Two-way Mixed ANOVA Design

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

2

Split plot design

Also known as

In a situation where the effect of two factors (one between-subjects and another within-subjects) on some dependent variable is investigated.

When to Use

Two-way Mixed ANOVA Design

3

Characteristics of Two-way Mixed ANOVA Design

Subjects are assigned to treatment conditions by using randomization and repeated measures concept.

Different treatments of within-subject factor are randomly assigned to the subjects in each level of the between-subjects factor.

All subjects in each level of the between-subjects factor are tested in each treatment condition of the within-subject factor. 

To test the differences between two or more independent groups while subjects are repeatedly measured on some dependent variable in each level of the within-subject factor.

Purpose

Features

4

AN ILLUSTRATION

Objective To investigate the effect of time of testing on memory retention among boys and girls.

Gender : Between-subjects factor Levels: male and female

Time : Within-subjects factor Levels: morning, afternoon and evening

Purpose of using this design

To check interaction

What Interaction means ?

Whether pattern of the memory retention during different testing time differs in male and female

5

This Presentation is based on

Chapter 6 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

Pre-post Control design - A particular case of two-way mixed ANOVA design

Subjects are randomly divided

Experimentalgroup

Control group

Pre testing

Post testing

Treatment

Pre testing

Post testing

Placebo

Subjects think that they are a part of experiment Subjects Don’t know whether they are in experimental or control group

hence bias reduction

Purpose of Placebo

Pre-post design can be solved by using two-way mixed ANOVA

But better way is To use ANCOVA

design

7

Advantages of Two-way Mixed ANOVA Design

Interaction between within-subjects and between-subjects factors can be investigated.

Between-subjects factor can be considered as a covariate.

  This design is efficient in comparison to single

factor RMD because between-subjects factor reduces error variance substantially.

  The design is very sensitive in detecting even the

slightest variation in the groups.   In mixed ANOVA design post-hoc test can be

applied for between-subjects factor.

8

Application

1. A human resource manager may investigate the effect of training intervention (onsite, offsite and mix of these two) on learning skills for their employees (male and female).

2. A psychologist may like to investigate the effect of cognitive therapy (three different types) on the stress level. Here sex may be taken as between-subjects factor.

3. An educational psychologists may investigate the effect of learning methods (traditional, audio-visual and self learning) and IQ(high and low) on memory retention.

4. A basketball coach may wish to investigate the effect of distance (3 mt., 4 mt and 5 mt.) and gender on shooting performance in basketball. Here distance is a within-subjects and gender is a between-subjects factor respectively.

5. A nutritionist may be interested to compare the effect of three diet programmes on weight reduction in a six week experiment. Subjects may be in different active, semi-active and sedentary groups.

9

Factor 2: Environment

S1

S2

S5

S6

S3

S4

Controlled

S3

S4

S1

S2

S5

S6

S5

S6

S3

S4

S1

S2

Testing protocol

HotCold

S7

S8

S11

S12

S9

S10

S9

S10

S7

S8

S11

S12

S11

S12

S9

S10

S7

S8

Subjects

Fact

or 1

: Sex

First phase testing

Second phase testing

Third phase testing

Male

Female

First phase testing

Second phase testing

Third phase testing

Figure 6.1 Layout of mixed ANOVA design Case I: Levels of the within-subjects variable are different treatment conditions

When to use Two-way Mixed ANOVA DesignUsed in Two Types of Situations

Order effect Tackled by counterbalancing Divide sample in each level into c

groups (c :number of levels in within-subjects factor.)

Allocate treatments randomly on these groups

Example: Investigate the effect of environment on mood behavior of the 12 subjects (male and female).

Within-subjects factor: EnvironmentBetween-subjects factor: Sex

10

2 weeks

S1

S2

S3

S4

S5

S1

S2

S3

S4

S5

S1

S2

S3

S4

S5

4 weeks 6 weeks

Factor 2: Time

Initial

S1

S2

S3

S4

S5

Male

Female

Subjects

S6

S7

S8

S9

S10

Testing protocol

S6

S7

S8

S9

S10

S6

S7

S8

S9

S10

S6

S7

S8

S9

S10

Fact

or 1

: Sex

Figure 6.2 Layout of mixed ANOVA design

Case II: levels of the within-subjects variable are different time periods

When to use Mixed ANOVA DesignUsed in Two Types of Situations

Example: To investigate the effect of Time on the effectiveness of an exercise therapy programme organized on 5 male and 5 female participants.

11

Steps in Mixed ANOVA DesignTest normality assumption in all treatment conditions

Describe design layout

Write research questions

Write different H0 to be tested

Decide family wise error rates (α)

Use SPSS to generate outputs

Descriptive statistics

F table for within-subjects effect and Interaction Cont

…..

