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Introduction to Multivariate AnalysisMPU2263
Zairul Nor Deana Md Desa, Ph.D.
Department of Educational Foundations and Social SciencesUniversiti Teknologi Malaysia
Email: [email protected]
Lecture 3: Multivariate ANOVA (MANOVA)
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Outline
Multivariate Analysis of Variance (MANOVA)
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Purpose of Multivariate ANOVA
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It is a generalization of ANOVA with several DVs.
IV can be different treatments or groups.
To discover whether behavior (i.e. the DVs) is
changed by manipulation of the IVs.
Sometimes the treatment (or groups) shown nodifference in separate ANOVA but when responses
to more than one DV group differences become
apparent.
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Research Questions in MANOVA
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What is it the changes in composite DV as a result of different treatments (or groups) and their interaction.
Is a combination of DVs varies as a function of IVs?
Is there any difference between the means and,
thus, is there any statistically difference among
groups?
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Research Questions in MANOVA
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Overall Model What is the proportion of the composite DV explained by each IV?
Main Effect
Separate test for each IV
Are there mean differences in composite DV associated with
differences in treatment (or group)? - without covariate Are there differences in composite DV associated with IV, after
adjustment for covariate prior to treatment (or condition)? - withcovariate (for next week)
Interaction Effect
Does change in the DV over levels of one IV depend on the levelof another IV?
Is the pattern of responses to the different types (or levels) of oneIV the same for different levels of another IV?
Given different types of treatment, does one treatment works
better than the other?
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Theoretical Issues
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How to choose IVs and/or DVs? Logic and research design!
Skills and area of study
A set of continuous DVs
Start from looking at more important DVs than others
Categorical IVs
How to choose covariates (CVs)? CVs are need to be controlled for or removed.
Identify a set of CVs that are uncorrelated with each other
Select CVs that adjust the DV for predictable butunwanted sources of variability.
Based on theoretical ground and literature.
Can look at CV with highest correlation with DV.
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Fundamental Interpretation in MANOVA
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See Example
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Example
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A study on proficiency assessment among Englishlanguage leaners.
See SPSS output
Data file: data3MANOVA.sav
Output file: outData3.spv
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Example
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How-to-SPSS Please refer to SPSS hand-out!
Analyze>Generalized Linear Model>Multivariate
Select the DVs (e.g. listen, read, speak, write)
Select the IVs (e.g. Nation)
In Options button:
Choose all IVs to Display Means for
Check Descriptive Statistics, Estimate of Effect Size, Homogeneity
test
In Post Hoc button: Choose pairwise comparison adjustment method (optional for this
example)
Use LSD for <4 pairwise comparisons, and use sequential
Bonferroni for more than 4 comparisons. Note.
# = −
2
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Example – One-way MANOVA
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Steps in interpreting MANOVA results in SPSS1. Examine the Box’s M Test (i.e. test for the assumption of
homogeneity variance-covariance of DVs across IVs)
We want non significant result (Why?) at α = .001 (why
conservative significance level?) – followed up with Wilks’
Lambda.
If the test is non-significant the follow-up analysis must use Pillai’s
Trace in order to use MANOVA!
2. Examine multivariate model (Omnibus Multivariate Test)
For each IV, examine its multivariate test at .001 level. Significant result Multivariate test is more powerful than
univariate in examine the DV as a composite (i.e. combination of
DVs)
It provide information about the proportion of generalized variance
in the DVs that is accounted for by the IVs.
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Example - One-way MANOVA
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Steps in interpreting MANOVA results in SPSS3. Levene’s Test
Test for the equality of error variances in each DV
Want significant results reject null hypothesis that variancesacross groups are unequal.
If significant for some DVs proceed with nonparametric or univariate ANOVA. We can also use MANOVA and assuming for unequal variances across groups!
4. Examine Test of Between Subjects Effects
We want to look at an individual effect of IV (one-way ANOVA)
We want to look at main and interaction effects (two- or three-way ANOVA).
We want significant results to support researcher’s hypotheses!
Note. Bonferroni adjustment of the significant level must be used!
=.5
#
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Example - One-way MANOVA
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Steps in interpreting MANOVA results in SPSS5. Follow-up Tests with LMATRIX (Pairwise comparisons)
LSD or Bonferroni (in Post Hoc button) or contrast in Contrast
button in default SPSS won’t provide F value and degree of
freedom for the pairwise comparisons.
LMATRIX will use contrast for the comparison which will testeach individual level of IV appropriately
Use the same procedure in Slide 9 then click Paste to use
Syntax for LMATRIX procedure.
Completed follow-up tests can be found in Contrast Results (K
Matrix)a andUnivariate Test Results tables.
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Example - Two-way MANOVA
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Now, consider two IVs in the model fromprevious example. Data file: data3MANOVA.sav
Output file: outData3-2.spv
We need to examine significant interaction, main
effect, and simple main effect.
Use sequential Bonferroni adjustment for follow-up
test with more than 4 pairs of comparison.
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Next Week
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Multivariate ANCOVA (MANCOVA)
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