Select the method or formula of your choice.

Minitab offers five different methods for comparing multiple factor means in one-way analysis of variance: Tukey's, Fisher's, Dunnett's, Hsu's MCB, and Games-Howell. The formulas for these tests are listed below.

Term | Description |
---|---|

the sample mean for the i^{th} factor level | |

the sample mean j^{th} factor level | |

the number of observations in level i | |

r | the number of levels |

s | the pooled standard deviation or sqrt(MSE) |

u | the degrees of freedom for error |

α | the simultaneous probability of making a Type I error |

α* | the individual probability of making a Type I error |

where Q = upper *α* percentile of the studentized range distribution with *r* and *n _{T }- r* degrees of freedom.

To find the individual error rate from the simultaneous error rate, use the following formula:

where *t* = upper *α*/2 point of the Student's t-distribution with *u* df.

To find the simultaneous confidence level from the individual error rate, use the following formula:

To see how *d* is calculated, refer to page 63 in Hsu^{1}.

We give formulas for the case where all group sizes are equal to *n*. Formulas for unequal group sizes are found in Hsu^{1}. Suppose you chose the best to be the largest mean, and you want the confidence interval for the *i*^{th} mean minus the largest of the others.

The lower endpoint is the smaller of zero and

The upper endpoint is the larger of zero and

To see how d is calculated, refer to page 83 in Hsu^{1}.

When the best is the smallest of the level means, the formulas are the same, except that max is replaced by min.

The Welch test statistic is computed as follows.

The p-value for the Welch test is an upper tail probability for an F distribution with numerator degrees of freedom *k* - 1, where *k* is the number of X levels, and denominator degrees of freedom given by:

The comparison interval for *μ _{i}* -

The critical value is based on the Studentized range (*Q*) for *k* groups, similar to the Tukey-Kramer intervals. But for Games-Howell, Minitab computes different degrees of freedom for each comparison:

The T-ratio used to compute the adjusted P-value equals:

Where:

The*Y _{ij}*,

The average response at the *i*^{th} level equals:

The sample variance equals:

The weight for level i equals:

The sum of all weights equals:

The overall weighted average of responses equals:

We are very grateful for assistance in the design and implementation of multiple comparisons from Jason C. Hsu.

[1] J.C. Hsu (1996). Multiple Comparisons, Theory and methods. Chapman & Hall.