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.csv
The program below shows the analysis of the rice data.
Download the SAS Program here: rice.sas
Note: In the upper right-hand corner of the code block you will have the option of copying ( ) the code to your clipboard or downloading ( ) the file to your computer.
options ls=78;
title "Two-Way MANOVA: Rice Data";
data rice;
infile "D:\Statistics\STAT 505\data\rice.csv" firstobs=2 delimiter=',';
input block variety $ height tillers;
run;
/*
* The class statement specifies block and variety as categorical variables.
* The model statement specifies responses height and tillers with predictors
* block and variety. lsmeans displays the least-squares means for variety.
* The manova statement requests the test for response mean vectors across
* levels of variety, and the printe and printh options display sums of
* squares and cross products for error and the hypothesis, respectively.
*/
proc glm data=rice;
class block variety;
model height tillers=block variety;
lsmeans variety;
manova h=variety / printe printh;
run;
Performing a two-way MANOVA
To carry out the two-way MANOVA test in Minitab:
- Open the ‘rice’ data set in a new worksheet.
- For convenience, rename the columns: block, variety, height, and tillers, from left to right.
- Stat > ANOVA > General MANOVA
- Highlight and select height and tillers to move them to the Responses window.
- Highlight and select block and variety to move them to the Model window.
- Choose 'OK'. The MANOVA results for the tests for block and variety are separately displayed in the results area.
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 the number of tillers within varieties \(( r = - 0.278 ; p = 0.3572 )\).