1. Multivariate Analysis of Variance (MANOVA)
Lecturer: Prof.Dr. Izani Ibrahim (CRM)
Presenter: Bijan Yavar (CRM)
Level of Study: PHD
Field of Study: Management (Crisis Management)
Course: Advanced Quantitative Techniques
SID: ZP01774
Researcher ID: A-3544-2010
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

2. PowerPoint Template
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The National University of Malaysia (UKM) Graduate School of Business (GSB)

3. Contents
1. Introduction (What is MANOVA?)
2. Differences Between MANOVA and ANOVA
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
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

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

5. Introduction (What is MANOVA?)
Technique for assessing group differences across multiple
metric 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
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

6. 2. Differences Between MANOVA and ANOVA
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

7. The Difference Between MANOVA & ANOVA
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December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

8. 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)
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

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

10. Introduction (What is MANOVA?)
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.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

11. Introduction (What is MANOVA?)
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
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

12. 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 toward
the product (metric – Likert scale and has more than 2
DVs)
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

13. 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)
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

14. 3. Geometry of MANOVA
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

15. 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.
Centroid
For Group 1
Y1
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December 2013
Centroid
For Group 2
Figure 11.1
Y2
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

16. To Explain More
MANOVA is concerned with determining whether
MD between group centroids is significantly
greater 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.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

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

18. To Explain More
Mahanalobis Distance (MD) between the 2 points
measures the distance between the centroids of the 2
groups. 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.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

19. 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.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

20. (1) Main effect of gender
In Panel I – the main effect of gender, is measured
by the distance between the 2 centroids.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

21. (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
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

22. (3) Interaction Effect of Gender and Party Affiliation
Panel III - if the effect of gender is independent
of party affiliation, the vectors joining the
respective centroids should be parallel.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

23. (4) Interaction Effect of Gender and Party Affiliation
Panel IV - if the effect of gender is not independent of party
affiliation, the vectors joining the respective points will not be parallel.
The magnitude of the interaction effect between 2 variables
is indicated by the extent to which the vector are nonparallel.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

24. 4. Analytic Computations for Two-Group MANOVA
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

25. Analytic Computations for Two-Group MANOVA
MULTIVARIATE
SIGNIFICANT TEST
SIGNIFICANT
TEST
MD2, T2,F
Eigenvalue
Wilks’ Lambda
Hotelling’s trace
Pillai’s criterion
Roy’s Largest Root
F-ratio
UNIVARIATE
SIGNIFICANT
TEST
MD2, T2,F
Analytic
computation
EFFECT
SIZE
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December 2013
Partial eta
square
Partial eta
square
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

26. Analytic Computations for Two-Group MANOVA
MULTIVARIATE
SIGNIFICANT TEST
UNIVARIATE
SIGNIFICANT
TEST
SIGNIFICA
NT TEST
MD2, T2,F
Eigenvalue
Wilks’ Lambda
Hotelling’s trace
Pillai’s criterion
Roy’s Largest Root
F-ratio
MD2, T2,F
EFFECT
SIZE
Partial eta square
Partial eta
square
Analytic
computation
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

27. 5, 6 and 7. Rest of the Topics
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

28. Multivariate Significance Test
The first step- determine if the 2 groups are significantly
different with respect to the variables
Are the centroids of the 2 groups significantly different?
The null and alternative hyphothesis for multivariates
statistical significance testing in MANOVA are ;
H0 = µv1 = µv2 = µv3 ……. = µvk
Null hypothesis formally states that the
difference between the centroid is zero
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

29. 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 a
linear combination are significantly different.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

30. Briefly
In the case of ONE IV, there is NO difference
between MANOVA and discriminant analysis
In the case of more than one IV, MANOVA
provide additional insights into the effects of IVs
on DVs that are not provided by discriminant
analysis.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

31. MANOVA vs Discriminant Analysis
MANOVA is applied to experimental situations
where all, or at least some, IVs are manipulated
and 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 not
usually the same size
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

32. 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.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

33. 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.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

34. MANOVA Advantages Over ANOVA
Using multiple ANOVAs inflates type 1 error
rates and MANOVA helps control for the inflation
Under certain (rare) conditions MANOVA may
find differences that do not show up under
ANOVA
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

35. 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.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

36. 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 not 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.
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

37. T2 and F
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December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

38. Pillai Trace and Hotelling Trace
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December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

39. F - Test
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December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

40. Questions
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December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

41. Reference
Sharma, Subhash (1996). Applied Multivariate Techniques,
U.S.A., John Wiley & Sons Inc.
(Chapter 11: Multivariate Analysis of Variance, PP. 342 – 373).
17th
December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)

42. For Listening
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December 2013
Bijan Yavar
The National University of Malaysia (UKM) Graduate School of Business (GSB)