This document presents information on multivariate analysis of variance (MANOVA). It discusses when MANOVA is appropriate to use and its advantages over univariate ANOVA. Specifically, it notes that MANOVA considers multiple dependent variables simultaneously and is more powerful than conducting separate univariate tests. The document provides an example of a two-factor mixed MANOVA design investigating the effects of sex and chocolate type on ratings of chocolate taste, crunchiness, and flavor.

1. Presented by
Dr.J.P.Verma
MSc (Statistics), PhD, MA(Psychology), Masters(Computer Application)
Professor(Statistics)
Lakshmibai National Institute of Physical Education, Gwalior, India
(Deemed University)
Email: vermajprakash@gmail.com

2. Latent variable Measured through components
Health blood pressure, heart beat and BMI
Personality openness, agreeableness and conscientiousness
Aggression anger, hostility and impulsivity
Quality of drinks sweetness, flavor and hardness
It investigates the Effect of two factors (between-subjects and within-subject)
on a group of dependent variables.
What it does
When to use
When group difference on a latent variable is required to be compared across
different levels of the between-subjects as well as within-subject factors.
LatentVariable A concept which can not be directly measured
2

3. To investigate whether multivariate effect across the interaction between
within-subject and between-subjects factors is significant or not.
Advantage
Focus in design
One can investigate multivariate as well as univariate effects of within-subject
and between-subjects factors along with the interaction on a group of
dependent variables.
3

4. MANOVA experiment controlsType-I error
Because
Univariate analysis is carried out only if the
multivariate effect is significant.
Why MANOVA experiment is more powerful?
 It considers a set of different dependent variables as one single entity
 Single entity works like a super-variable, meta-variable
4

5. 5
This Presentation is based on
Chapter 8 of the book
Repeated Measures Design
for Empirical Researchers
Published by Wiley, USA
Complete Presentation can be accessed on
Companion Website
of the Book

6. These tests are equivalent to F test in univariate ANOVA
MANOVA creates
meta-variable
by
using
a linear
combination of
the dependent
variables
so as to maximize the
group difference.
Meta variable is compared in different groups
using
Multivariate tests Wilks’ Lambda or Pillai’sTrace
6

7. MultivariateAnalysis
Data type
IVs – two categorical ,one between-subjects and the other within-subject.
DVs – two or more, measured on metric scale
Sample Size
At least higher than the number of dependent variables
Minimum sample of size 20.
Independence of Observation
The observations obtained on each subject must be independent.
Missing Data
Complete data of all subjects is required in this design
Outlier
No outlier should exist in any group
7

8. MultivariateAnalysis
Linear relationship
All dependent variables should be reasonably related to each other linearly in
each cell.
Normality
The data in each cell must be normally distributed.
Multicollinearity
No multicollinearity should exist. Correlation among dependent variable
should not exceed 0.9.
Homogeneity ofVariance Covariance Matrices
Assumption of homogeneity is tested by Box’s M test
Due to sensitivity α is taken as .001.
8

9. Univariate Analysis
Sphericity
There should be no sphericity in the data.
Homogeneity ofVariances
Variance for the data obtained on each dependent variable must be
same in all the levels of the between-subjects variable separately in
each level of the within-subject variable.
 Sphericity is tested by Mauchly's test
 Homogeneity ofVariance is tested by Levene’s test
How to test these Assumptions
9

10. Case I: Levels of the within-subject variable are different treatment conditions
Example: To study the effect of hypertension and caffeine on aggression in an experiment
organized on six hypertensive subjects.
When to useTwo-factor Mixed MANOVA
Each subject of different levels of between subjects-factor is tested
on multiple dependent variables in each treatment condition
Issues in the Design
Carryover effect – Controlled by having sufficient gap between any two treatments
Order effect – Controlled by counterbalancing
IVs : Between-subjects: hypertension(hypertensive and non-hypertensive)
Within-subject: caffeine intensity(low, medium and high)
DV : Aggression(anger, hostility and impulsivity)
10

11. Figure 8.1 Layout design in two-factor mixed MANOVA
H2
H5
H3
H6
H1
H4
High
First phase
testing
H2
H5
H3
H6
H1
H4
H2
H5
H3
H6
H1
H4
Second phase testing
Third phase
testing
Testing protocol
Factor 2: Caffeine
Anger Hostility Impulsivity
H3
H6
H1
H4
H2
H5
H3
H6
H1
H4
H2
H5
H3
H6
H1
H4
H2
H5
H1
H4
H2
H5
H3
H6
H1
H4
H2
H5
H3
H6
H1
H4
H2
H5
H3
H6
MediumLow
Factor1:Hypertensionstatus
Hypertension
Anger Hostility Impulsivity Anger Hostility Impulsivity
N1
N3
N2
N6
N4
N5
First phase
testing
N12
N3
N2
N6
N4
N5
N1
N3
N2
N6
N4
N5
Second phase testing
Third phase
testing
N2
N6
N4
N5
N1
N3
N2
N6
N4
N5
N1
N3
N2
N6
N4
N5
N1
N3
N4
N5
N1
N3
N2
N6
N4
N5
N1
N3
N2
N6
N4
N5
N1
N3
N2
N6
Non Hypertension
11

