# 10.3 - Pairwise Comparisons

While the results of a one-way ANOVA will tell you if there is what is known as a main effect of the explanatory variable, the initial results will not tell you which groups are different from one another. In order to determine which groups are different from one another, a post-hoc test is needed. Post-hoc tests are conducted after a one-way ANOVA to determine which groups differ from one another. There are many different post-hoc analyses that could be performed following a one-way ANOVA. Here, we will learn about one of the most common tests known as Tukey's Honestly Significant Differences (HSD) Test.

Most statistical software, including Minitab Express, will compute Tukey's pairwise comparisons for you. This specific post-hoc test makes all possible pairwise comparisons. In this class we will be relying on statistical software to perform these analyses, if you are interested in seeing how the calculations are performed, this information is contained in the notes for STAT 502: Analysis of Variance and Design of Experiments. This analysis takes into account the fact that multiple tests are being performed and makes the necessary adjustments to ensure that Type I error is not inflated.

In the examples later in this lesson you will see a number of Tukey post-hoc tests. Next, you will also learn how to obtain these results using Minitab Express.

For each pairwise comparison, $$H_0: \mu_i - \mu_j=0$$ and $$H_a: \mu_i - \mu_j \ne 0$$.