How to execute multiple comparison

Student’s t-test would be executed to compare average between two groups. Then if you would like to compare average between 3 or more groups, what do you do? The test needs 2 steps process.

1. Analysis of variance (ANOVA)
2. Compare between each 2 groups

1. Analysis of variance

On analysis of variance, null hypothesis is that all groups belong to one population. Therefore, if null hypothesis has been rejected, all groups would not belong to one population.

If all groups belong to one population, the average of all groups, called as grand mean (MG), would be close to average of population. Furthermore, if all groups belong to one population, each average of each groups, for example, M1, M2, M3, would be close to grand mean. However, if any group doesn’t belong to one population, the average of other population would be far from MG. Then we need the indicator that represents how far each average of each groups from grand mean, corrected with number of sample, n. It is called as mean square among groups (MSA).

MSA; mean square among groups. n; number of sample in each groups. i; number of group (incremental variable). k; number of groups.

Then calculate variances of each samples, correct with number of each sample and you would take index of variances in samples. It is mean square of error (MSE), average of variance in group.

MSE; mean square of error. n; number of sample in each groups. i; number of group (incremental variable). k; number of groups. V; variance.

x; each value of samples in each groups. n; number of sample.

F statistics, calculated as ratio MSA to MSE, follows F distribution. When F statistics would be over a value, null hypothesis would be rejected and you could compare average between each groups.

2. Compare between each 2 groups

If null hypothesis would be rejected with ANOVA, you could compare between each groups with following method.

• Bonferroni method
• Tukey’s HSD
• Dunnet’s procedure
• Hsu’s MCB tests
• Scheffe’s procedure

Bonferroni method may be easy to understand and use. Divided significance level by k, number of pairs, would be Bonferroni corrected significance level. See following chart.