Investigation of Some Features Inherent in Analysis of Variance and one of its Variants in the Multivariate Setting

Abstract

It is common to have experiments conducted with the quest to seek an option in the use of analysis of variance (ANOVA) as incontrovertible choice among experimenters. The validity of this quest has been responsible for the considerable attention that has been given to the use of ANOVA. Different Types of experiments may throw up one or more response variable. With one response variable, the ANOVA option is indeed appropriate. However, for the simplicity of ANOVA the experimenter may have to contend with the somewhat a variant of it, even when the use of more than one response variable becomes expedient. The foregoing situation will be in violation of some well establishment ground norm of the need to overcome burdensome computations. This paper therefore presents a number of situations for the purpose of unveiling the inherent forms of statistics that are required to be computed. Although these statistics may appear to be burdensome or complicated in forms as they may present themselves in the setting of Multivariate ANOVA (MANOVA), which in reality has the other side of considerable reduction in the number of various statistics that are required to be computed. Such statistics are the sums of squares and sum of products for the two distinct independent components, in both the treatment and error, that the total sums of squares and products (SSPM) in the MANOVA has to be split into. Ditto, for the ANOVA that has its total sums of squares (SS), to be split into two orthogonal parts of treatment and error respectively. Furthermore, the appropriate number of independent components in which the treatment component will be partitioned is indeed determined by the number of factors say k, with the corresponding labels A1, A2, … , Ak, that the experimenter may have to adopt. A number of examples were presented for the illustration of distinct features inherent in both the ANOVA and MANOVA.