1. MANOVA (and DISCRIMINANT ANALYSIS) Alan Garnham, Spring 2005
STATISTICAL ANALYSIS
MANOVA
(and DISCRIMINANT ANALYSIS)
Alan Garnham, Spring 2005

2. What is MANOVA?
Like ANOVA, applied to regimented experimental designs.
But in cases where there is more than one DEPENDENT variable
Example: text comprehension experiment with three dependent variables
clause reading time
question answering time
question answering accuracy
usually analysed in separate ANOVAs, but could do MANOVA).

3. Carreiras et al. 1996 Stereotyping Experiment
The electrician examined the light fitting.
He needed a special attachment to fix it. OR
She needed a special attachment to fix it.
Was the electrician mending a stereo?
Design: 2 (male/female stereotype) x 2 (pronoun matches or mismatches stereotype)

4. Carreiras et al. 1996 Stereotyping Experiment
In the paper we actually analysed the data using multiple univariate ANOVAs
We could have used MANOVA
This tells you something about typical practice in the field of psycholinguistics

5. MANOVA - further examples
Questionnaire data with subtest scores (the DVs) and respondents classified as e.g. male vs female, old vs young etc.
Any other type of study with multiple tests (e.g. reading, writing, maths) and participants of different kinds (boys / girls; 6 year olds / 8 year olds etc.)

6. What is MANOVA?
Like ANOVA, MANOVA is a special case of the General Linear Model.
y = Xb + e
Where y is a vector of criterion variables (DVs), X is a matrix of predictors (IVs, reflecting the study’s design), b is a vector of regression coefficients (weightings), and e is a vector of error terms.
So, in SPSS: Analyse, GLM, Multivariate

7. What is MANOVA?
Looks to see if there are differences between groups on a linear combination of standardised DVs
Which is effectively a single new DV
This new DV is the linear combination of DVs which maximises group differences
Different combinations of DVs are selected for each main effect or interaction in the design

8. Statistical Reasons for MANOVA
Fragmented univariate ANOVAs lead to type 1 errors
seeing effects that aren’t really there.
Because MANOVA effectively uses a single DV it protects against type 1 errors arising by chance from performing multiple tests
Univariate ANOVAs throw away info - correlation among dependent variables.

9. Statistical Reasons for MANOVA
Can get differences on a "combined" MANOVA measure, when none of the differences on the individual ANOVA measures are significant (so avoiding type 2 errors)
in particular if treatments have different effects on the dependent variables, but the dependent variables are strongly correlated within any particular treatments (giving a small multivariate error term).
(Extension of above) can avoid cancelling out effects
However, in practice this advantage is rarely realised

10. MANOVA - Disadvantages
More complex
Additional assumptions
Outcome can be ambiguous
Usually lower power than ANOVA   

11. Null hypothesis in MANOVA
Groups (experimental conditions) have the same mean for all the dependent (criterion) variables

12. MANOVA - Restriction
Cannot have too many DVs (fewer than cases)

13. MANOVA: When and How
May not be a good idea to put all dependent variables in one MANOVA. Better to put those that there is a good rationale for including in the main MANOVA and perhaps doing another on speculative variables.
Reason: if there are no effects on the speculative dependent variables, they will just add noise to the analysis.

14. Assumptions of MANOVA
Independence of observations (as in univariate ANOVA)
Multivariate normality - all dependent variables and linear combinations of them are distributed normally
Equality of covariance matrices (cf homogeneity of variance in univariate). (Box's test to check, but set alpha to .001).

15. Assumptions of MANOVA
Second and third assumptions are more stringent than corresponding univariate assumptions in univariate ANOVA.

16. MANOVA Stats
Generalisation of Student's t (replaces scalars by vectors/matrices) leads to Hotelling's T2 - only for 2 group case, though.
For the multigroup case, no single agreed statistic. Best known is Wilk's lambda.

17. MANOVA Stats
Significance means: there is a linear combination of the dependent variables (the discriminant function) that distinguishes the groups.
Need post hoc tests to find out which dependent variables make significant contributions to discriminant function.
For the multigroup case it is possible to use Hotelling's T2 tests for post hoc pairwise multivariate analyses.
Hotelling's T2 can be followed up in this and the simple 2 group multivariate case by univariate t's.

18. MANOVA STATISTICS Pillai-Bartlett Trace Hotelling's Trace
Wilk's Lambda
Roy's Greatest Root
ALL 4 are reported by SPSS

19. MANOVA STATISTICS Each will have an F value associated with it
These Fs are typically different (for the different tests) in the case of a "within" factor and any interaction including a within factor.

20. MANOVA AND REPEATED MEASURES
Repeated measures on a single individual, usually treated as a “within” factor in a univariate ANOVA can be thought of as measures on multiple dependent variables.
So, repeated measures designs can be alternatively analysed using MANOVA.
Recent versions of SPSS report MANOVA statistics for repeated measures designs.

21. MANOVA AND REPEATED MEASURES
Advantage: Avoids assumptions about equality of covariances required in repeated measures ANOVA.
Violation of this assumption may be particularly problematic for specific comparisons.
Problem: MANOVA may have less power.

22. Discriminant Analysis
As we have seen, MANOVA produces discriminant functions
Linear combinations of DVs that best separate the levels of an IV (or an interaction of IVs)
Discriminant Analysis can be regarded as the inverse of (one-way) MANOVA

23. Discriminant Analysis and MANOVA
In discriminant analysis we ask if group membership can be predicted by a set of variables
E.g. Can party voted for at General Election be predicted from age, income, social class etc.

24. Discriminant Analysis and MANOVA
So, the IVs in MANOVA (specifically the levels of the single factor in one-way MANOVA) become the groups to which an individual might belong (Labour voter, Conservative voter etc.)
And the DVs in the MANOVA become the predictors
Whether one thinks of a study as requiring MANOVA or discriminant analysis depends on extra-statistical considerations.

25. Discriminant Analysis and MANOVA
The mathematics is equivalent, just as ANOVA and multiple regression are equivalent, and all of them (ANOVA, MANOVA, MR, Discriminant Analysis) are special cases of the GLM.

26. Discriminant Analysis and Logistic Regression
Logistic Regression can also be used to predict group membership from a set of other variables.
It has a different set of assumptions from Discriminant Analysis and is preferred by many authorities.
In particular it unproblematically allows binary (in particular, and discontinuous, in general) predictors (as well as continuous ones).

27. MANOVA - Summary
An apparently attractive extension of ANOVA to the case of multiple dependent variables - included in a single analysis
It has more complex assumptions and less is known about robustness in relation to violations of assumptions
In practice, its advantages are rarely realised

28. Discriminant Analysis -Summary
MANOVA produces discriminant functions
Looked at in a different way, one can ask whether the “DVs” in a MANOVA can predict “group membership” of the levels of the IV in the MANOVA
Logistic Regression, an alternative approach to such prediction, has advantages over discriminant analysis