Scheffés mixed model for multivariate repeated measures:a relative efficiency evaluation

Scheffé’s mixed model, generalized for application to multivariate repeated measures, is known as the multivariate mixed model (MMM). The primary advantages the MMM are (1) the minimum sample size required to conduct an analysis is smaller than for competing procedures and (2) for certain covariance structures, the MMM analysis is more powerful than its competitors. The primary disadvantage is that the MMM makes a very restrictive covariance assumption; namely multivariate sphericity. This paper shows, first, that even minor departures from multivariate sphericity inflate the size of MMM based tests. Accordingly, MMM analyses, as computed in release 4.0 of SPSS MANOVA (SPSS Inc., 1990), can not be recommended unless it is known that multivariate sphericity is satisfied. Second, it is shown that a new Box-type (Box, 1954) Δ-corrected MMM test adequately controls test size unless departure from multivariate sphericity is severe or the covariance matrix departs substantially from a multiplicative-Kronecker structure. Third, power functions of adjusted MMM tests for selected covariance and noncentrality structures are compared to those of doubly multivariate methods that do not require multivariate sphericity. Based on relative efficiency evaluations, the adjusted MMM analyses described in this paper can be recommended only when sample sizes are very small or there is reason to believe that multivariate sphericity is nearly satisfied. Neither the e-adjusted analysis suggested in the SPSS MANOVA output (release 4.0) nor the adjusted analysis suggested by Boik (1988) can be recommended at all.