MANOVA and ANCOVA

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# MANOVA and ANCOVA - PowerPoint PPT Presentation

MANOVA and ANCOVA. Martin Dempster. Review. Analysis of Variance (ANOVA) examines the difference between 2 or more groups in terms of their scores on a single dependent variable It does this by looking at the ratio of the differences between the groups against the differences within the groups

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### MANOVA and ANCOVA

Martin Dempster

Review
• Analysis of Variance (ANOVA) examines the difference between 2 or more groups in terms of their scores on a single dependent variable
• It does this by looking at the ratio of the differences between the groups against the differences within the groups
• This may be a very simplistic representation of reality

Martin Dempster

ANOVA

Drugs

CBT

Anxiety Assessment

No treatment

Martin Dempster

Objectives
• Introduce the MANOVA model – the ANOVA with additional dependent variable(s)
• Introduce the ANCOVA model - the ANOVA with covariate(s)
• Introduce the MANCOVA model – the ANOVA with additional dependent variable(s) and covariate(s)

Martin Dempster

### Introduction to Multivariate Analysis of Variance (MANOVA)

ANOVA vs MANOVA
• In all cases ANOVAs have only 1 dependent variable (they are univariate tests)
• When you have more than 1 related dependent variables you need to conduct a MANOVA
• MANOVA can be one-way, two-way, between-groups, repeated measures and mixed

Martin Dempster

Example
• A researcher wished to compare those who had registered as an organ donor with those who had not.
• He wanted to compare them on: attitudes to organ donation, feelings about organ donation, and previous exposure to issue.
• These 3 dependent variables are conceptually related

Martin Dempster

Appropriate Analysis
• We could take each of the dependent variables separately and conduct a one-way between-groups ANOVA (or independent t-test)
• This means conducting 3 tests (one for each DV)
• However, every time we conduct a test we take a risk of an incorrect conclusion

Martin Dempster

Solution
• Conduct 1 significance test which assesses the differences between the groups on all DVs
• This is a multivariate test
• Returning to our example…

Martin Dempster

Example
• A researcher wished to compare those who had registered as an organ donor with those who had not.
• He wanted to compare them on: attitudes to organ donation, feelings about organ donation, and previous exposure to issue.
• These 3 dependent variables are conceptually related

Martin Dempster

One-Way Between-Groups MANOVA

Pillai’s Trace = 0.033; F(3,373) = 4.255, p = .006

Interpretation
• The MANOVA result indicates that there is a significant difference between those on the organ donor register and those not on the register, in terms of their scores on at least one of the DVs
• Which one of the DVs? All of the DVs?
• Univariate tests

Martin Dempster

Interpretation
• There is a significant difference between those on the organ donor register and those not on the register, in terms of their scores on attitude towards organ donation and feelings towards organ donation.
• However, feelings towards organ donation has the strongest influence on registering as an organ donor.
• What is the nature of the influence?

Martin Dempster

Post Hoc Tests
• If there are more than 2 levels of the IV, then post hoc tests will be required to examine the nature of the findings
• Proceed from this point as for a univariate ANOVA

Martin Dempster

Assumptions of MANOVA
• Independent variable is categorical.
• Dependent variables should be measured at the interval / ratio level.
• There should be more cases in each cell than there are DVs
• Multivariate distribution is approximately normal.
• Linearity
• Distributions have approximately equal variances
• Homogeneity of intercorrelations.

### Analysis of Covariance (ANCOVA)

Covariates
• A covariate is a (continuous) variable that is not part of the main experimental manipulation but has an effect on the dependent variable
• Including covariates enables us to:

Explain more within-group variance, thereby increasing the power of our test

Remove the bias of a confounding variable

Martin Dempster

ANOVA

Drugs

CBT

Anxiety Assessment

No treatment

Martin Dempster

ANCOVA

Non-CBT

CBT

Anxiety Assessment

No treatment

Depression

Martin Dempster

ANOVA Result

Martin Dempster

ANCOVA Result

Martin Dempster

What Next?
• Post hoc tests or planned comparisons to pinpoint differences
• Graph can be useful

Martin Dempster

Pretest - Post-test Designs
• When random allocation to groups does not take place, it is possible that the groups are unequal
• These differences at the pretest period can confound the results at the post-test period
• Solution: treat the pretest scores as a covariate, thereby removing the effects of differences at baseline

Martin Dempster

Homogeneity of Regression Slopes
• Means that the relationship between the covariate and the dependent variable is approx the same for all groups
• In other words, there should be no interaction between the groups and the covariate

Martin Dempster

Checking Assumption

Martin Dempster

MANCOVA
• Combination of previous 2 analyses
• Allows us to examine differences on more than one DV, while controlling for covariate(s)
• Interpretation combines information from before – multivariate test result is the result after removal of the covariate
• No further assumptions

Martin Dempster

Summary
• ANOVA is unlikely to be useful for “real life” studies
• If covariates cannot be physically controlled, they should be measured and subsequently controlled statistically
• When several DVs are being measured, a MANOVA or MANCOVA procedure will be required

Martin Dempster