STATISTICAL ANALYSIS

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STATISTICAL ANALYSIS

• MANOVA
• (and DISCRIMINANT ANALYSIS)
• Alan Garnham, Spring 2005

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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).

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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)

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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

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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.)

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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
  studys design), b is a vector of regression
  coefficients (weightings), and e is a vector of
  error terms.
• So, in SPSS Analyse, GLM, Multivariate

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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

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Statistical Reasons for MANOVA

• Fragmented univariate ANOVAs lead to type 1
  errors
• seeing effects that arent 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.

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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

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MANOVA - Disadvantages

• More complex
• Additional assumptions
• Outcome can be ambiguous
• Usually lower power than ANOVA   

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Null hypothesis in MANOVA

• Groups (experimental conditions) have the same
  mean for all the dependent (criterion) variables

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MANOVA - Restriction

• Cannot have too many DVs (fewer than cases)

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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.

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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).

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Assumptions of MANOVA

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

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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.

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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.

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MANOVA STATISTICS

• Pillai-Bartlett Trace
• Hotelling's Trace
• Wilk's Lambda
• Roy's Greatest Root
• ALL 4 are reported by SPSS

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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).

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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

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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