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MANOVA

The wonderfully complex world of the multivariate analysis of means!

OK, I realize the thought of another dose of stats may have you feeling like this...

Or maybe even like this....

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However, it's not that bad...

Just because the word "multivariate" enters the equation does not mean life suddenly changes! As you'll see, MANOVA and RMANOVA are variations of ANOVA...which you already know!

Why the confusion/anxiety?

You read: A 4 (group) x 2 (time) repeated measures multivariate analysis of variance revealed a significant multivariate main effect for Time, Wilks' 8 = .406, F (4, 113) = 41.28, p < .001, 02 = .59, but no significant effect for Group, Wilks' 8 = .865, F (12, 229) = 1.41, p > .05.

What's it all mean?

Multivariate Analysis of Variance (MANOVA) assesses the statistical significance of the effect of one or more independent variables (IV's) on a set of two or more dependent variables (DV's)

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Why MANOVA?

Ability to examine more than 1 DV at once, or the simultaneous effects of the IV's on multiple DV's The major benefit of MANOVA over multiple ANOVAs = controlling Type I error rate

How's it do that?

Overall (Omnibus) MANOVA is run to determine if significant. If so, then univariate ANOVA's are conducted for each DV DV's must be correlated in MANOVA! MANOVA includes the correlation between DV's...therefore, the analysis removes/ accounts for any redundancy in DV's

Think "systems of variables"

MANOVA allows you to detect when groups differ on a system of variables (Note. This implies the variables have some meaningful relationship) Individually, groups may not differ on DV's, but systems of variables may have a significant combined effect

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Rules, just like in ANOVA

DV's must be continuous IV's are categorical Covariates may be included and must be continuous All assumptions of ANOVA hold with a few minor exceptions...

New Assumptions of MANOVA

Multivariate Normality: All IV's must be normally distributed, any linear combination of the DV's must be normally distributed, and all subsets of the variables must have a multivariate normal distribution However, MANOVA is robust to violations of assumptions (surprise!).

Assumptions con't

Homogeneity of Covariance Matrices Variance for all DV's must be equal across the experimental groups defined by the IV's Covariances (variance shared between 2 variables) for all unique pairs of DV's are equal for all experimental groups. Box's M is used to test this assumption

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Final Assumption

Independence of Observations: S's scores on the DV's are not influenced by other S's in his/her experimental group. For example, what if school-aged kids were asked to answer questions aloud in a group! This is the most critical assumption...not open to violations (however, repeated measures are a special exception)

Example

Compare 6 verbal subscales of WAIS-R between men and women No longer compares 2 means, instead two sets of six means are compared (vectors) Null = the vector of means for men is equal to the vector of means for women

With me so far?

Some new tests and new ideas kicked around but it all equates to something you already know:

ANOVA...yet again

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So...what needs to be reported?

Based on today's lecture and past experience, what will be important to report to describe MANOVA results? To the lab to test see how this works!

More on MANOVA

Additional tests and concerns using Marcoulides computer example

Additional SPSS "tricks"

In SPSS, go to "edit" --> "options" --> "viewer". If you click on "display commands in log", you will see the actual syntax associated with all commands sent to the SPSS engine Many times it is useful to have a copy of the syntax to understand your output.

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And more

Go to "edit" --> "options" --> "viewer", then click "display commands in log" (also can click on "notes" in output) You can request the syntax window to open on start-up...syntax is being written to a log named "spss.jnl" without you even knowing it! If you forget what you have done, you can always check this log

Variance Explained in MANOVA

Recall in ANOVA, eta-squared provides a measure of the variance in the DV accounted for by the IV(s) (also called effect size) The multivariate analog to this can be derived by Wilks' criterion: (02 = 1 - 7) For example, where 7 = .437, multivariate 02 = 1 - .437 or .55...55% of variance in DVs accounted for by type of instruction!

Variance Explained con't

Use the previous formula or examine the etasquare values provided in the "Test of Between Subjects Effects" table (and the model summary) to understand the variance explained in the model Compare the value of 55% to the Wilks' criterion to see that knowledge adds little to the model

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Reporting Values in MANOVA

Widely variable, but at minimum it is essential to include the test statistic used, the F test approximation, the univariate analyses of variance, post-hoc comparisons, and the measures of association Run "analyze" --> "descriptives" --> "explore" on knowledge and application

What about normality?

Recall that skewness is an index of symmetry of a distribution A normally distributed variable will have a skewness value of zero...positive skew means values are to left of distribution (application), negative skew = values to the right (knowledge)

Decision Time

Are the distributions normal? How can you tell in MANOVA? Yet another assumption: If the univariate distributions are normal, then bivariate distributions should also be normally distributed!

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So what about this data?

Work through the example and reach some conclusions It's all just another ANOVA, albeit a more complex one!

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