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Pertemuan 13

Matakuliah : I0214 / Statistika MultivariatTahun : 2005Versi : V1 / R1

Analisis Ragam Peubah Ganda(MANOVA I)

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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

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Outline Materi

• Konsep dasar analisis ragam peubah ganda (manova)

• Analisis ragam peubah ganda satu klasifikasi

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

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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)

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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

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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)

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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

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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?

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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

1 - 2

The t statistic = ----------------------SE1 2

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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

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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

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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?

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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

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Null Hypothesis Testing

ANOVA

• H0: 1 = 2 =...k

• H0: All the group means are equal, that is, they come from the same populations

MANOVA

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21

p1

12

22

p2

1k

2k

pk

= =...=

H0: All the group mean vectors are equal, that is, they come from the same populations

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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)

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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

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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

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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)

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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 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