Which matrix is described as the multivariate analogue of the F-statistic in MANOVA?

Which matrix is described as the multivariate analogue of the F-statistic in MANOVA?

• A. HE−1
• B. E−1H
• C. H−1E
• D. H E

Answer: (A)

In MANOVA you’re comparing explained variation to unexplained variation across multiple dependent variables. The between-group variation is captured by H and the within-group (error) variation by E. To get a single multivariate measure that reflects how strong the group differences are, you form a matrix that standardizes the between variation by the within variation. The natural choice is the product of the between-group matrix with the inverse of the error matrix: HE^{-1}. This matrix encodes how much the between-group differences stand out when scaled by the within-group variation across all dependent variables. Its eigenvalues (and related statistics like Roy’s test) provide a multivariate analogue to the univariate F-statistic: larger values indicate stronger group separation. Note that HE^{-1} and E^{-1}H share the same eigenvalues, so either form serves the same role; the convention here uses HE^{-1}.

In MANOVA you’re comparing explained variation to unexplained variation across multiple dependent variables. The between-group variation is captured by H and the within-group (error) variation by E. To get a single multivariate measure that reflects how strong the group differences are, you form a matrix that standardizes the between variation by the within variation. The natural choice is the product of the between-group matrix with the inverse of the error matrix: HE^{-1}. This matrix encodes how much the between-group differences stand out when scaled by the within-group variation across all dependent variables. Its eigenvalues (and related statistics like Roy’s test) provide a multivariate analogue to the univariate F-statistic: larger values indicate stronger group separation. Note that HE^{-1} and E^{-1}H share the same eigenvalues, so either form serves the same role; the convention here uses HE^{-1}.