Box’s M Test for homogeneity

Levene’s test of equality of variances

Test assumption of homogeneity

F table for between-subjects effect

12

Steps in Mixed ANOVA Design

Is Sphericity Significan

t

No

Test F by Assuming Sphericity

If F significant do pair-wise comparison of means by using Tukey/using Bonferroni

Yes

Apply correction and test F

Use SPSS to generate outputs

Means plots

Cont …..

pair-wise comparison tables for effects if F significant

Mauchly's test of sphericity

13

Steps in Mixed ANOVA Design

Do following using SPSS

Is Interacti

on Significa

nt

No

Discuss Main Effect If Significant

Discuss pair-wise comparison of means and means plot

Yes

Test Simple Effect Of each IV

Test simple effects of between-subjects as well as within-subjects factor.

Report findings

Report findings

Cont …..

14

Steps in Mixed ANOVA DesignCheck sphericity assumption while testing main or simple

effect

Is p<.0

5Test F ratio by

assuming sphericity N

Y

Check

<.75 Test F by using Huynh-

Feldt correctionNTest F by using

Greenhouse-Geisser correction

Y

If F is significant apply t tests for comparison of means using Bonferroni

correction.

Report findings

Cont …..

15

Movie Romantic Social

ActionTeens 1 65 50 57

2 65 56 623 59 46 534 67 50 545 66 52 606 62 51 63

 Mid age 7 65 62 48

8 60 67 539 57 52 4410 61 55 4311 62 64 4612 62 65 47

 Old age 13 61 67 50

14 58 62 5215 62 68 4616 60 66 4817 55 65 5318 60 72 56

Age

Cate

gory

Mixed ANOVA Design – AN ILLUSTRATION

To investigate the effect of age and movie types on the enjoyment of audience.

Objective

Age category : Teens, Mid age and Old ageMovie type : Romantic, Social and Action

Table 6.1 Score on enjoyment reported by the subjects after watching movies

16

S1 S2

S5S6

S3S4

Action

First testing

Second testing

Third testing

SocialRomantic

Teens

Subjects

S3 S4

S1S2

S5S6

S5 S6

S3S4

S1S2

S7 S8

S11S12

S9S10

First testing

Second testing

Third testing

Mid Age

S9 S10

S7S8

S11S12

S11 S12

S9S10

S7S8

S13 S14

S17S18

S15S16

First testing

Second testing

Third testing

Old Age

S15 S16

S13S14

S17S18

S17 S18

S15S16

S13S14

Fact

or 1

: Age

Factor 2: Movie

Testing protocol

Figure 6.3 Layout of the mixed ANOVA design in the illustration

Design Layout of Two-way Mixed ANOVA Design

17

r = number of levels of Movie factor(within-subjects) = 3c = number of levels of Age factor(between-subjects) = 3n = number of subjects in each of the r levels of factor Age = 6

Movie

Romantic Social ActionTeens 1 65 50 57

2 65 56 623 59 46 534 67 50 545 66 52 606 62 51 63

 Mid age 7 65 62

488 60 67 539 57 52 4410 61 55 4311 62 64 4612 62 65 47

 Old age 13 61 67

5014 58 62 5215 62 68 4616 60 66 4817 55 65 5318 60 72 56

 

Age

Cate

gory

Distribution of SS in Two-way Mixed ANOVA Design

Total SS = SSSubjects + SSWithing Subjects

  = (SSAge + SSError_Age) + (SSMovie + SSAge×Movie + SSError_Movie)

18

SSBetween_Subjects df=nr-1

Total SS df = nrc-1

SSWithin_Subjects df= nr(c-1)

53

17 36

SSError_AgeSSAge SSAge× MovieSSMovie SSError_Movie

r-1=2 r(n-1)=15 c-1=2 (r-1)(c-1)=4 r(n-1)(c-1)=30

Distribution of SS and df in Two-way Mixed ANOVA Design

Figure 6.4 Scheme of distributing total SS and df in the mixed ANOVA design

19

Whether enjoyment in watching movie depends upon the age of the subjects.

Whether enjoyment in watching movie depends upon the type of movie seen by the subject.

Whether interaction between age and movie type affects the enjoyment in watching movie.

Research Issues and Hypothesis Construction

 against H1: At least one group mean differs 

Research Questions

Hypotheses Construction

Effect of Movie

against H1: At least one group mean differs 

Effect of AgeInteraction Effect (Age × Movie)  H0: There is no interaction between Age and Music against H1: The interaction between Age and Music is significant

ActionSocialRomantic0 :H

age_Oldage_MidTeens0 :H

20

Level of Significance

 Bonferroni correction shall be used for

correcting level of significance for pair wise

comparison of means

Family wise error rate(α) is .05

In case interaction is significant multiple ANOVA (independent and repeated) shall be done to test the simple effect.

α for testing significance of F in simple effect would be .017(=.05/3) level.

21

Data File for Two-way Mixed ANOVA Design in SPSS

Figure 6.5 Data format in the two-way mixed ANOVA design

22

Analyze General Linear Model Repeated Measures

Figure 6.5 Screen for initiating commands for the two-way mixed ANOVA design

While being in Data View click on the following command sequence

SPSS Commands for initiating Two-way Mixed ANOVA Design

23

To buy the book

Repeated Measures Design for Empirical Researchers

and all associated presentations

Click Here

Complete presentation is available on companion website of the book