- What is the difference between a repeated measures ANOVA over some factor (say experimental condition) and a MANOVA?
- In particular one website I stumbled across suggested that MANOVA does not make the same assumption of sphericity that repeated measures ANOVA does, is that true?
- If so, why would one not just always use MANOVA?
- I am trying to conduct a repeated measures ANOVA with multiple DVs, what is the appropriate approach?
Having several repeated-measures DVs one can apply a univariate approach (also called Repeated Measures sensu stricto or split-plot approach) or multivariate approach (or MANOVA). In univariate approach, RM levels are treated as deviations from one variable, their average level. In multivariate approach, RM levels are treated as covariates of each other. Univariate approach requires sphericity assumption while multivariate approach does not, and because of this it is becoming more popular indeed. However, it spends more df and thus needs larger sample size. Also, univariate approach retains its popularity because it generalizes to Mixed models. When sphericity assumption (and beyond expectation more general compound symmetry assumption too) holds results by both approaches are very similar, as far as I know.
Geometrically, MANOVA rejects iff the mean (difference) vector lays outside of an ellipsoid. Repeated measures ANOVA, say, with $d$ repeated measures per subject, rejects iff the $d$-dimensional mean (difference) vector lays outside of a sphere. The shape of the ellipsoid is determined by the covariance matrix. It can be very excessive or nearly spherical.
The consequence is that ANOVA and MANOVA "favour" different alternatives. So use MANOVA if you want to reject great Mahalanobis-lengths of the mean vector while use ANOVA if you want to reject great Euclidean lengths.
But if the covariance matrix is spherical, both criteria conincide, so that in this case the results of ANOVA and MANOVA also conincide (though only asymptotically) as ttnphns pointed out.
I prefer a repeated measures model. Not only is it easier to interpret the results, it is more flexible in that you can specify a covariance structure.
This reference may be of use as it works through an example: Mixed or MANOVA