An Illustration of the Principal Ideas of MANOVA

Abstract

The succession of graphs overleaf illustrates the steps necessary for the determination of eigenvalues and -vectors and the subsequent transformations back into the original system in connection with MANOVA. The 8 figures are to be linked in a clockwise direction. In Figure 1 a special case of MANOVA is demonstrated. There are 9 groups. Their means are indicated by a cross; the individual deviations from the means of the corresponding groups are represented by the 9 ellipses of dispersion. These ellipses are defined by their means and the covariance matrix of the respective group. The mean of the means, e.g. the total mean, is indicated by an encircled dot. In Figure 2 an ellipse of dispersion of the means is constructed. It is defined by the total mean and the covariance matrix of the means; this matrix is called the matrix between the groups (ellipse A). Around the total mean there is drawn a second smaller ellipse which is defined by the total mean and the covariance matrix within the groups (ellipse E). It is representative of the dispersion within the groups. Corresponding to an F-ratio we want to compare the deviation of the groups from the total mean with the deviation within the groups in any direction. This comparison will be very easy if we transform the ellipse to a circle. The further steps are demonstrated in Figures 3 through 5. Figure 5 is the result of a transformation into the canonical base. It demonstrates that there is one direction, represented by the new axis K1, ,in which there is a maximal ratio a/X1 of covariance (equivalent to dispersion of ellipse A* from total mean = V/X1) between groups to covariance within groups (equivalent to dispersion of ellipse I from total mean = 1). A second ratio V/X2 demonstrates the least ratio of covariance-between to covariance-within. K1 is the first canonical axis, K2 the second. Projections on K1 and K2, respectively, will yield canonical variates. In Figures 5 through 8 the transformation to the canonical base is reversed so that we get in Figure 8 the original system containing the two canonical axes. These are a pair of conjugate axes of both the ellipses between and within. Note that these canonical axes, which are sometimes called generalized principal axes, are different from the special principal axes of both ellipses. In Figure 8 the coordinates of the points of intersection between the