Two-Factor Mixed MANOVA
Presented by
Dr.J.P.VermaMSc (Statistics), PhD, MA(Psychology), Masters(Computer Application)
Professor(Statistics)
Lakshmibai National Institute of Physical Education, Gwalior, India
(Deemed University)Email: [email protected]
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Latent variable Measured through componentsHealth blood pressure, heart beat and BMIPersonality openness, agreeableness and conscientiousnessAggression anger, hostility and impulsivityQuality of drinks sweetness, flavor and hardness
Two-factor Mixed MANOVA
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.
Latent Variable A concept which can not be directly measured
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To investigate whether multivariate effect across the interaction between within-subject and between-subjects factors is significant or not.
Two-factor Mixed MANOVA
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.
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How MANOVA Experiment is Performed
MANOVA experiment controls Type-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
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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
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These tests are equivalent to F test in univariate ANOVA
How MANOVA Experiment is Performed
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 groupsusin
g
Multivariate tests Wilks’ Lambda or Pillai’s Trace
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Assumptions in Two-factor Mixed MANOVA
Multivariate Analysis
Data typeIVs – 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 variablesMinimum 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
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Assumptions in Two-factor Mixed MANOVA
Multivariate Analysis
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 of Variance Covariance Matrices Assumption of homogeneity is tested by Box’s M test Due to sensitivity α is taken as .001.
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Two-factor Mixed MANOVA
Univariate Analysis
Sphericity There should be no sphericity in the data. Homogeneity of Variances
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 of Variance is tested by
Levene’s test
How to test these Assumptions
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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 use Two-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
DesignCarryover effect – Controlled by having sufficient gap between any two treatmentsOrder 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)
11 Figure 8.1 Layout design in two-factor mixed MANOVA
Layout in Two-factor Mixed MANOVA
H2
H5
H3
H6
H1
H4
High
First phase testing
H2
H5
H3
H6
H1
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H4
Second phase testing
Third phase testing
Testing protocolFactor 2: Caffeine
Anger Hostility Impulsivity
H3
H6
H1
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H5
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H6
H1
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MediumLow
Fact
or 1
:Hyp
erte
nsio
n st
atusHypertension
Anger Hostility Impulsivity Anger Hostility Impulsivity
N1
N3
N2
N6
N4
N5
First phase testing
N12
N3
N2
N6
N4
N5
N1
N3
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N6
N4
N5
Second phase testing
Third phase testing
N2
N6
N4
N5
N1
N3
N2
N6
N4
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N1
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N1
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Non Hypertension
12Figure 8.2 Layout of the mixed design
When to Use Two-factor Mixed MANOVACase 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
Fact
or 1
:Sex
Male
M1
M2
M3
M4
M5
M6
M1
M2
M3
M4
M5
M6
Male
Female
F1
F2
F3
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F1
F2
F3
F4
F5
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F1
F2
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M1
M2
M3
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M6
Cardio Strength Flexibility
2 Weeks
M1
M2
M3
M4
M5
M6
M1
M2
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M4
M5
M6
M1
M2
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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
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F1
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F1
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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
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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.
Application of Mixed MANOVA
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Steps in Two-factor Mixed MANOVATest 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 sphericity for each dependent variable
Cont …..
Box’s M Test For homogeneity
ANOVA table for bet-sub variable on each DV
MANOVA table containing Wilks’ Lambda
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Steps in Two-factor Mixed MANOVA
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.
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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
Steps in Two-factor Mixed MANOVA
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____________________________________________________________________Sub Dark chocolate Milk chocolate White chocolate
Taste Crunch Flavour Taste Crunch Flavour Taste Crunch Flavour1 5 4 5 7 6 6 5 5 62 4 5 4 5 5 7 6 4 53 6 5 6 7 6 7 5 5 44 5 4 7 8 7 8 7 5 55 4 5 6 6 8 7 5 6 66 5 6 4 7 7 8 6 5 57 4 5 6 7 6 8 6 6 58 6 5 5 8 8 7 5 5 69 7 5 6 7 7 8 5 4 510 5 6 4 7 7 7 6 5 4
1 7 6 7 4 5 6 7 5 52 6 8 6 3 4 5 5 5 43 8 7 6 3 3 5 8 4 54 6 8 8 5 4 6 7 3 65 5 9 6 4 4 5 5 5 66 7 8 5 6 6 4 6 4 57 7 9 8 6 6 5 6 3 68 5 9 6 5 8 6 5 4 59 6 7 5 3 6 4 7 4 510 8 7 6 4 4 5 4 5 6
____________________________________________________________________
Sex
Mal
eFe
mal
e
Two-factor Mixed MANOVA
Table 8.1 Response on chocolate characteristics
Objective: To investigate the effect of gender and chocolate types on chocolate characteristics (taste, crunchiness and flavor).
- An Illustration with SPSS
M1 M4M8
M3M5M9
M2M6M7M10
White
First phase testing M1 M4M8
M3M5M9
M2M6M7M10
M1 M4M8
M3M5M9
M2M6M7M10
Second phase testing
Third phase testing
Testing protocolFactor 2: Chocolate
Taste Crunch Flavour
M3 M5M9
M2M6M7M10
M1M4M8
M3 M5M9
M2M6M7M10
M1M4M8
M3 M5M9
M2M6M7M10
M1M4M8
M2 M6M7M10
M1M4M8
M3M5M9
M2 M6 M7M10
M1M4M8
M3M5M9
M2 M6M7 M10
M1M4M8
M3M5M9
MilkDark
Fact
or 1
: Sex
Taste Crunch Flavour Taste Crunch Flavour
F2 F5S9
F1F3F8F10
S2S6S8
F2 F5F9
F1F3F8F10
S2S6S8
F2 F5F9
F1F3F8F10
S2S6S8
F1 F3F8F10
F4F6F7
F2F5F9
F1 F3F8F10
F4F6F7
F2F5
F9
F1 F3F8F10
F4F6F7
F2F5
F9
F4 F6F7
F2F5F9
F1F3F8F10
S4 F6 F7
F2F5F9
F1F3F8F10
F4 F6F7
F2F5F9
F1F3F8F10
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
Two-factor Mixed MANOVA
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
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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.
Research Questions
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Hypotheses Construction
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
Hypotheses Construction
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 questionTest the following hypotheses for each chocolate characteristics in male and female
group separately.lateWhiteChocoChocolate_MilkChocolate_Dark0 :H
e. To investigate the fifth research questionTest 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
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If Wilks’ test for interaction is significant then two rANOVA for Gender (within-subject) and three independent measures ANOVA
for Chocolate shall be applied
Level of Significance
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
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Figure 8.4 Data format in mixed MANOVA
Data File for Two-factor Mixed MANOVA in SPSS
Defining VariablesTaste_DarkCrunch_DarkFlavour_DarkTaste_MilkCrunch_MilkFlavour_MilkTaste_WhiteCrunch_WhiteFlavour_White
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