two-way repeated measures anova
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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
3
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?
5
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
6
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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Repeated Measures Design for Empirical Researchers
and all associated presentations
Click Here
Complete presentation is available on companion website of the book
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