Post Hoc Tests > Tukey Test / Honest Significant Difference

## What is the Tukey Test / Honest Significant Difference?

The Tukey Test (or Tukey *procedure*), also called Tukey’s Honest Significant Difference test, is a post-hoc test based on the studentized range distribution. An ANOVA test can tell you if your results are significant overall, but it won’t tell you exactly where those differences lie. After you have run an ANOVA and found significant results, then you can run Tukey’s HSD to find out which specific groups’s means (compared with each other) are different. The test compares all possible pairs of means.

To test all pairwise comparisons among means using the Tukey HSD, calculate HSD **for each pair of means **using the following formula:

Where:

- M
_{i}– M_{j}is the difference between the pair of means. to calculate this, M,_{i}should be larger than M_{j} - MS
_{w}is the Mean Square Within, and n is the number in the group or treatment.

## General Steps

Step 1: Perform the ANOVA test. Assuming your F value is significant, you can run the post hoc test.

Step 2: Choose two means from the ANOVA output. Note the following:

- Means,
- Mean Square Within,
- Number per treatment/group,
- Degrees of freedom Within.

Step 3: Calculate the HSD statistic for the Tukey test using the formula.

Step 4: Find the score in Tukey’s critical value table.

Step 5: Compare the score you calculated in Step 3 with the tabulated value you found in Step 4. If the calculated value from Step 3 is bigger than the critical value from the critical value table, the two means are significantly different.

## Assumptions for the test

- Observations are independent within and among groups.
- The groups for each mean in the test are normally distributed.
- There is equal within-group variance across the groups associated with each mean in the test (homogeneity of variance).

## Tukey-Kramer Method

If you have unequal sample sizes, you have to calculate the estimated standard deviation for each pairwise comparison. This is called the **Tukey-Kramer Method**.

**Reference**:

Brillinger, D. “The Collected Works of John W. Tukey”. 1984.

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