Minitab offers four different confidence interval methods for comparing multiple factor means in one-way analysis of variance when you assume equal variances between the groups: Tukey's, Fisher's, Dunnett's, and Hsu's MCB. The formulas for the confidence intervals follow.
|sample mean for the ith factor level |
|ni||number of observations in level i |
|r ||number of factor levels |
|s||pooled standard deviation or sqrt(MSE) |
|nT||total number of observations|
|α||probability of making a Type I error |
where Q = (1 − α) percentile of the studentized range distribution with r number of factor levels and nT- r degrees of freedom.
where t = (1 − α/2) percentile of the Student's t-distribution with nT − r degrees of freedom.
To see how d is calculated, refer to page 63 in Hsu1.
We give formulas for the case where all group sizes are equal to n. Formulas for unequal group sizes are found in Hsu1. Suppose you chose the best to be the largest mean, and you want the confidence interval for the ith mean minus the largest of the others.
The lower endpoint is the smaller of zero and the formula that follows:
The upper endpoint is the larger of zero and the formula that follows:
To see how d is calculated, refer to page 83 in Hsu1.
When the best is the smallest of the level means, the formulas are the same, except that max is replaced by min.
We are very grateful for assistance in the design and implementation of multiple comparisons from Jason C. Hsu.
- J.C. Hsu (1996). Multiple Comparisons, Theory and methods. Chapman & Hall.