12. Figure 8.2 Layout of the mixed design
Case II: Levels of the within-subject variable are different time periods
Example: To investigate the effect of sex and time on fitness status during a 6-weeks exercise
programme.
IVs : Between-subjects: Sex (Male, Female)
Within-subject: Time(zero, 4, 8 and 12 week)
M1
M2
M3
M4
M5
M6
Testing protocol
Factor 2: Time
Cardio Strength Flexibility
Initial
Factor1:Sex
Male
M1
M2
M3
M4
M5
M6
M1
M2
M3
M4
M5
M6
Male
Female
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
M1
M2
M3
M4
M5
M6
Cardio Strength Flexibility
2 Weeks
M1
M2
M3
M4
M5
M6
M1
M2
M3
M4
M5
M6
M1
M2
M3
M4
M5
M6
4 Weeks
M1
M2
M3
M4
M5
M6
M1
M2
M3
M4
M5
M6
M1
M2
M3
M4
M5
M6
Cardio Strength Flexibility
6 Weeks
M1
M2
M3
M4
M5
M6
M1
M2
M3
M4
M5
M6
Cardio Strength Flexibility
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
F1
F2
F3
F4
F5
F6
Female
DV : Fitness condition (cardio, strength and flexibility)
Purpose: To investigate
response pattern of the
subjects on a group of
dependent variables in
different durations
during treatment
12

13.  A medical researcher may like to see the response of
tuberculosis drug on the conditions of the male and female
patients over the period of time during the treatment.
 A market researcher may wish to investigate the effect of sex
and toothpaste brand on the buying behavior of customers on
the basis of toothpaste features (therapeutic, taste and
fragrance).
 A nutritionist may wish to investigate the effect of gender and
duration on the change in lifestyle indicators (fat%, cholesterol
and weight) in a six weeks health awareness programme.
13

14. Test assumptions of design
Describe layout design
Specify research questions to be investigated
Formulate multivariate and univariate
hypotheses to be tested
Decide familywise error rates (α)
Use SPSS to generate outputs
Levene’s test
for equality
of variances
Mauchly's test of
sphericityfor each
dependent variable
Cont …..
Box’s M Test
For
homogeneity
ANOVA table for
bet-sub variable
on each DV
MANOVA table
containing Wilks’
Lambda
14

15. Use SPSS to generate outputs
Marginal means for
bet-sub main effect
comparisons
Marginal means
plots
Cont …..
rANOVA table for
significance of with-
sub and interaction
Marginal means for
with-sub main
effect comparisons.
15

16. Is Interaction
significant
No
Test significance of F by
Assuming Sphericity
Yes
Report the effect of bet-sub
& with-sub factors
Perform factorial
rANOVA for each DV to
investigate main effects
Find simple effect of between-
subjects and within-subject
factors for each DV separately
Simple effect of with-sub
factor is obtained by applying
one-way rANOVA after
splitting the data file
Simple effect of bet-sub factor
is obtained by applying one-
way one-way ANOVA without
splitting the data file
16

17. ____________________________________________________________________
Sub Dark chocolate Milk chocolate White chocolate
Taste Crunch Flavour Taste Crunch Flavour Taste Crunch Flavour
1 5 4 5 7 6 6 5 5 6
2 4 5 4 5 5 7 6 4 5
3 6 5 6 7 6 7 5 5 4
4 5 4 7 8 7 8 7 5 5
5 4 5 6 6 8 7 5 6 6
6 5 6 4 7 7 8 6 5 5
7 4 5 6 7 6 8 6 6 5
8 6 5 5 8 8 7 5 5 6
9 7 5 6 7 7 8 5 4 5
10 5 6 4 7 7 7 6 5 4
1 7 6 7 4 5 6 7 5 5
2 6 8 6 3 4 5 5 5 4
3 8 7 6 3 3 5 8 4 5
4 6 8 8 5 4 6 7 3 6
5 5 9 6 4 4 5 5 5 6
6 7 8 5 6 6 4 6 4 5
7 7 9 8 6 6 5 6 3 6
8 5 9 6 5 8 6 5 4 5
9 6 7 5 3 6 4 7 4 5
10 8 7 6 4 4 5 4 5 6
____________________________________________________________________
Sex
MaleFemale
Table 8.1 Response on chocolate characteristics
Objective: To investigate the effect of gender and chocolate types on chocolate
characteristics (taste, crunchiness and flavor).
17
- An Illustration with SPSS

