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Repeated Measures ANOVA
Cal State Northridge
Andrew Ainsworth PhD
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Major Topics
What are repeated-measures?
An example
Assumptions
Advantages and disadvantages
Effect size
Psy 320 - Cal State Northridge
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Repeated Measures?
Between-subjects designs
- different subjects serve in different
treatment levels
Repeated-measures (RM) designs
- each subject receives all levels of at
least one independent variable
- (what weโre learning today) Psy 320 - Cal State Northridge
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Repeated Measures
All subjects get all treatments.
All subjects receive all levels of the
independent variable.
Different n โs are unusual and cause
problems (i.e. dropout or mortality).
Treatments are usually carried out
one after the other (in serial).
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Example: Counseling For PTSD
Foa, et al. (1991)
- Provided supportive counseling (and other
therapies) to victims of rape
- Do number of symptoms change with
time?
- Thereโs no control group for comparison
- Not a test of effectiveness of supportive counseling Psy 320 - Cal State Northridge
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Example: Counseling For PTSD
9 subjects measured before therapy,
after therapy, and 3 months later
We are ignoring Foaโs other treatment
conditions.
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Partitioning Variability
This partitioning is reflected in the summary table. Psy 320 - Cal State Northridge
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Sums of Squares
The total variability can be partitioned
into Between Groups (e.g. measures),
Subjects and Error Variability
( ) ( ) ( ) ( ) ( )
2 2 2 (^2 )
i GM j j GM i GM i j i GM Total BetweenGroups Subjects Error T BG S Error
Y Y n Y Y g Y Y Y Y g Y Y SS SS SS SS SS SS SS SS
+ ๏ฃฎ๏ฃฏ^ โ โ โ ๏ฃน๏ฃบ
โ โ โ โ โ
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Deviation Sums of Squares
( )
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
(___ ____) (___ ____) (___ ____)
SS Total = Yi โ YGM =
= โ + โ + โ + โ + โ + โ + โ + โ + โ + โ + โ + โ +
โ
L 3 17.593)^2
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Deviation Sums of Squares
( )
2 2 2 2
[*(____ ____) ] [*(____ ____) ]
[__*(13.222 17.593) ] 562.
SS BG = n j Y j โ YGM =
โ
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Deviation Sums of Squares
( )
2 2 2 2 2 2 2 2 2
[*(____ ____) ] [*(____ ____) ]
[*(____ ____) ] [*(20.333 17.593) ]
[*(21.667 17.593) ] [*(21.333 17.593) ]
[*(12.333 17.593) ] [*(19.667 17.593) ]
[__*(10.
SS (^) Subject = g Yi โ YGM =
= โ + โ +
- โ + โ +
- โ + โ +
- โ + โ +
โ
โ 17.593) ]^2 =397.
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Deviation Sums of Squares
( ) ( ) ( )
2 2 (^2 2 2 ) 2 2 2 2 2 2 2 2 2
(___ ____) (___ ____) (___ ____) (26 23.889) (32 23.889) (27 23.889) (21 23.889) (25 23.889) (18 23.889) (15 15.667) (15 15.667) (17 15.667) (
Error i j i GM i j
SS Y Y g Y Y Y Y
= ๏ฃฎ๏ฃฏ^ โ โ โ ๏ฃน๏ฃบ= ๏ฃฐ ๏ฃป โ = โ + โ + โ +
- โ + โ + โ +
- โ + โ + โ +
- โ + โ + โ +
โ โ โ
L 8 13.222)^2 (15 13.222)^2 (3 13.222)^2
552.444 397.852 154.
โ + โ + โ =
= โ = Psy 320 - Cal State Northridge
Degrees of Freedom
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1 3 1 2 ( 1) 27 3 24 1 9 1 8 ( 1) ( 1) 8 2 16 1 27 1 26
bg ws s error total
df g df g n N g df n df n g df N
= โ = โ = = โ = โ = โ = โ = โ = โ = โ = โ โ = = = โ = โ =
From BGANOVA
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Summary Table
Fcrit with 2 and 16 degrees of freedom
is 3.63 we would reject h 0
Source SS df MS F
Time 562.074 2 281.037 29.
WS (from
BG design) 552.444^24
Subject 397.852 8 49.
Error 154.593 16 9.
Total 1114.519 26
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Plot of the Data
0
5
10
15
20
25
30
PreTest PostTest FollowUp
Reported Symptoms
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Interpretation
Note parallel with diagram
Note subject differences not in error
term
Note MSerror is denominator for F on
Time
Note SStime measures what we are
interested in studying
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Assumptions
Correlations between trials are all
equal
- Actually more than necessary, but close
- Matrix shown below Pre Post Followup Pre 1.00 .637. Post 1.00. Followup 1. Psy 320 - Cal State Northridge
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Assumptions
Previous matrix might look like we
violated assumptions
- Only 9 subjects
- Minor violations are not too serious.
Greenhouse and Geisser (1959)
correction (among many)
- Adjusts degrees of freedom Psy 320 - Cal State Northridge
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Effect Size
Simple extension of what we said for t
test for related samples.
Stick to pairs of means.
OR
- ฮท^2 can be used for repeated measures
data as well
- Some adjustments can make it more
meaningful
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