This document provides information on two-way repeated measures designs, including when to use them, their structure, and how to analyze the data. A two-way repeated measures design is used to investigate the effects of two within-subjects factors on a dependent variable simultaneously. All subjects are tested at each level of both factors. This design allows comparison of mean differences between groups split on the two within-subject factors. The document describes the analysis process, including testing for main effects, interactions, and simple effects using SPSS. An example is provided to illustrate a two-way repeated measures design investigating the effects of music and environment on work performance.

1. Presented by
Dr.J.P.Verma
MSc (Statistics), PhD, MA(Psychology), Masters(Computer Application)
Professor(Statistics)
Lakshmibai National Institute of Physical Education, Gwalior, India
(Deemed University)
Email: vermajprakash@gmail.com

2.  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
2

3.  All subjects are tested in each level of both the factors.
 Mean differences between groups, split on two within-subjects factors
are compared.
Structure
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, A1B3
A2B1, A2B2, A2B3
A randomly drawn sample is then tested in all the six treatment conditions
3

4.  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?
4

5. 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. Objective: To compare the effect of teaching methods on learning
< 20 years(b1) 21 - 40 years (b2)
18
19
21
35
32
29
24
28
34
18
19
22
Traditional(a1)
Audio-visual(a2)
Factor A: Teaching
Methods
Factor B: Age
Main Effect of A : Effect ofTeaching 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)
20
22
29
20
21
17
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
6

7. < 20 (b1) 21 – 40 (b2)
18
19
21
35
32
29
24
28
34
18
19
22
Traditional(a1)
Audio-visual(a2)
Factor A:
Teaching
Methods
Factor B: Age
>4o (b3)
20
22
29
20
21
17
Interaction Joint effect ofTeaching 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
7

8.  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
8

9. Testing protocol
Factor1:Caffeine
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 factors
1. 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 RMD
Used in Two Types of Situations
Figure 5.1 Layout design
9

10. 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
Factor1:Caffeine
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 RMD
Used 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
10

11.  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
11

12. Steps in Two-way RMD
Test 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 …..
12

13. 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
13

14. Check 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
correction
NTest F by using Greenhouse-
Geisser correction
Y
If F is significant apply t tests for comparison of
means using Bonferroni correction.
Report findings 14

15. Table 5.1 Number of match box prepared per hour in a day
Environment
Hot Humid Cold
_____________________________________________
20 16 27
18 17 24
No music 22 16 26
16 19 17
18 20 26
20 22 23
22 21 23
20 25 21
Jazz 24 27 22
19 21 20
22 27 25
20 26 25
24 26 21
26 22 20
Instrumental 25 22 18
26 21 24
24 19 18
25 22 21
_______________________________________________
Music
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 cold
Types of music : Instrumental, Classical
Jazz and no music
15

16. Testing protocolFactor1:Music
Factor 2: Environment
S1
S2
S3
S4
S5
S6
Cold
First testing
Second testing
Third testing
HumidHot
No Music
Subjects
S5
S6
S1
S2
S3
S4
S3
S4
S5
S6
S1
S2
S5
S6
S1
S2
S3
S4
First testing
Second testing
Third testing
Jazz
S3
S4
S5
S6
S1
S2
S1
S2
S3
S4
S5
S6
S3
S4
S5
S6
S1
S2
First testing
Second testing
Third testing
Instrumental
S1
S2
S3
S4
S5
S6
S5
S6
S1
S2
S3
S4
Figure 5.3 Layout of the repeated measures design with two factor
16

17. Total SS = SSSubjects + SSWithing Subjects
= SSSubjects
+ (SSMusic + SSError_Music)
+ (SSEnvir + SSError_Envir )
+ (SSMusic×Envir+ SSError_Music×Envir)
Hot Humid Cold
20 16 27
18 17 24
No music 22 16 26
16 19 17
18 20 26
20 22 23
22 21 23
20 25 21
Jazz 24 27 22
19 21 20
22 27 25
20 26 25
24 26 21
26 22 20
Instru 25 22 18
26 21 24
24 19 18
25 22 21
Music
Environment
17

18. 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
Figure 5.4 Scheme of distributing total SS and df in two-way repeated measures design
18

19. 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.
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
19

20. Bonferroni correction shall be applied for
correcting the level of significance
Family wise error rate(α) shall be taken as .05
20

21. NOM_Hot
NOM_Humid
NOM_Cold
Jz_Hot
Jz_Humid
Jz_Cold
Inst_Hot
Inst_Humid
Inst_Cold
How to Prepare Data File in SPSS?
In VariableView define the following nine treatment combinations as variables
21

22. 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
While being in DataView click on the following command sequence
22

23. 23
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