- 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?
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asked Jul 18, 2011 at 15:20
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2$\begingroup$ The multivariate approach to repeated measures does not treat each factor level as a separate DV. Instead, it treats all unique differences between factor levels as separate DVs and then tests the hypothesis that the theoretical centroid of these DVs is the 0-vector. If there are $p$ levels, there are p over 2 differences, and $p-1$ unique differences (involving $p-1$ different factor levels). $\endgroup$– caracalCommented Jul 18, 2011 at 20:43
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$\begingroup$ I've edited the question to remove the offending phrase, but I'm not sure I understand your comment entirely, and it seems like it might be a relevant point to make clear as an answer to the first bullet point question. $\endgroup$– russellpierceCommented Jul 19, 2011 at 15:03
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3$\begingroup$ Chapter 13 of Maxwell & Delaney (2004) "Designing Experiments and Analyzing Data" provides an in-depth treatment of exactly the answers you are looking for in your first two bullet points. $\endgroup$– caracalCommented Jul 19, 2011 at 17:13
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1$\begingroup$ A very clear and concise discussion is given in A Bluffer’s Guide to ... Sphericity by Andy Field. See also An introduction to sphericity by Thom Baguley. $\endgroup$– amoebaCommented Feb 26, 2014 at 14:45
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3 Answers
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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.
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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.
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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
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1$\begingroup$ I suppose by "repeated measures model" you mean a mixed model (as in the link you provided). It is really important to be specific here: You do NOT seem to prefer repeated measures ANOVA (as in the question), you prefer mixed models for repeated measures. And as pointed out in the blog post, mixed models really are preferable in most cases. $\endgroup$ Commented Jul 19, 2011 at 17:05
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1$\begingroup$ The link to the reference has changed; it can now be found here. On a different note, I think it's fair to think of RM ANOVA as a special case of linear mixed models. $\endgroup$ Commented Jun 26, 2012 at 16:54
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$\begingroup$ Yes a repeated measures model is a mixed model. One can see the chapter in SAS for Mixed Models. $\endgroup$– GlenCommented Jul 6, 2012 at 1:52
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2$\begingroup$ A repeated measures model is a special case of mixed model. But, I think it is very important to emphasize that they are not the same. PROC MIXED in SAS can implement models that are noticeably different from repeated measures ANOVA. SAS tends to gloss over these differences in their output leading users to interpret mixed models no differently than they would repeated measures ANOVA. I'm just chiming in here to say that caution is warranted and users of PROC MIXED should be careful to be sure they know precisely what they are doing. $\endgroup$ Commented May 17, 2014 at 12:33