18. M1
M4
M8
M3
M5
M9
M2
M6
M7
M10
White
First phase testing M1
M4
M8
M3
M5
M9
M2
M6
M7
M10
M1
M4
M8
M3
M5
M9
M2
M6
M7
M10
Second phase testing
Third phase testing
Testing protocol
Factor 2: Chocolate
Taste Crunch Flavour
M3
M5
M9
M2
M6
M7
M10
M1
M4
M8
M3
M5
M9
M2
M6
M7
M10
M1
M4
M8
M3
M5
M9
M2
M6
M7
M10
M1
M4
M8
M2
M6
M7
M10
M1
M4
M8
M3
M5
M9
M2
M6
M7
M10
M1
M4
M8
M3
M5
M9
M2
M6
M7
M10
M1
M4
M8
M3
M5
M9
MilkDark
Factor1:Sex
Taste Crunch Flavour Taste Crunch Flavour
F2
F5
S9
F1
F3
F8
F10
S2
S6
S8
F2
F5
F9
F1
F3
F8
F10
S2
S6
S8
F2
F5
F9
F1
F3
F8
F10
S2
S6
S8
F1
F3
F8
F10
F4
F6
F7
F2
F5
F9
F1
F3
F8
F10
F4
F6
F7
F2
F5
F9
F1
F3
F8
F10
F4
F6
F7
F2
F5
F9
F4
F6
F7
F2
F5
F9
F1
F3
F8
F10
S4
F6
F7
F2
F5
F9
F1
F3
F8
F10
F4
F6
F7
F2
F5
F9
F1
F3
F8
F10
First phase testing
Second phase testing
Third phase testing
Male
Female
Figure 8.3 Layout of the mixed design with two factors in the illustration
 Divide subjects into three groups randomly.
 Allocate treatments randomly on these groups.
 One can design the study by allocating treatments randomly to each subject independently.
 Order effect is controlled through counterbalancing.
 Learning/ fatigueness is controlled by giving sufficient gap between two treatments.
Procedure
18

19. 1. Whether chocolate type affects the subject’s response on the
overall chocolate characteristics irrespective of the sex?
2. Whether sex affects the subject’s response on the overall
chocolate characteristics irrespective of the chocolate types?
3. Whether interaction of sex and chocolate type affects the
subject’s response on the overall chocolate characteristics?
4. Whether the chocolate type affects the subject’s response on
each of the chocolate characteristics in each sex?
5. Whether the male and female response differs on each of the
chocolate characteristics in each type of chocolate.
19

20. H0: There is no difference between group mean vectors of the subject’s
response in three types of chocolate irrespective of the sex.
H1: At least one group mean vector differs.
a.To investigate the first research question
H0: There is no difference between group mean vectors of the subject’s
response in two different sexes irrespective of the chocolate.
H1: At least one group mean vector differs.
b.To investigate the second research question
Chocolate_WhitFlavour
sCrunchines
Taste
Chocolate_MilkFlavour
sCrunchines
Tastes
Chocolate_DarkFlavour
sCrunchines
Taste
0 :H









































FemaleFlavour
sCrunchines
Tastes
MaleFlavour
sCrunchines
Taste
0 :H



























20

21. H0 : There is no interaction between sex and chocolate type on group mean
vectors of the subject’s response. `
H1 : The interaction between sex and chocolate type on group mean vectors
of the subject’s response is significant.
c.To investigate the third research question
H1: At least any one group mean differs
d.To investigate the fourth research question
Test the following hypotheses for each chocolate characteristics in male and female group separately.
lateWhiteChocoChocolate_MilkChocolate_Dark0 :H 
e.To investigate the fifth research question
Test the following hypotheses for each of the chocolate characteristics in each chocolate type separately.
FemaleMale0 :H 
FemaleMale1:H 
Remark: If interaction is significant then the fourth and fifth set of hypotheses shall be tested by means of
univariate analysis for each dependent variable separately. 21

22. IfWilks’ test for interaction is significant then two rANOVA for Gender (within-subject)
and three independent measures ANOVA for Chocolate shall be applied
The family wise error rate(α) shall be taken as .05
This will inflate the family wise error rate (α).
To compensate this, α shall be adjusted
22

23. Figure 8.4 Data format in mixed MANOVA
Defining
Variables
Taste_Dark
Crunch_Dark
Flavour_Dark
Taste_Milk
Crunch_Milk
Flavour_Milk
Taste_White
Crunch_White
Flavour_White
23

24. 24
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