STATISTICAL ANALYSIS

MANOVA

(and DISCRIMINANT ANALYSIS)

Alan Garnham, Spring 2005

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).

Carreiras et al. 1996Stereotyping 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)

Carreiras et al. 1996Stereotyping 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

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.)

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

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

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.

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

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

Null hypothesis in MANOVA Groups (experimental conditions) have

the same mean for all the dependent (criterion) variables

MANOVA - Restriction Cannot have too many DVs (fewer than

cases)

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.

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).

Assumptions of MANOVA

Second and third assumptions are more stringent than corresponding univariate assumptions in univariate ANOVA.

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.

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.

MANOVA STATISTICS Pillai-Bartlett Trace Hotelling's Trace Wilk's Lambda Roy's Greatest Root

ALL 4 are reported by SPSS

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.

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.

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.

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

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.

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.

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.

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).

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

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