## MANOVA vs Univariate ANOVAs

A Multivariate Analysis of Variance (MANOVA) takes into account inter-correlations between a group of outcomes. For example we could use a MANOVA if we have two or more highly correlated scores which measure attention and wish to see if these differ en bloc across patient groups.

If these scores were independent OR do not measure the same construct then a series of multiple univariate tests may be more applicable with each score representing a different outcome measure and score means compared across groups using separate anovas.

MANOVA has three advantages over univariate analyses. Firstly, it does not make the strong assumption of sphericity amongst levels of the repeated measures variable. Secondly, it takes into account inter-correlations between sets of outcome variables which are highly correlated. Everitt and Hay (1992,pp 77-81) state that a MANOVA, unlike the less refined straight summation, is a weighted sum of subscales with non-unit weights based upon shared variance between the subscales which has maximum discriminability between the groups being compared. Thirdly it reduces the number of statistical tests by handling multiple outcome variables in the one analysis thus reducing type I error.

MANOVA does assume that variances and covariances in each of the groups are equal. This assumption can be tested by requesting a homogeneity of variance test labelled as Box's M test in the SPSS output.

Jeremy Miles also mentions

Cole et al. (1994) show that the more highly correlated your outcome variables, the higher (i.e.less significant) your multivariate p-values will be, which is kind of the opposite of what you might expect. You get the most power when your outcomes are correlated negatively - but many psychologists (who kind of blindly say "I've got more than one outcome, I'd better do a multivariate test") will have measured two highly correlated outcomes - like anxiety and depression, and then do a multivariate test, because there are two of them, but that guarantees (almost) that you won't get a significant result.

For multiple comparisons (e.g. between groups considered by the MANOVA) Huberty and Morris (1989) use a Bonferroni corrected p-value. Marzillier & Davey (2005), on the other hand also use a more conservative approach by adopting a 0.01 level of significance for all comparisons. Bird and Hadzi-Pavlovic (2014) give SPSS and SAS syntax applying Sidak corrections (two dependent variables) and a critical value based on Tukey HSD tests for more than three responses.

Understanding and interpreting a MANOVA using ''only'' between subjects differences

Understanding and interpreting a MANOVA in the SPSS Repeated Measures procedure

References

Bird KD and Hadzi-Pavlovic DH (2014) Controlling the maximum familywise type I error rate in analyses of multivariate experiments. *Psychological Methods* **19(2)** 265-280.

Cole DA, Maxwell SE, Arvey R, & Salas E (1994) How the power of MANOVA can both increase and decrease as a function of the intercorrelations among the dependent variables. *Psychological Bulletin* **115(3)** 465-474.

Everitt B and Hay D (1992) Talking about statistics: A psychologist's guide to design and analysis. Edward Arnold:London.

Field A (2005, 2013) Discovering statistics using SPSS. 2nd/4th Editions. Sage:London. Excellent primers with worked examples on both MANOVA and ANOVA. The fourth edition has a slightly different title of *Discovering statistics using IBM SPSS statistics.* (Both in CBSU library).

Field A, Miles J, & Field Z (2012) Discovering Statistics Using R (p. 957). London: SAGE Publications. Retrieved from http://books.google.com/books?id=jsoVnwEACAAJ&pgis=1 (Mentions fitting separate regression models between each of the scales A-C and the covariates to further explain a MANCOVA = MANOVA with a covariate)

Huberty CJ, Morris, JD (1989) Multivariate Analysis Versus Multiple Univariate Analyses. *Psychological Bulletin* **105(2)** 302-308.

Marzillier S & Davey G (2005) Anxiety and disgust: Evidence for a unidirectional relationship. *Cognition & Emotion* **19(5)**, 729-750.