Pertemuan 13

1. Pertemuan 13 Analisis Ragam Peubah Ganda(MANOVA I) Matakuliah : I0214 / Statistika Multivariat Tahun : 2005 Versi : V1 / R1

2. Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa dapat menerangkan konsep dasar analisis ragam peubah ganda (manova)  C2 • Mahasiswa dapat menghitung manova satu klasifikasi  C3

3. Outline Materi • Konsep dasar analisis ragam peubah ganda (manova) • Analisis ragam peubah ganda satu klasifikasi

4. <<ISI>> MANOVA ~ the history • Developed as a theoretical construct by S.S. Wilks in 1932 • Published in Biometrika • Wide availability of computers made these methods practical for researchers

5. <<ISI>> MANOVA ~ the definition • Technique for assessing group differences across multiple metric dependent variables (DV’s) simultaneously, based on a set of categorical (non-metric) variables acting as independent variables (IV’s)

6. <<ISI>> ANOVA vs MANOVA • ANOVA ~ only 1 dependent variable • MANOVA ~ 2 or more dependent variables • Both are used with experimental designs in which researchers manipulate or control one or more independent variables to determine the effect on one (ANOVA) or more (MANOVA) dependent variables

7. <<ISI>> Equations • ANOVA Y1 = X1 + X2 + X3 +...+ Xn (metric DV) (non-metric IV’s) • MANOVA Y1 + Y2 + ... + Yn = X1 + X2 + X3 +...+ Xn (metric DV’s) (non-metric IV’s)

8. <<ISI>> MANOVA and Regression • Note the different terminology • In multiple regression, univariate and multivariate/multiple refer to the number of IV’s • In ANOVA and MANOVA discussions, univariate and multivariate refer to the number of DV’s

9. <<ISI>> Univariate Research Example • Subjects shown different advertising messages • Emotional or Informational or ?? • Viewers rate appeal of the message using scores from 1 to 10 Ad appeal?

10. <<ISI>> Univariate Review ~ t Test • Two commercials shown (emotional~informational) • Single treatment/factor with two levels • Use a t Test: One IV’s, one DV, two treatment groups M1 - M2 • The t statistic = ---------------------- SEM1 M2

11. <<ISI>> Univariate Review ~ ANOVA • Two or more commercials (emotional~informational~funny~etc) • Single treatment/factor with two or more levels • Use ANOVA : Multiple IV’s, one DV, two or more treatment groups MSB • F statistic = ------ MSW

12. <<ISI>> Univariate Hypothesis Testing • Null Hypothesis (H0) ~ That there is no difference between the DV means of the treatment groups • Alternate Hypothesis (HA) ~ That there is a statistically significant difference between the DV means of the treatment groups

13. <<ISI>> Multivariate Research Example • Subjects shown different advertising messages • Emotional or Informational or ?? • Viewers rate appeal of the message using scores from 1 to 10 Ad appeal? Will I buy?

14. <<ISI>> Multivariate Procedures • Hotelling’s T2 One IV, multiple DV’s, two groups • The k Group Case: MANOVA Multiple IV’s, multiple DV’s, more than two treatment groups • Null Hypothesis ~ that there is no difference between vectors of means of multiple DV’s across the treatment groups

15. ANOVA H0: M1 = M2 =...Mk H0: All the group means are equal, that is, they come from the same populations MANOVA <<ISI>> Null Hypothesis Testing M11 M21 Mp1 M12 M22 Mp2 M1k M2k Mpk =...= = • H0: All the group mean vectors are equal, that is, they come from the same populations

16. <<ISI>> Hotelling’s T2 • Direct extension of the t test, used when there are only two groups, but multiple DV’s to be measured • Accounts for the fact that DV’s may be related to one another (correlated) • Provides a statistical test of the variate, formed from the DV’s, that produces the greatest group difference

17. <<ISI>> Hotelling’s T2 ~ how it works • Maximize group differences, using the equation below: C = W1Y1 + W2Y2 +...+ WnYn where C = composite or variate score for a respondent Wi = weight for dependent variable i Yi = dependent variable I • Square the obtained t statistic to get T2 and check statistical significance

