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Tukey HSD for Post-Hoc Analysis

Step 2: Tukey HSD for Post-Hoc Analysis

If your ANOVA test shows that the means aren’t all equal, your next step is to determine whichmeans are different, to your level of significance. You can’t just perform a series of t tests, because that would greatly increase your likelihood of a Type I error. So what do you do?

John Tukey gave one answer to this question, the HSD (Honestly Significant Difference) test. You compute something analogous to a t score for each pair of means, but you don’t compare it to the Student’s t distribution. Instead, you use a new distribution called the studentized range or q distribution.

Caution:Perform post-hoc analysis only if the ANOVA test shows a p-value less than your α. If p>α, you don’t know whether the means are all the same or not, and you can’t go fishing for unequal means.

You generally want to know not just which means differ, but by how much they differ (theeffect size). The easiest thing is to compute the confidence interval first, and then interpret it for a significant difference in means (or no significant difference). You’ve already seen this relationship between a test of significance at the α level and a 1−α confidence interval:

  • If the endpoints of the CI have the same sign (both positive or both are negative), then 0 is not in the interval and you can conclude that the means are different.
  • If the endpoints of the CI have opposite signs, then 0 is in the interval and you can’t determine whether the means are equal or different.