# MANOVA and correlations between dependent variables: how strong is too strong?

The dependent variables in a MANOVA should not be "too strongly correlated". But how strong a correlation is too strong? It would be interesting to get people's opinions on this issue. For instance, would you proceed with MANOVA in the following situations?

• Y1 and Y2 are correlated with $r=0.3$ and $p<0.005$

• Y1 and Y2 are correlated with $r=0.7$ and $p=0.049$

Update

Some representative quotes in response to @onestop:

• "MANOVA works well in situations where there are moderate correlations between DVs" (course notes from San Francisco State Uni)

• "The dependent variables are correlated which is appropriate for Manova" (United States EPA Stats Primer)

• "The dependent variables should be related conceptually, and they should be correlated with one another at a low to moderate level." (Course notes from Northern Arizona University)

• "DVs correlated from about .3 to about .7 are eligible" (Maxwell 2001, Journal of Consumer Psychology)

n.b. I'm not referring to the assumption that the intercorrelation between Y1 and Y2 should be the same across all levels of independent variables, simply to this apparent grey area about the actual magnitude of the intercorrelation.

• Who says they shouldn't be "too strongly correlated", i.e. what's the source of that quote? – onestop Jan 3 '11 at 17:38
• Taking a wild guess: If zero correlation, you may as well conduct separate anovas and thus simplify your task. If very high correlation, you may as well conduct anova on just one of the Y variables since results will be largely the same for all the others. – rolando2 Feb 26 '11 at 3:53
• Just a note: the reason I haven't accepted an answer is that, as Prof Lee says, there doesn't seem to be a clear one. So everyone's contribution is useful. – Freya Harrison May 31 '11 at 9:42
• I agree with @rolando2 (and others) that in case of a very high correlation MANOVA does not add much to an ANOVA on one of the variables (or e.g. on their average), but important issue not covered in any of the existing answers is: why would MANOVA be in any way worse in this situation? – amoeba says Reinstate Monica Oct 27 '14 at 12:00

There's no clear answer. The idea is that if you have a correlation that approaches 1 then you essentially have one variable and not multiple variables. So you could test against the hypotheses that r=1.00. With that said, the idea of MANOVA is to give you something more than a series of ANOVA tests. It helps you find a relationship with one test because you're able to lower your mean square error when combining dependent variables. It just won't help if you have highly correlated dependent variables.

I would recommend to conduct a MANOVA whenever you are comparing groups on multiple DVs that have been measured on each observation. The data are multivariate, and a MV procedure should be used to model that known data situation. I do not believe in deciding whether to use it on the basis of that correlation. So I would use MANOVA for either of those situations. I would recommend reading the relevant portions of the following conference paper by Bruce Thompson (ERIC ID ED429110).

p.s. I believe the 'conceptually related' quote comes from the Stevens book.

Why not use Cohen's (1988, 1992) guidelines for effect size values? He defines a "small" $(0.1 \leq r \leq 0.23)$, "medium" $(0.24 \leq r \leq 0.36)$ and "large" $(r \geq 0.37)$ effect. This would suggest to use MANOVA with variables whose $r$ is below $0.37$.

### References

Cohen, J. (1988) Statistical Power Analysis for the Behavioral Sciences. 2nd Ed. Routledge Academic, 567 pp.

Cohen, J (1992). A power primer. Psychological Bulletin 112, 155–159.

Claims about what correlations should or shouldn't be used in MANOVA are basically "myths" (see Frane, 2015, "Power and Type I error control fro univariate comparisons in multivariate two-group designs"). But of course, if your DVs are almost perfectly correlated (i.e., near 1 or -1), you should ask yourself why you're treating them as different variables in the first place.