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MANOVA and ANCOVA

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

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  1. MANOVA and ANCOVA Martin Dempster

  2. 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

  3. ANOVA Drugs CBT Anxiety Assessment No treatment Martin Dempster

  4. 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

  5. Introduction to Multivariate Analysis of Variance (MANOVA)

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. One-Way Between-Groups MANOVA Pillai’s Trace = 0.033; F(3,373) = 4.255, p = .006

  12. 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

  13. Univariate Tests

  14. 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

  15. Plot

  16. 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

  17. 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.

  18. Analysis of Covariance (ANCOVA)

  19. 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

  20. ANOVA Drugs CBT Anxiety Assessment No treatment Martin Dempster

  21. ANCOVA Non-CBT CBT Anxiety Assessment No treatment Depression Martin Dempster

  22. ANOVA Result Martin Dempster

  23. ANCOVA Result Martin Dempster

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

  25. 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

  26. 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

  27. Checking Assumption Martin Dempster

  28. 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

  29. 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

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