### Chapter 13 Repeated Measures ANOVA

```Repeated Measures
Chapter 13
Chapter Objectives
• Rationale of Repeated Measures ANOVA
– One- and two-way
– Benefits
• Partitioning Variance
• Statistical Problems with Repeated Measures
Designs
– Sphericity
– Overcoming these problems
• Interpretation
Benefits of Repeated Measures Designs
• Sensitivity
– Unsystematic variance is reduced.
– More sensitive to experimental effects.
– More powerful than independent designs
• Economy
– Less participants are needed.
– But, be careful of fatigue.
Problems with Repeated Measures Designs
• Same participants in all conditions.
– Scores across conditions correlate.
– Violates assumption of independence.
• Assumption of Sphericity.
– Crudely put: the correlation across conditions
should be the same.
– The variance of the differences of each pair must
be equal.
What is Sphericity?
Group A
Group B
Group C
A–B
A–C
B-C
10
12
8
-2
2
5
15
15
12
0
3
3
25
30
20
-5
5
10
35
30
28
5
7
2
30
27
20
3
10
7
Variance
15.7
10.3
10.3
Correlation Matrix
A
B
Machly’s Test of Sphericity W(2) = .919, p =.881
B
C
.929
.977
.929
Effects of a 6 Week Agility Training Program
• 8 subjects were pre-tested for agility.
• The subjects then participated in a 6 week agility
training program.
• Their agility was measured at the following time
points: Pre, 3 weeks, 6 weeks, and a 3 week
• A single factor repeated measures ANOVA with
one within-subjects factor Time (pre, 3 weeks, 6
weeks, follow-up) was used to determine the
effects of the training program on agility.
No Grouping Variable
Repeated measures
factors are placed in
columns, no grouping
variable is needed since
there are no between
subjects factors in this
experiment.
If there is a repeated measures factor in your
experiment you must go to the Repeated
Measures under GLM.
Type the name of the
factor and enter the
number of levels for the
Click define.
This experiment does NOT have a between subjects
factor, checking Homogeneity tests will cause an
error on the printout.
Plots
Add Time on the horizontal axis.
This experiment does NOT have a between subjects
factor, checking Homogeneity tests will cause an
error on the printout.
Sphericity is Violated W(5) = .136, p =.047
If GG > .75 then use HF, otherwise use GG.
Use the GG Row (Sphericity Violated)
If sphericity was not violated we would report F(3,21) = 3.794, p = .026.
But since there was a problem with sphericity we will use the GG adjustments to the F
ratio. F(1.599, 11.19) = 3.794, p = .063
Some researchers might use HF: F(1.997, 13.981) = 3.794, p = .048
3x3 With Two Within Subjects Factors
The purpose of this experiment was to test the effects of diathermy,
ultrasound and ice on joint stiffness. Twenty subjects with arthritis received
treatments in all three conditions (diathermy, ultrasound and ice). The order
of treatment was counterbalanced. The subjects joint stiffness was measured
at the following time points for each treatment: pre, post and followup.
A 3 x 3 repeated measures ANOVA with two within-subjects factors treatment
(diathermy, ultrasound, ice) and time (pre, post, follow up) was used to
compare the effects of the treatments on joint stiffness.
3 x 3 Within – Within Design
Click Paste
Always check
this carefully.
Time and modality have a problem with
sphericity.
Time F(1.181,22.448) = 5.106, p = .029, Modality F(1.594, 30.28) = 122.565, p = .000
Time x Modality F(4,76) = 17.155, p = .000
Time x Modality F(4,76) = 17.155, p = .000
Pre
Post
Follow-Up
Modality
Diathermy
21.05 ± 13.01
25.35 ± 6.74
17.40 ± 7.07
21.27 ± 4.37
Ultrasound
4.45 ± 17.30
-12.00 ± 6.18
-9.20 ± 6.80
-5.58 ± 7.39
Ice
10.00 ± 10.23
11.65 ± 6.24
2.35 ± 6.84
8.00 ± 4.33
Time Means
11.83 ± 11.72
8.33 ± 2.57
3.52 ± 5.13
Time F(1.181,22.448) = 5.106, p = .029
Modality F(1.594, 30.28) = 122.565, p = .000
Time Main Effect Post hoc Tests
Modality Main Effects Post hoc Tests
Simple Effects for Interaction
* Post Diathermy different from Follow-Up Diathermy; † Pre Ultrasound different from: Post Ultrasound
and Follow-Up Ultrasound; ‡ Follow-Up Ice different from: Pre Ice and Post Ice; § Pre Diathermy is
different from: Pre Ultrasound and Pre Ice; || Post Diathermy different from: Post Ultrasound and Post
Ice, Post Ultrasound different from Post Ice; ¶ Follow-Up Diathermy different from: Follow-Up
Ultrasound and Follow-Up Ice, Follow-Up Ultrasound different from Follow-Up Ice; p < 0.05.
Homework
Analyze the Task 1 the book (TutorMarks.sav), see page 504.
Analyze Experiment 3.
Do a Sidak post hoc test instead of the planned contrast
suggested in the book.
Use simple effects testing for a significant interaction.
Use the Sample Methods and Results section as a guide to
write a methods and results section for your homework.
```