An ANOVA test is used to find out if there is a significant difference between three or more group means. However, the ANOVA analysis simply indicates there is a difference between two or more group means, but it does not tell you which means there is a significant difference between. For example, in the table below there are five groups listed. An ANOVA indicated there is a significant difference! Is it between mean I & II; means II & IV? Means III and IV? In order to find out between which means there is a significant difference, a post hoc test needs to be done. The Tukey Test is a post hoc test designed to perform a pairwise comparison of the means to see where a significant difference lies!
The first step in doing a Tukey Test is to arrange the means in ascending order in a comparison table and to calculate the difference between each pair of means.
The next step is to calculate the minimum pairwise difference needed. Use the following formula.
Dmin= Qt (sqrt(MSwithin group))/(sqrt(n))
Dmin is the minimum pairwise
difference needed for significance
n is the sample size of each group (number of data points in each group in the data set or, if the groups have different numbers of data points, use the smallest group number for a conservative analysis)
Qt is the t statistic you will need. It requires the following:
Dmin= Qt (sq rt(MSwithin group))/(sq rt(n))
Dmin= 3.92(sq rt(219.28))/(sqrt(25))
The final step is to compare the difference between the means in the
table you constructed to the minimum pairwise difference. The ones that are larger than
the minimum are the means pairs that are significantly different. In this example, group 4 is statistically significantly different than every other group.
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