## Thursday, May 29, 2014

### Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA

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

Overview of Single-Factor ANOVA

Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013

Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group

Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013

Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison

Single-Factor ANOVA - All Excel Calculations

Overview of Post-Hoc Testing For Single-Factor ANOVA

Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA

Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA

Overview of Effect Size For Single-Factor ANOVA

ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013

ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013

ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013

Power of Single-Factor ANOVA Test Using Free Utility G*Power

Welch’s ANOVA Test in 8 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

Brown-Forsythe F-Test in 4 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

# Tukey-Kramer Post-Hoc Test in Excel For Single- Factor ANOVA

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 qCritical . 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:

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:

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:

q2,3 = ABS(x_bar2 – x_bar3) / SE

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

ABS(x_bar2 – x_bar3) = 1.205

MSWithin = 2.306

n2 = 23

n3 = 18

q2,3 = 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

qCritical = 3.399

According to the Tukey-Kramer test the largest difference (pair 2,3) is significant because q(2,3) = 3.566 and is larger than qCritical (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 qCritical 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)

Excel Master Series Blog Directory

Statistical Topics and Articles In Each Topic

• Histograms in Excel
• Bar Chart in Excel
• Combinations & Permutations in Excel
• Normal Distribution in Excel
• t-Distribution in Excel
• Binomial Distribution in Excel
• z-Tests in Excel
• t-Tests in Excel
• Hypothesis Tests of Proportion in Excel
• Chi-Square Independence Tests in Excel
• Chi-Square Goodness-Of-Fit Tests in Excel
• F Tests in Excel
• Correlation in Excel
• Pearson Correlation in Excel
• Spearman Correlation in Excel
• Confidence Intervals in Excel
• Simple Linear Regression in Excel
• Multiple Linear Regression in Excel
• Logistic Regression in Excel
• Single-Factor ANOVA in Excel
• Two-Factor ANOVA With Replication in Excel
• Two-Factor ANOVA Without Replication in Excel
• Randomized Block Design ANOVA in Excel
• Repeated-Measures ANOVA in Excel
• ANCOVA in Excel
• Normality Testing in Excel
• Nonparametric Testing in Excel
• Post Hoc Testing in Excel
• Creating Interactive Graphs of Statistical Distributions in Excel
• Solving Problems With Other Distributions in Excel
• Optimization With Excel Solver
• Chi-Square Population Variance Test in Excel
• Analyzing Data With Pivot Tables
• SEO Functions in Excel
• Time Series Analysis in Excel
• VLOOKUP