Treatment by Subject Design

1
Treatment by Subject Design
Structural Approach
2
Simple Model
Source E(MS) Subject Condition (T) Error
This is not the model that we use!
3
Complex Model
Source E(MS) Subject Condition (T) Error
This IS the model that we use!
4
Treatment by Subject Design
Treatment
Subject 1 2 j k
1 y11 y12 y1j y1k
2 y21 y22 y2j y2k
i yi1 yi2 yij yik
n yn1 yn2 ynj ynk
GM
5
Correcting for Dependence
Treatment
Subject 1 2 3 T
1 4 4 7 15 5
2 10 9 14 33 11
3 7 8 12 27 9
4 13 14 18 45 15
5 7 6 8 21 7
T 41 41 59 141
8.2 8.2 11.8 9.4
Equate Subject Means
Treatment
Subject 1 2 3 T
1 -1 -1 2 0 0
2 -1 -2 3 0 0
3 -2 -1 3 0 0
4 -2 -1 3 0 0
5 0 -1 1 0 0
T -6 -6 12 0
-1.2 -1.2 2.4 0
6
Treatment
Subject I II III
1 4 4 7 5
2 10 9 14 11
3 7 8 12 9
4 13 14 18 15
5 7 6 8 7
8.2 8.2 11.8 GM9.4
Remove GM effect
Treatment
Subject I II III
1 -5.4 -5.4 -2.4 -4.4
2 0.6 -0.4 4.6 1.6
3 -2.4 -1.4 2.6 -0.4
4 3.6 4.6 8.6 5.6
5 -2.4 -3.4 -1.4 -2.4
-1.2 -1.2 2.4 GM0
7
Treatment by Subject Design
Treatment
Subject 1 2 j k
1 y11 y12 y1j y1k
2 y21 y22 y2j y2k
i yi1 yi2 yij yik
n yn1 yn2 ynj ynk
GM
Treatments are no longer independent. We must
remove (partial out) the dependence.
Treatment
Subject 1 2 3
1 4 4 7
2 10 9 14
3 7 8 12
4 13 14 18
5 7 6 8
8
Remove Subject Effects
Treatment
Subject I II III
1 -1 -1 2 0
2 -1 -2 3 0
3 -2 -1 3 0
4 -2 -1 3 0
5 0 -1 1 0
-1.2 -1.2 2.4 GM0
9
Remove Treatment Effects
Treatment
Subject I II III
1 0.2 0.2 -0.4 0
2 0.2 -0.8 0.6 0
3 -0.8 0.2 0.6 0
4 -0.8 0.2 0.6 0
5 1.2 0.2 -1.4 0
0 0 0 GM0
SSTreat ?
SSbs ?
10
TxS or Repeated Measures
df
11
Repeated Measures Design
Partitioning SSTotal
SSTotal
SSbs
SSws
12
SSws
sum across subjects in the jth condition
continued
13
SSws (continued)
now, sum across treatments or conditions.
SSws

SSTreat SSres
(k-1)df (n-1)(k-1)df
n(k-1)df

14
SSws df
Subject
1
2
3
4
5
T
15 5
33 11
27 9
45 15
21 7





The mean for each subject is fixed. With 3
treatments we have 2df within each subject. We
have 5 subjects.
Thus, 2 x 5 10df
15
SSTreat df
Treatment
Subject
1
2
3
4
5
T

1 2 3





41 41 59
8.2 8.2 11.8 GM9.4
The GM is fixed. With 3 treatments only two are
free to vary. Thus, 3 1 2df
16
SSres df
Treatment
Subject
1
2
3
4
5
T
15 5
33 11
27 9
45 15
21 7
1 2 3





41 41 59
8.2 8.2 11.8
T

GM9.4
The mean for each subject is fixed. Thus, (k-1)
The mean for each treatment is fixed. Thus,
(n-1)
The GM is fixed. But nothing is lost since all
treatment means are fixed already. Thus,
(k-1)(n-1)
17
Treatment by Subject Design
SSws
SSbs
SSTreat
SSres(error)
18
Treatments
Subjects 1 x2 2 x2 3 x2 T
1 4 16 4 16 7 49 15
2 10 100 9 81 14 196 33
3 7 49 8 64 12 144 27
4 13 149 14 196 18 324 45
5 7 49 6 36 8 64 21
T 41 41 59 141

SSTotal

227.6

SSbs

177.6

SSTreat

43.2
SS(res)error


6.8
SSws SSTotal SSbs or SSTreat SSerror