Multivariate Analaysis of Variance (MANOVA): Sharma, Chapter 11 - Bijan Yavar

Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)17th December 2013

Multivariate Analysis of Variance (MANOVA)Lecturer: Prof.Dr. Izani Ibrahim (CRM)Presenter: Bijan Yavar (CRM)Level of Study: PHDField of Study: Management (Crisis Management)Course: Advanced Quantitative TechniquesSID: ZP01774Researcher ID: A-3544-2010


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17th December 2013 Bijan Yavar

The National University of Malaysia (UKM) Graduate School of Business (GSB)

Contents

1. Introduction (What is MANOVA?)

3. Geometry of MANOVA

4. Analytic Computations for Two-Group MANOVA

5. Two-Group MANOVA

6. Multiple-Group MANOVA

7. MANOVA for Two Independent Variables of Factors

2. Differences Between MANOVA and ANOVA


1. Introduction



Introduction (What is MANOVA?)

� Technique for assessing group differences across multiplemetric dependent variables (DVs) simultaneously, based on a set of categorical independent variables (IVs)

� Procedure used to test the significance of the effects of one or more IVs (categorical) on two or more DVs (continuous)

� The Objective of MANOVA is Very similar to some of the objectives of Discriminant Analysis . Remember, in Discriminant Analysis one of the objective was to Determine if the Groups are Significantly Different with respect to a Given set of variables


2. Differences Between MANOVA and ANOVA



The Difference Between MANOVA & ANOVA



The Difference Between MANOVA & ANOVA

� MANOVA is a Multivariate extension of ANOVA with the only Difference being that in MANOVA there are Multiple Dependent Variables

� ANOVA - 1 DV� MANOVA - 2 or more DVs

� Both are used with experimental designs in which researchers manipulate or control one or more IVs to determine the effect on one DV (ANOVA) or more DVs (MANOVA)


1. Introduction




� In MANOVA the Independent Variables (IV’s) are Categorical

� In MANOVA the Dependent Variables (DV’s) are Continues

� MANOVA is an extension of ANOVA in which main effects and interactions are assessed on a combination of DVs

� MANOVA tests whether mean differences among groups on a combination of DVs is likely to occur by chance .




� A new DV is created that is a linear combination of the individual DVs that maximizes the difference between groups.

� In Factorial designs a different linear combination of the DVs is created for each main effect and interaction that maximizes the group difference separately.

� Also when the IVs have more than one level , the DVs can be recombined to maximize paired comparisons



Scenarios

� Scenario Number 1

� A marketing manager is interested in determining if geographical region (NEWS - 1 categorical IVs at 4 levels), has an effect on consumers ’ Taste preferences, Purchase Intentions, and Attitude towardthe product (metric – Likert scale and has more than 2 DVs)



Scenarios

� Scenario Number 2

� A political analyst is interested in determining if party affiliation (Democratic, Republican, Independent and gender (male & female) - 2 categorical IV’s - have any effect on voters’ views on a number of issues such as a abortion, taxes, economy deficits (multiple metric DVs)


3. Geometry of MANOVA



Cases – case 1: 1 IV (at 2 Levels) & 1 DV

� The Centroid (Or the mean) of each group (i.e Y1 and Y2) can be presented as a point in the One-Dimensional Space .

� If IV has an Effect on the DV, then the Means of the two groups are Different and the effect of IV is measured by the Difference Between the Two Means .

Figure 11.1

Centroid For Group 1

Centroid For Group 2

Y2Y1



� MANOVA is concerned with determining whether MD between group centroids is significantlygreater than zero.

� In ANOVA case , because there are only 2 groups and 1 DV, the problem reduces to comparing the means of two group using t-test.

To Explain More



Case II: 1 IV (at 2 levels) and 2 or more DVs

� Since the IV is at 2 levels, there are 2 groups � 2 DVs - Y and Z and for group 1 and for group

2, be the centroid for the 2 groups � Fig. 11.2 - shows the centroid of each group . It can be

presented a point or a vector in 2-dimensional space defined by DV (Y and Z)

Figure 11.2



� Mahanalobis Distance (MD) between the 2 pointsmeasures the distance between the centroids of the 2groups. The larger the distance, the greater the difference between the 2 groups (vice versa)� MANOVA reduces to computing the distance between

centroids of the 2 groups and determining if the distance is statistically significant.� In the case of p variables, the centroids of the 2 groups

can be represented as 2 points in the p-dimensional space and the problem reduces to determining whether the distance between the 2 points is different from zero.

To Explain More



Case III - more than 1 IV and p DV

� Refer to example party affiliation and gender� 2 DVs - tax increase (Y) and gun control (Z)� 3 IVs (at 2 levels) - Democrats (M,F) ,

Republicans (M,F), Independents (M,F)� Refer to Table 11.1 (pg. 344)� There are 3 types of effects:� Main effect of gender� Main effect of party affiliation� The interaction effect of gender and party affiliation.



(1) Main effect of gender

In Panel I – the main effect of gender, is measuredby the distance between the 2 centroids.



(2) Main effect of party affiliation

•In Panel II - the main effect of party affiliation is measured by the distances between pairs of the 3 centroids

•There will be 3 distances, each representing the distance between pairs of group



(3) Interaction Effect of Gender and Party Affiliation

Panel III - if the effect of gender is independentof party affiliation, the vectors joining therespective centroids should be parallel.



