Two-way Repeated Measures 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

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Two-Way Repeated Measures Design

Where the effect of two within-subjects factor on

a dependent variable needs to be investigated simultaneously

Where individual variations of the subjects cannot be controlled

Recruiting large sample in the study is difficult

within-within design, two-way repeated measures design(RMD) or

two-way ANOVA with repeated measures.

Also known as

When to Use

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Features of Two-way RMD

All subjects are tested in each level of both the factors. Mean differences between groups, split on two within-subjects

factors are compared. Structu

re

Highlights

If Factor A has two levels A1 and A2 and Factor B has three levels B1, B2 and B3

Then there will be six treatment conditions

A1B1, A1B2, A1B3A2B1, A2B2, A2B3 

A randomly drawn sample is then tested in all the six treatment conditions

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What research questions we investigate?

Whether the factor A affects the dependent variable?

Whether the factor B affects the dependent variable?

Investigated through main effects

Investigated through simple effects Whether interaction between the factor A and

B is significant?

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This Presentation is based on

Chapter 5 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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Main and Simple Effect

Objective: To compare the effect of teaching methods on learning

< 20 years(b1)

21 - 40 years (b2)18

1921

353229

242834

181922

Traditional(a1)

Audio-visual(a2)

Factor A: Teaching Methods

Factor B: Age

Main Effect of A : Effect of Teaching methods on learning across all levels of factor B (Age)

Simple Effect of A Effect of Factor A on learning in each level of factor B

>4o years (b3)

202229

202117

Main Effect of B : Effect of Age on learning across all the levels of factor A (Teaching methods)

Simple Effect of B Effect of Factor B on learning in each level of factor A

Investigated only when Interaction is significant

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Understanding Interaction

< 20 (b1)

21 – 40 (b2)18

1921

353229

242834

181922

Traditional(a1)

Audio-visual(a2)

Factor A:Teaching Methods

Factor B: Age >4o

(b3)202229

202117

Interaction Joint effect of Teaching method and Age (A×B) on learning

If Interaction (A×B) is significant

Association exists between teaching method and age

Pattern of learning response differs in each teaching methods

b1

b2

b3

a1

a2

b1

b2

b3

a1

a2

No Interaction

There is Interaction

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Characteristics of Two-way RMD

If factor levels are large, subjects get tired/bored resulting inaccurate observations

Design becomes less efficient if variability among subjects becomes insignificant

Advantage

Disadvantage

 Requires limited number of subjects Study can be completed quickly Increased efficiency in comparison to independent

measures ANOVA Can be used for the longitudinal studies

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Testing protocol

Fact

or 1

: C

affe

ine

Factor 2: Environmental

S1

S2

S5

S6

S3

S4

Evening

First phase testing

S3

S4

S1

S2

S5

S6

S5

S6

S3

S4

S1

S2

Second phase testing

Third phase testing

AfternoonMorning

S3

S4

S1

S2

S5

S6

S1

S2

S5

S6

S3

S4

S5

S6

S3

S4

S1

S2

Coffee

Placebo

Subjects

First phase testing

Second phase testing

Third phase testing

Case I: Levels of the within-subjects variable are different treatment conditions

Example: Investigate the effect of caffeine (coffee and placebo) and time of testing on the mathematical ability on six subjects.

Layout Procedure

Within-subjects factors1. Caffeine 2. Time

Divide subjects into 3(number of levels) groups

Allocate treatments randomly on these groups

Like (1,1,1), 2,1,3 and 3,1,2 as shown in figure

(1,1,1): Group will undergo the first treatment condition thereafter second and then the third

When to use Two-way RMDUsed in Two Types of Situations

Figure 5.1 Layout design

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3 weeks

S1

S2

S3

S4

S5

S6

S1

S2

S3

S4

S5

S6

S1

S2

S3

S4

S5

S6

6 weeks 9 weeks

Factor 2:Time

Initial

S1

S2

S3

S4

S5

S6

Coffee

Placebo

Subjects

Testing protocol

Fact

or 1

: Caf

fein

e

S1

S2

S3

S4

S5

S6

S1

S2

S3

S4

S5

S6

S1

S2

S3

S4

S5

S6

S1

S2

S3

S4

S5

S6

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

When to use Two-way RMDUsed in Two Types of Situations

Example: To see the effect of caffeine on mathematical ability in four different time duration i.e. before experiment, after 3 weeks, 6 weeks and 9 weeks. Let us have the sample of size six.

