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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

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manova and ancova

MANOVA and ANCOVA

Martin Dempster

review
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
ANOVA

Drugs

CBT

Anxiety Assessment

No treatment

Martin Dempster

objectives
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

anova vs 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
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
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
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

example1
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
One-Way Between-Groups MANOVA

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

interpretation
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

interpretation1
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
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
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.
covariates
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

anova1
ANOVA

Drugs

CBT

Anxiety Assessment

No treatment

Martin Dempster

ancova
ANCOVA

Non-CBT

CBT

Anxiety Assessment

No treatment

Depression

Martin Dempster

anova result
ANOVA Result

Martin Dempster

ancova result
ANCOVA Result

Martin Dempster

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

Martin Dempster

pretest post test designs
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
Homogeneity of Regression Slopes
  • Additional assumption of ANCOVA
  • 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
Checking Assumption

Martin Dempster

mancova
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
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