Select the method or formula of your choice.

The test statistic, Wilks' lambda, is:

with pq and (rt – 2u) df.

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

H | hypothesis matrix |

E | error matrix |

p | number of responses |

q | df of the hypothesis |

v | df for E |

s | min (p, q) |

m | .5 ( | p – q | – 1) |

n | .5 (v – p – 1) |

r | v – 0.5 (p – q + 1) |

u | 0.25(pq – 2) |

t | = Sqrt ([p^{2} q^{2} - 4] / p^{2} + q^{2} - 5, if p^{2} + q^{2} - 5 > 0 |

t | 1 |

Let λ_{1}≥λ_{2}≥λ_{3}≥ . . . ≥λ_{p} be the eigenvalues of (E** - 1) * H. The first three test statistics can be expressed in terms of either H and E or these eigenvalues.

The H matrix is a p x p matrix that contains the "between" sum of squares on the diagonal for each of the p variables. The H matrix is calculated as:

The E matrix is a p x p matrix that contains the "within" sum of squares on the diagonal for each of the p variables. The E matrix is calculated as:

In the first three tests, the F statistic is exact if s = 1 or 2, otherwise it is approximate. Minitab tells you when the test is approximate.

The test statistic, the Lawley-Hotelling trace, is:

with s (2m + s + 1) and 2 (sn + 1) df.

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

H | hypothesis matrix |

E | error matrix |

p | number of responses |

q | df of the hypothesis |

v | df for E |

s | min (p, q) |

m | .5 ( | p – q | – 1) |

n | .5 (v – p – 1) |

r | v – 0.5 (p – q + 1) |

u | 0.25(pq – 2) |

t | = Sqrt ([p^{2} q^{2} - 4] / p^{2} + q^{2} - 5, if p^{2} + q^{2} - 5 > 0 |

t | 1 |

Let λ_{1}≥λ_{2}≥λ_{3}≥ . . . ≥λ_{p} be the eigenvalues of (E** - 1) * H. The first three test statistics can be expressed in terms of either H and E or these eigenvalues.

The H matrix is a p x p matrix that contains the "between" sum of squares on the diagonal for each of the p variables. The H matrix is calculated as:

The E matrix is a p x p matrix that contains the "within" sum of squares on the diagonal for each of the p variables. The E matrix is calculated as:

In the first three tests, the F statistic is exact if s = 1 or 2, otherwise it is approximate. Minitab tells you when the test is approximate.

The test statistic, Pillai's trace, is:

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

H | hypothesis matrix |

E | error matrix |

p | number of responses |

q | df of the hypothesis |

v | df for E |

s | min (p, q) |

m | .5 ( | p – q | – 1) |

n | .5 (v – p – 1) |

r | v – 0.5 (p – q + 1) |

u | 0.25(pq – 2) |

t | = Sqrt ([p^{2} q^{2} - 4] / p^{2} + q^{2} - 5, if p^{2} + q^{2} - 5 > 0 |

t | 1 |

Let λ_{1}≥λ_{2}≥λ_{3}≥ . . . ≥λ_{p} be the eigenvalues of (E** - 1) * H. The first three test statistics can be expressed in terms of either H and E or these eigenvalues.

The H matrix is a p x p matrix that contains the "between" sum of squares on the diagonal for each of the p variables. The H matrix is calculated as:

The E matrix is a p x p matrix that contains the "within" sum of squares on the diagonal for each of the p variables. The E matrix is calculated as:

In the first three tests, the F statistic is exact if s = 1 or 2, otherwise it is approximate. Minitab tells you when the test is approximate.

The largest eigenvalue, λ_{1}. To finish the test, you must use special charts, called Heck charts, along with the parameters s, m, and n, to find the significance level.

See Heck^{1} for these charts.

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

s | min (p, q) |

m | .5 ( | p – q | – 1) |

n | .5 (v – p – 1) |

_{1}≥λ_{2}≥λ_{3}≥ . . . ≥λ_{p} be the eigenvalues of (E** - 1) * H. The first three test statistics can be expressed in terms of either H and E or these eigenvalues.

- D.L. Heck (1960), "Charts of Some Upper Percentage Points of the Distribution of the Largest Characteristic Root," The Annals of Statistics, 625–642.