, see multiple comparisons built into the ANOVA methods, outlined below.

Multiple Comparisons

Multiple comparisons in Aabel are simple comparisons (also know as pair-wise comparisons), and the tests are built into the ANOVA methods.

The default α level (the chance taken by researchers to make a Type I error) is set to 0.05. However, you can edit the α value.

The output results are presented in table format, and tables are editable. If the number of generated tables is more than what can be displayed on a single viewer page, Aabel will place the data on multiple pages. The image below shows the results of the Bonferroni test for the two-way ANOVA report illustrated above.

Tests That can be performed with equal groups sizes:

Tukey's HSD test
Tukey B test on ordered means
Fisher's LSD test
The Newman-Keuls (Neuman-Keuls) test on ordered means

Tests that can be performed with both equal and unequal group sizes:

Tukey Kramer test
Scheffe test
Bonferroni test