Multivariate Analysis of Variance for Repeated Measures
Multivariate analysis of variance analysis is a test of the
A*B*C = D, where
B is the p-by-r matrix
of coefficients. p is the number of terms, such
as the constant, linear predictors, dummy variables for categorical
predictors, and products and powers, r is the number
of repeated measures, and n is the number of subjects.
an a-by-p matrix, with rank a ≤ p,
defining hypotheses based on the between-subjects model.
an r-by-c matrix, with rank c ≤ r ≤ n
– p, defining hypotheses based on the within-subjects
D is an a-by-c matrix,
containing the hypothesized value.
manova tests if the model terms are significant
in their effect on the response by measuring how they contribute to
the overall covariance. It includes all terms in the between-subjects
manova always takes
zero. The multivariate response for each observation (subject) is
the vector of repeated measures.
manova uses four different methods to measure these contributions: Wilks’
lambda, Pillai’s trace, Hotelling-Lawley trace, Roy’s maximum root statistic. Define
where X is a design matrix containing the factor values for the MANOVA. Then, the hypotheses sum of squares and products matrix is
and the residuals sum of squares and products matrix is
The matrix Qh is analogous
to the numerator of a univariate F-test, and
Qe is analogous to the error sum of
squares. Hence, the four statistics
manova uses are:
where λi are the solutions of the characteristic equation |Qh – λQe| = 0.
where θi values are the solutions of the characteristic equation Qh – θ(Qh + Qe) = 0.
Roy’s maximum root statistic
 Charles, S. D. Statistical Methods for the Analysis of Repeated Measurements. Springer Texts in Statistics. Springer-Verlag, New York, Inc., 2002.