(4) Interaction Effect of Gender and Party Affiliation

Panel IV - if the effect of gender is not independent of partyaffiliation, the vectors joining the respective points will not be parallel.The magnitude of the interaction effect between 2 variablesis indicated by the extent to which the vector are nonparallel.


4. Analytic Computations for Two-Group MANOVA



Analytic Computations for Two-Group MANOVA

Partial eta square

Analyticcomputation

SIGNIFICANT TEST

EFFECT SIZE

MULTIVARIATE SIGNIFICANT TEST

UNIVARIATE SIGNIFICANT

TEST

MD2, T2,FEigenvalueWilks’ LambdaHotelling’s tracePillai’s criterionRoy’s Largest RootF-ratio

MD2, T2,F

Partial eta square



Analytic Computations for Two-Group MANOVA

MULTIVARIATESIGNIFICANT TEST

UNIVARIATE SIGNIFICANT

TEST

Analyticcomputation

SIGNIFICANT TEST

MD2, T2,FEigenvalueWilks’ LambdaHotelling’s tracePillai’s criterionRoy’s Largest RootF-ratio

MD2, T2,F

EFFECT SIZE

Partial eta square Partial eta square


5, 6 and 7. Rest of the Topics



Multivariate Significance Test

� The first step- determine if the 2 groups are significantlydifferent with respect to the variables

� Are the centroids of the 2 groups significantly different?

� The null and alternative hyphothesis for multivariatesstatistical significance testing in MANOVA are ;

H0 = µv1 = µv2 = µv3 ……. = µvk

� Null hypothesis formally states that thedifference between the centroid is zero



Similarities Between MANOVA and Discriminant Analysis

� Objective of discriminant analysis� To identify a linear combination of (discriminant function) of

the variables that would give the maximum separation between 2 groups.� Statistical test is performed to determine if the groups are

significantly different with respect to the linear combination (discriminant scores)

� In MANOVA, we test whether the centroids of the 2 groups are significantly different .� Although a linear combination of , which provides the maximum

separation between the 2 groups, is not computed in MANOVA, the multivariate significance tests implicitly test whether the mean score of the 2 groups obtained from such alinear combination are significantly different.



Briefly

� In the case of ONE IV, there is NO differencebetween MANOVA and discriminant analysis

� In the case of more than one IV , MANOVAprovide additional insights into the effects of IVs on DVs that are not provided by discriminant analysis.



MANOVA vs Discriminant Analysis

� MANOVA is applied to experimental situations where all, or at least some, IVs are manipulatedand subjects are randomly assigned to group, usually with equal cell sized

� Discriminant function analysis developed in the context of non -experimental research where groups are formed naturally and are notusually the same size



MANOVA vs Discriminant Analysis (cont’d)

� MANOVA asks if mean differences among groups on the combined DV are larger than expected by chance.

� Discriminant function analysis asks if there is some combination of variables that reliably separates groups.

� Should be noted that there is NO mathematical distinction between MANOVA and discriminant function analysis.



MANOVA Advantages Over ANOVA

� By measuring multiple DVs you increase your chances for finding a group difference

� With a single DV you “put all of your eggs in one basket”

� Multiple measures usually do not “cost” a great deal more and you are more likely to find a difference on at least one.



MANOVA Advantages Over ANOVA

� Using multiple ANOVAs inflates type 1 errorrates and MANOVA helps control for the inflation

� Under certain (rare) conditions MANOVA may find differences that do not show up under ANOVA



Assumptions of MANOVA

� Normal Distribution : - The dependent variable should be normally distributed within groups. Overall, the F test is robust to non-normality, if the non-normality is caused by skewness rather than by outliers. Tests for outliers should be run before performing a MANOVA, and outliers should be transformed or removed.

� Linearity - MANOVA assumes that there are linear relationships among all pairs of dependent variables , all pairs of covariates, and all dependent variable-covariate pairs in each cell. Therefore, when the relationship deviates from linearity, the power of the analysis will be compromised.



Assumptions of MANOVA (Cont’d)

� Homogeneity of Variances : - Homogeneity of variances assumes that the dependent variables exhibit equal levels of variance across the range of predictor variables. Remember that the error variance is computed (SS error) by adding up the sums of squares within each group. If the variances in the two groups are different from each other, then adding the two together is no t appropriate , and will not yield an estimate of the common within-group variance. Homoscedasticity can be examined graphically or by means of a number of statistical tests.

� Homogeneity of Variances and Covariances : - In multivariate designs, with multiple dependent measures, the homogeneity of variances assumption described earlier also applies. However, since there are multiple dependent variables, it is also required that their intercorrelations (covariances) are homogeneous across the cells of the design. There are various specific tests of this assumption.



T2 and F



Pillai Trace and Hotelling Trace



F - Test



Questions



Reference

Sharma, Subhash (1996). Applied Multivariate Techniques, U.S.A., John Wiley & Sons Inc.(Chapter 11: Multivariate Analysis of Variance, PP. 342 – 373).


For Listening

Design

Multivariate Analaysis of Variance (MANOVA): Sharma, Chapter 11 - Bijan Yavar