The partitioning of the total sum of squares and cross products matrix may be summarized in the multivariate analysis of variance table as shown below:
| Source | d.f. | SSP |
| Blocks | b - 1 | B |
| Treatments | a - 1 | H |
| Error | (a - 1)(b - 1) | E |
| Total | ab - 1 | T |
SSP stands for the sum of squares and cross products discussed above.
To test the null hypothesis that the treatment mean vectors are equal, compute a Wilks Lambda using the following expression:
\(\Lambda^* = \dfrac{|\mathbf{E}|}{|\mathbf{H+E}|}\)
This is the determinant of the error sum of squares and cross products matrix divided by the determinant of the sum of the treatment sum of squares and cross products plus the error sum of squares and cross products matrix.
Under the null hypothesis, this has an F-approximation. The approximation is quite involved and will not be reviewed here. Instead, let's take a look at our example where we will implement these concepts.
Example 8-11: Rice Data Section
Rice data can be downloaded here: rice.txt
Analysis
- We reject the null hypothesis that the variety mean vectors are identical \(( \Lambda = 0.342 ; F = 2.60 ; d f = 6,22 ; p = 0.0463 )\). At least two varieties differ in means for height and/or number of tillers.
- Results of the ANOVAs on the individual variables:
Variable F SAS p-value Bonferroni p-value Height 4.19 0.030 0.061 Tillers 1.27 0.327 0.654 Each test is carried out with 3 and 12 d.f. Because we have only 2 response variables, a 0.05 level test would be rejected if the p-value is less than 0.025 under a Bonferroni correction. Thus, if a strict \(α = 0.05\) level is adhered to, then neither variable shows a significant variety effect. However, if a 0.1 level test is considered, we see that there is weak evidence that the mean heights vary among the varieties (F = 4.19; d. f. = 3, 12).
- The Mean Heights are presented in the following table:
Variety Mean Standard Error A 58.4 1.62 B 50.6 1.62 C 55.2 1.62 D 53.0 1.62 -
Variety A is the tallest, while variety B is the shortest. The standard error is obtained from:
\(SE(\bar{y}_{i.k}) = \sqrt{\dfrac{MS_{error}}{b}} = \sqrt{\dfrac{13.125}{5}} = 1.62\)
- Looking at the partial correlation (found below the error sum of squares and cross products matrix in the output), we see that height is not significantly correlated with number of tillers within varieties \(( r = - 0.278 ; p = 0.3572 )\).