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Multiple Comparisons/Post-Hoc Tests

In an analysis of variance, if the null hypothesis is rejected, i.e., at least two of the group means differ significantly, we still may not know which group means differ and how many group means differ from each other. The multiple comparison/post-hoc tests are designed to answer these questions. There are two types of comparison tests, i.e., simple comparisons and complex comparisons.

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

The α Level

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.

Analysis Output

The output results are presented in table format, and tables are editable (see the example shown at the bottom of this page).
If the output results include more tables that can be displayed on a single viewer page, Aabel will automatically create new pages as needed.

Multiple Comparisons Accompanying Analysis of Variance (ANOVA)

Tests That Can Be Performed With Equal Group Sizes:

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

Tests That Can Be Performed With Both Equal and Unequal Group Sizes:

Tukey Kramer test
Scheffé test
Bonferroni-Dunn test

The more conservative a test, the less powerful, and the lower the risk of making a Type I error (i.e., incorrectly deducing a difference, when in fact there is no significant difference). However, reducing the risk of making a Type I error increases the chance of making a type II error (i.e., incorrectly deducing no difference, when in fact there is a significant difference).

Scheffé test is commonly described as the most conservative of the unplanned tests:

Testing Independent Groups vs. a Given Control Group

Aabel provides the Dunnett test for pair-wise comparisons where all independent groups are compared with the same control group.

Example The response value ranges from four independent groups of subjects, where three groups have received an experimental treatment (E1, E2, E3), and a control group has received placebo, are compared in the box & whisker plot below. The corresponding one-way between-subjects ANOVA report indicates that at least two groups have means that differ significantly. The Dunnett test reveals that it is only the experimental group E1 that differs significantly from the control group.

Multiple Comparisons Accompanying Analysis of Covariance (ANCOVA)

The multiple comparisons/post-hoc tests accompanying the analysis of covariance are conducted on The Adjusted Group/Treatment Means, and include:

Tukey's HSD test
Scheffé test

The expressions employed to compute the test statistic for the tests accompanying ANCOVA are different from those used for tests accompanying ANOVA. In the former, computing adjusted group/treatment means determines what the scores on dependent variable would be if the groups did not differ on the covariate.