8.12 - Summary

In this lesson we learned about:

• The 1-way MANOVA for testing the null hypothesis of equality of group mean vectors;
• Methods for diagnosing the assumptions of the 1-way MANOVA;
• Bonferroni corrected ANOVAs to assess the significance of individual variables;
• Construction and interpretation of orthogonal contrasts;
• Wilks lambda for testing the significance of contrasts among group mean vectors; and
• Simultaneous and Bonferroni confidence intervals for the elements of contrast.

In general, a thorough analysis of data would be comprised of the following steps:

1. Step 1:

   Perform appropriate diagnostic tests for the assumptions of the MANOVA. Carry out appropriate normalizing and variance-stabilizing transformations of the variables.

2. Step 2:

   Perform a one-way MANOVA to test for equality of group mean vectors. If this test is not significant, conclude that there is no statistically significant evidence against the null hypothesis that the group mean vectors are equal to one another and stop. If the test is significant, conclude that at least one pair of group mean vectors differ on at least one element and go on to Step 3.

3. Step 3:

   Perform Bonferroni-corrected ANOVAs on the individual variables to determine which variables are significantly different among groups.

4. Step 4:

   Construct up to g-1 orthogonal contrasts based on specific scientific questions regarding the relationships among the groups.

5. Step 5:

   Use Wilks lambda to test the significance of each contrast defined in Step 4.

6. Step 6:

   For the significant contrasts only, construct simultaneous or Bonferroni confidence intervals for the elements of those contrasts. Draw appropriate conclusions from these confidence intervals, making sure that you note the directions of all effects (which treatments or group of treatments have the greater means for each variable).

In this lesson we also learned about:

• How to perform multiple factor MANOVAs;
• What conclusions may be drawn from the results of a multiple-factor MANOVA;
• The Bonferroni corrected ANOVAs for the individual variables.

Just as in the one-way MANOVA, we carried out orthogonal contrasts among the four varieties of rice. However, in this case, it is not clear from the data description just what contrasts should be considered. If a phylogenetic tree were available for these varieties, then appropriate contrasts may be constructed.