This is one of the following sixteen articles on Single-Factor ANOVA in Excel

The Tukey-Kramer Post-Hoc test is performed when group variances are equal and group sizes are unequal. The Tukey-Kramer test is normally performed in place of Tukey’s HSD when group sizes are the same because both Post-Hoc tests produce the same answer.

The Tukey-Kramer test calculates Test Statistic q for each pair of means. This Test Statistic is compared to q_Critical. The critical q values are found on the Studentized Range q table using the Excel lookup function, INDEX(array, row number, column number).

The difference between the two means is designated as significant if its test statistic q is larger than the critical q value from the table.

The Test Statistic q is calculated as follows:

q = (Max Group Mean – Min Group Mean) / SE

where SE (standard error) is calculated as follows:

(Click Image To See a Larger Version)

df = Degrees of freedom = (total number of samples) – (total number of groups)

The first step when performing the Tukey-Kramer test is to list all unique mean pairs and the differences between the means. All of this information can be found from the Excel ANOVA output as follows:

(Click Image To See a Larger Version)

Group 1 Mean = x_bar₁ = 19.045

Group 2 Mean = x_bar₂ = 19.261

Group 3 Mean = x_bar₃ = 18.056

Three unique group pairings exist: (1,2), (1,3), and (2,3)

The differences in means of each pair are as follows:

Pair (1,2) Mean Difference = ABS(19.045-19.261) = 0.216

Pair (1,3) Mean Difference = ABS(19.045-18.056) = 0.989

Pair (2,3) Mean Difference = ABS(19.261-18.056) = 1.205

The pair of groups having the largest difference in means occurs in groups 2 and 3. This group pair will therefore be the first evaluated to determine if its difference is large enough to be significant.

Test Statistic q for this group pair will be calculated as follows:

q_2,3 = ABS(x_bar₂ – x_bar₃) / SE

where SE = SQRT(1/2 * MS_Within * (1/n₂ + 1/n₃) )

ABS(x_bar₂ – x_bar₃) = 1.205

MS_Within = 2.306

n₂ = 23

n₃ = 18

q_2,3 = 1.205 / SQRT((0.5)*2.306*(1/23+1/18)) = 3.566

df = (total number of samples) – (total number of groups)

df = 63 – 3 = 60

From the Studentized Range q table

q_Critical= 3.399

(Click Image To See a Larger Version)

According to the Tukey-Kramer test the largest difference (pair 2,3) is significant because q_(2,3) = 3.566 and is larger than q_Critical(3.399). The differences between the other pairs are not significant because q_(1,2) = 0.6745 and q_(1,3) = 2.898. The Games-Howell test will shortly be shown to produce very similar results.

Looking Up q_Critical on the Studentized Range q Table With the Excel INDEX() Function

The Studentized Range q table and the Excel Index() function appear as follows:

=INDEX( array, relative row number, relative column number )

A relative address is the address relative to the cell in upper left corner of the array. In the INDEX() function is attempting to locate a value in a cell that is in the third column over (to the right of) and third row down from the cell in the upper left corner of the array, the relative row number equals 3 and the relative column number equals 3.

The array is the absolute address of the array. This is given by:

(upper left corner cell:lower right corner cell). In this case it would be (D5:K103)

(Click Image To See a Larger Version)

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