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# Multiple comparison testing in ANOVA

Multiple comparison testing in ANOVA offers different methods that help solving the unresolved scenario which is returned by a rejection of H0.

## Purpose of multiple comparison testing in ANOVA

We have seen in one-way ANOVA and in two-way ANOVA that, when we reject a null hypothesis, we only conclude that not all population means are equal. We are not saying that they are all different from each other. Thus, some of them can still be equal. In fact, we can have scenarios with several means being equals, and only one being different from all the others. We can have scenarios like these which are not revealed by the ANOVA:

These scenarios will still return test statistic results that lead to rejection of H0. So, a rejection of H0 does not tell us ‘the whole story’:

• Are there still some of the population means that can be expected to be equal?
• In that case, which ones?
• And how confident can we be of these calculations?
• Maybe the means from group 1 and 2 are equal and only different from the ones of group 3 and 4?

We may wish to explore further and compare the mean of one group with the mean of another, and this is what multiple comparison testing in ANOVA is for.

## Methods for multiple comparison testing in ANOVA

There are different methods for multiple comparison testing in ANOVA:

• Fishers Least Significant Difference (LSD)
• Tukey method
• Bonferroni (Dunn) method
• Newman-Keuls method
• Scheffee method
• Dunnett method

I will be working out theory page and cases for these methods. Meanwhile, I refer to this page: Multiple comparison analysis testing in ANOVA

#### Carsten Grube

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##### ANOVA & the F-distribution

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