Which matrix in MANOVA is functionally equivalent to the hypothesis SSCP divided by the error SSCP, representing the ratio of systematic to unsystematic variance and serving as a multivariate analogue of the F-statistic?

Which matrix in MANOVA is functionally equivalent to the hypothesis SSCP divided by the error SSCP, representing the ratio of systematic to unsystematic variance and serving as a multivariate analogue of the F-statistic?

• A. HE−1
• B. E−1H
• C. H+E
• D. H−E

Answer: (A)

In MANOVA you separate variation into what the model explains (H, the hypothesis SSCP) and what remains within groups (E, the error SSCP). To get a multivariate analogue of the F-ratio, you form a matrix that represents how much systematic (between-group) variance exceeds unsystematic (within-group) variance. This is done by multiplying the hypothesis matrix by the inverse of the error matrix, yielding H times E inverse. This product encodes the ratio in a way that, through its eigenvalues, drives the multivariate test statistics (like Wilks’ lambda and Pillai’s trace). The larger the eigenvalues, the stronger the evidence that the group means differ across the response variables. Note that E inverse times H would yield the same eigenvalues in this context, so both expressions capture the same ratio of systematic to unsystematic variance. The other options do not represent this ratio: they either combine the matrices through sums or differences, which do not convey the ratio of explained to unexplained variance.

In MANOVA you separate variation into what the model explains (H, the hypothesis SSCP) and what remains within groups (E, the error SSCP). To get a multivariate analogue of the F-ratio, you form a matrix that represents how much systematic (between-group) variance exceeds unsystematic (within-group) variance. This is done by multiplying the hypothesis matrix by the inverse of the error matrix, yielding H times E inverse. This product encodes the ratio in a way that, through its eigenvalues, drives the multivariate test statistics (like Wilks’ lambda and Pillai’s trace). The larger the eigenvalues, the stronger the evidence that the group means differ across the response variables. Note that E inverse times H would yield the same eigenvalues in this context, so both expressions capture the same ratio of systematic to unsystematic variance. The other options do not represent this ratio: they either combine the matrices through sums or differences, which do not convey the ratio of explained to unexplained variance.