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: vermajprakash@gmail.com

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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

H4

H2

H5

H3

H6

H1

H4

Second phase testing

Third phase testing

Testing protocolFactor 2: Caffeine

Anger Hostility Impulsivity

H3

H6

H1

H4

H2

H5

H3

H6

H1

H4

H2

H5

H3

H6

H1

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H6

H1

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H5

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H1

H4

H2

H5

H3

H6

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

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

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N5

N1

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N2

N6

N4

N5

N1

N3

N2

N6

N4

N5

N1

N3

N2

N6

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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F5

F6

F1

F2

F3

F4

F5

F6

F1

F2

F3

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F5

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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

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F1

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F1

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F1

F2

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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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