Multivariate Analysis of Variance

1. Introduction In many agricultural experiments, generally the data on more than one character is observed. One common example is grain yield and straw yield. The other characters on which the data is generally observed are the plant height, number of green leaves, germination count, etc. The analysis is normally done only on the grain yield and the best treatment is identified on the basis of this character alone. The straw yield is generally not taken into account. If we see the system as a whole, the straw yield is also important either for the cattle feed or for mulching or manuring, etc. Therefore, while analyzing the data, the straw yield should also be taken into consideration. Similarly, in varietal trials also the data is collected on several plant characteristics and quality parameters. In these experimental situations also the data is generally analyzed separately for each of the characters. The best treatment or genotype is identified separately for each of the characters. In these situations, Multivariate Analysis of Variance (MANOVA) can be helpful. Before discussing about MANOVA, a brief description about Analysis of Variance (ANOVA) is given in Section 2. A general procedure of performing MANOVA on the data generated from RCB design is given in Section 3. The procedure of MANOVA has been illustrated with the help of an example in Section 4. 2. Overview of ANOVA The ANOVA looks at the variance within classes relative to the overall variance. The dependent variable must be metric, and the independent variables, which can be many, must be nominal. ANOVA is used to uncover the main and interaction effects of categorical independent variables (called "factors") on an interval dependent variable. A main effect is the direct effect of an independent variable on the dependent variable. An interaction effect is the joint effect of two or more independent variables on the dependent variable. Whereas regression models cannot handle interaction unless explicit crossproduct interaction terms are added, ANOVA uncovers interaction effects on a built-in basis. The key statistic in ANOVA is the F-test of difference of group means, testing if the means of the groups formed by values of the independent variable (or combinations of values for multiple independent variables) are different enough not to have occurred by chance. If the group means do not differ significantly then it is inferred that the independent variable(s) did not have an effect on the dependent …