18. <<ISI>> MANOVA • Extension of ANOVA • Extension of Hotelling’s T2 • Establish dependent variable weights to produce a variate for each respondent • Adjust weights to maximize F statistic computed on variate scores of all groups

19. <<ISI>> MANOVA ~ how it works • The first variate (called a discriminate function) maximizes differences between groups and therefore also the F value • With maximum F, calculate greatest characteristic root (grc) and check its significance to reject null hypothesis (or not) • Subsequent discriminant functions are orthogonal and seek to explain remaining variance

20. <<ISI>> When to use MANOVA • When you have multiple dependent variables • Control of Experimentwide Error Rate • Repeated univariate procedures can dramatically increase Type I errors • DV’s that are not highly correlated with one another will cause the most trouble • Differences among a Combination of Dependent Variables • Multiple univariate procedures do not equal a multivariate procedure • Multicollinearity is ignored

21. <<ISI>> Discriminant Analysis • MANOVA ~ sort of a mirror image of discriminant analysis • DV’s in MANOVA become IV’s of DA • DV of DA becomes IV of MANOVA

22. <<ISI>> Decision Process for MANOVA • Powerful analytic tool suitable to a wide array of research questions • Six step process • Logical progression through all six will yield best results

23. <<ISI>> Step #1: Objectives of MANOVA • Determine research question • Multiple Univariate Questions ~ MANOVA used to control experimentwide error rate before further univariate analysis • Structured Multivariate Questions ~ MANOVA used to address multiple DV’s with known relationships • Intrinsically Multivariate Questions ~ MANOVA used with multiple DV’s where the principal concern is how they differ/change as a whole...or how they remain consistent across time

24. <<ISI>> Step #1: continued • Select Dependent Variables carefully • There is a danger of including too many DV’s and a tendency to do so...simply because you can • One bad variable can skew all results • Ordering of variables can also be important and can lead to sequential effects • MANOVA step-down analysis can help here • Researcher responsibility to use tools properly

25. <<ISI>> Stage #2: Research Design • MANOVA requires greater sample sizes than ANOVA ~ overall and by group (must exceed specific thresholds in each cell) • Factorial Designs ~ two or more IV’s or treatments in the design • Sometimes treatments are added post hoc • Blocking factors (example: gender)

26. <<ISI>> ANOVA Cereal Example • Three colors (red, blue, green) • Three shapes (stars, cubes, balls) • 3 x 3 factorial design • With ANOVA, you would evaluate main effect for color, main effect for shape, and interaction effect of color and shape • Each would be tested with an F statistic

27. <<ISI>> Ordinal and Disordinal • With MANOVA, we can establish the nature of the interaction between two treatments • No interaction • Ordinal Interaction ~ effects of treatment are not equal across all levels of another treatment...but magnitude is in the same direction • Disordinal Interaction ~ Effects of one treatment are positive for some levels and negative for other levels of the other treatment

28. <<ISI>> MANOVA Interaction • If significant interactions are ordinal, researcher must interpret the interaction term carefully • If significant interaction is disordinal however, main effects of the treatments cannot be interpreted and study must be redesigned (treatments do not represent a consistent effect)

29. <<ISI>> Covariates • Metric independent variables, called covariates can be used to eliminate systemic errors • ANOVA becomes ANCOVA • MANOVA becomes MANCOVA • Procedures similar to linear regression are used to remove variation in the DV associated with covariates and then standard ANOVA and MANOVA can be used • Ideal covariate is highly correlated with DV and not correlated with the IV

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33. << CLOSING>> • Sampai dengan saat ini Anda telah mempelajari kosep dasar analisis ragam peubah ganda, dan manova satu klasifikasi • Untuk dapat lebih memahami konsep dasar analisis ragam peubah ganda dan manova satu klasifikasi tersebut, cobalah Anda pelajari materi penunjang, website/internet dan mengerjakan latihan