The test statistic, the Lawley-Hotelling trace, is:
with s (2m + s + 1) and 2 (sn + 1) df.
|H||hypothesis 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 ([p2 q2 - 4] / p2 + q2 - 5, if p2 + q2 - 5 > 0|
Let λ1≥λ2≥λ3≥ . . . ≥λp 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.