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Matakuliah Tahun Versi : I 0214 / Statistika Multivariat : 2005 : V 1 Matakuliah Tahun Versi : I 0214 / Statistika Multivariat : 2005 : V 1 / R 1 Pertemuan 13 Analisis Ragam Peubah Ganda (MANOVA I) 1

Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa dapat Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa dapat menerangkan konsep dasar analisis ragam peubah ganda (manova) C 2 • Mahasiswa dapat menghitung manova satu klasifikasi C 3 2

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

<<ISI>> MANOVA ~ the history • Developed as a theoretical construct by S. S. <> 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

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

<<ISI>> ANOVA vs MANOVA • ANOVA ~ only 1 dependent variable • MANOVA ~ <> 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

<<ISI>> Equations • ANOVA Y 1 = (metric DV) X 1 + X 2 <> Equations • ANOVA Y 1 = (metric DV) X 1 + X 2 + X 3 +. . . + Xn (non-metric IV’s) • MANOVA Y 1 + Y 2 +. . . + Yn = X 1 + X 2 + X 3 +. . . + Xn (metric DV’s) (non-metric IV’s)

<<ISI>> MANOVA and Regression • Note the different terminology • In multiple regression, univariate <> 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

<<ISI>> Univariate Research Example • Subjects shown different advertising messages • Emotional or Informational <> 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?

<<ISI>> Univariate Review ~ t Test Two commercials shown (emotional~informational) Single treatment/factor with two <> 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 M 1 - M 2 The t statistic = -----------SEM 1 M 2

<<ISI>> Univariate Review ~ ANOVA Two or more commercials (emotional~informational~funny~etc) Single treatment/factor with two <> 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 F statistic = MSB -----MSW

<<ISI>> Univariate Hypothesis Testing • Null Hypothesis (H 0) ~ That there is no <> Univariate Hypothesis Testing • Null Hypothesis (H 0) ~ 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

<<ISI>> Multivariate Research Example • Subjects shown different advertising messages • Emotional or Informational <> 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?

<<ISI>> Multivariate Procedures Hotelling’s T 2 One IV, multiple DV’s, two groups The k <> Multivariate Procedures Hotelling’s T 2 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

<<ISI>> Null Hypothesis Testing MANOVA • H 0: M 1 = M 2 =. <> Null Hypothesis Testing MANOVA • H 0: M 1 = M 2 =. . . Mk • H 0: All the group means are equal, that is, they come from the same populations H 0: M 11 M 12 =. . . = = M 21 M 22 M 1 k M 2 k Mp 1 Mpk Mp 2 All the group mean vectors are equal, that is, they come from the same populations

<<ISI>> Hotelling’s T 2 • Direct extension of the t test, used when there <> Hotelling’s T 2 • 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

<<ISI>> Hotelling’s T 2 ~ how it works • Maximize group differences, using the <> Hotelling’s T 2 ~ how it works • Maximize group differences, using the equation below: C = W 1 Y 1 + W 2 Y 2 +. . . + Wn. Yn 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 T 2 and check statistical significance

<<ISI>> MANOVA • Extension of Hotelling’s T 2 Establish dependent variable weights to produce <> MANOVA • Extension of Hotelling’s T 2 Establish dependent variable weights to produce a variate for each respondent Adjust weights to maximize F statistic computed on variate scores of all groups

<<ISI>> MANOVA ~ how it works • The first variate (called a discriminate function) <> 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

<<ISI>> When to use MANOVA • When you have multiple dependent variables • Control <> 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

<<ISI>> Discriminant Analysis • MANOVA ~ sort of a mirror image of discriminant analysis <> 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

<<ISI>> Decision Process for MANOVA • Powerful analytic tool suitable to a wide array <> 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

<<ISI>> Step #1: Objectives of MANOVA • Determine research question Multiple Univariate Questions ~ <> 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

<<ISI>> Step #1: continued Select Dependent Variables carefully There is a danger of including <> 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

<<ISI>> Stage #2: Research Design • MANOVA requires greater sample sizes than ANOVA ~ <> 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)

<<ISI>> ANOVA Cereal Example • • Three colors (red, blue, green) Three shapes (stars, <> 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

<<ISI>> Ordinal and Disordinal • With MANOVA, we can establish the nature of the <> 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

<<ISI>> MANOVA Interaction • If significant interactions are ordinal, researcher must interpret the interaction <> 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)

<<ISI>> Covariates • Metric independent variables, called covariates can be used to eliminate systemic <> 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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<< CLOSING>> • Sampai dengan saat ini Anda telah mempelajari kosep dasar analisis ragam << 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 33