Figure 5.2 Layout design

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Application of Two-Way RMD

To study the effect of caffeine(coffee and placebo) on memory retention over a period of time(0, 1 and 2 weeks)

To see the impact of fat consumption(no fat, medium fat and high fat) and time(morning afternoon and evening) of the day on the performance in a comprehension test

  A market researcher may like to investigate the effect

of time and season on the sale in grocery outlets of a company

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Steps in Two-way RMDTest 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

Mauchly's test of sphericity

F table for within-subjects effect

Pair-wise comparison of means for IVs if found significant

Different Means plots

Marginal Means for each cell and IV

Cont …..

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Steps in Two-way RMD

Generate following results using SPSS

Descriptive statistics

Mauchly's test of sphericity

F table for within-subjects effect

Pair-wise comparison of means for IVs if found significant

F table for within-subjects effect

Interaction Significant

No

Test Main Effect if Significant

Do pair-wise comparison of means

Yes

Test Simple Effect of each IV

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Steps in Two-way RMDCheck sphericity assumption while testing main or simple

effect

p<.05

Test 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

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Table 5.1 Number of match box prepared per hour in a day Environment

Hot Humid Cold _____________________________________________

20 16 2718 17 24

No music 22 16 2616 19 1718 20 2620 22 23

22 21 2320 25 21

Jazz 24 27 2219 21 2022 27 2520 26 25

24 26 2126 22 20

Instrumental 25 22 1826 21 2424 19 1825 22 21

_______________________________________________

Mus

ic

Solving Two-way RMD with SPSS

To investigate the effect of environment and music on the performance of six employees in a cottage industry of packaging.

Objective

Environment : hot, humid and coldTypes of music : Instrumental, Classical Jazz and no music

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Testing protocolFa

ctor

1: M

usic

Factor 2: Environment

S1 S2

S3S4

S5S6

Cold

First testing

Second testing

Third testing

HumidHot

No Music

Subjects

S5 S6

S1S2

S3S4

S3 S4

S5S6

S1S2

S5 S6

S1S2

S3S4

First testing

Second testing

Third testing

Jazz

S3 S4

S5S6

S1S2

S1 S2

S3S4

S5S6

S3 S4

S5S6

S1S2

First testing

Second testing

Third testing

Instrumental

S1 S2

S3S4

S5S6

S5 S6

S1S2

S3S4

Two-Way RMD with SPSS

Figure 5.3 Layout of the repeated measures design with two factor

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Distribution of SS in Two-way RMD

  Total SS = SSSubjects + SSWithing

Subjects

  = SSSubjects + (SSMusic + SSError_Music) + (SSEnvir + SSError_Envir ) + (SSMusic×Envir+

SSError_Music×Envir)

Hot Humid Cold

20 16 2718 17 24

No music 22 162616 19 1718 20 2620 22 23

22 21 2320 25 21

Jazz 24 27 2219 21 2022 27 2520 26 25

24 26 2126 22 20

Instru 25 22 1826 21 2424 19 1825 22 21

Mus

ic

Environment

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SSBetween_Subjects n-1

Total SS df = nrc-1

SSWithin_Subjects n(rc-1)

53

5 48

SSError_Music (r-1)(n-1) 10SSMusic r-1

SSError_Music×Envir (r-1)(c-1)(n-1)SSMusic×Envir (r-1)(c-1)

SSError_Envir (c-1)(n-1) 10SSEnvir c-1

204

2 2

Distribution of SS and df in Two-way RMD

Figure 5.4 Scheme of distributing total SS and df in two-way repeated measures design

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1.  Whether back ground music affects the performance of workers.

2. Whether performance of workers is affected by the environment.

3. Whether interaction between background music and type of environment affects the worker’s performance.

Research Issues and Hypothesis Construction

 against H1: At least one group mean differs 

Research Questions

Hypotheses Construction

alInstrumentJazzMusic_No0 :H Main effect of Music

 against H1: At least one group mean differs 

Main effect of Environment

ColdHumidHot0 :H

Interaction Effect (Music × Environment)  H0: There is no interaction between Music and Environment against H1: The interaction between Music and Environment is significant

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Level of Significance

 Bonferroni correction shall be

applied for correcting the level of

significance

Family wise error rate(α) shall be taken as .05

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Two-way RMD with SPSS

NOM_HotNOM_HumidNOM_ColdJz_HotJz_HumidJz_ColdInst_HotInst_HumidInst_Cold

How to Prepare Data File in SPSS?In Variable View define the following nine treatment

combinations as variables

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Figure 5.5 Data format in the repeated measures design with two factors

Figure 5.5 Data format in the repeated measures design with two factors

  Analyze General Linear Model Repeated Measures 

Data File for Two-way RMD in SPSS

While being in Data View click on the following command sequence

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