Pillai-Bartlett trace (V) is a statistic used in which multivariate analysis?

• A. A measure of explained variance across discriminant functions.
• B. A statistic used only for univariate ANOVA.
• C. The determinant of the within-group covariance matrix.
• D. Pillai-Bartlett trace is a MANOVA statistic equal to the sum of the proportion of explained variance on the discriminant function variates.

Answer: (D)

Pillai-Bartlett trace measures how much the group differences explain variance when you’re looking at multiple dependent variables together in a MANOVA. It does this by using the eigenvalues from the matrix product W^-1 B (within-group vs. between-group sums of squares and cross-products). For each discriminant function, the portion of variance explained by the group differences is λ_i / (1 + λ_i). Pillai’s trace sums these proportions across all discriminant functions, giving V = sum_i λ_i / (1 + λ_i). A larger value means stronger multivariate separation among groups. This exact interpretation—being the sum of the explained-variance proportions across the discriminant variates—is why it’s the best description. It isn’t a univariate ANOVA statistic, it isn’t merely the determinant of the within-group matrix, and while it relates to explained variance, the defining point is the summed proportions across discriminant functions.