1. Treatment by Subject Design
Structural Approach

2. This is not the model that we use!
Simple Model
Source E(MS)
Subject
Condition (T)
Error
This is not the model that we use!

3. This IS the model that we use!
Complex Model
Source E(MS)
Subject
Condition (T)
Error
This IS the model that we use!

4. Treatment by Subject Design1 2 j k
y11
y12
y1j
y1k
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 4 7 15 5 10 9 14 33 11 8 12 27 13 18 45 6 21 41 59
141
8.2
11.8
9.4
Equate Subject Means
Treatment
Subject 1 2 3 T -1 -2 4 5 -6 12
-1.2
2.4

6. Remove GM effect Treatment Subject I II III 1 4 7 5 2 10 9 14 11 3 812 13 18 15 6
8.2
11.8
GM=9.4
Remove GM effect
Treatment
Subject I II
III 1
-5.4
-2.4
-4.4 2
0.6
-0.4
4.6
1.6 3
-1.4
2.6 4
3.6
8.6
5.6 5
-3.4
-1.2
2.4
GM=0

7. Treatment by Subject Design1 2 j k
y11
y12
y1j
y1k
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 4 7 10 9 14 8 12 13 18 5 6

8. Remove Subject Effects
Treatment
Subject I II
III 1 -1 2 -2 3 4 5
-1.2
2.4
GM=0

9. Remove Treatment Effects
Subject I II
III 1
0.2
-0.4 2
-0.8
0.6 3 4 5
1.2
-1.4
GM=0
SSTreat = ?
SSbs = ?

10. TxS or Repeated Measuresdf

11. Repeated Measures Design
Partitioning SSTotal
SSTotal
SSws
SSbs

12. sum across subjects in the jth condition
SSws
sum across subjects in the jth condition
continued…

13. now, sum across treatments or conditions.
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 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 The GM is fixed.
Treatment
Subject 1 2 3 4 5 T 1 2 3 41 59
8.2
11.8
GM=9.4
The GM is fixed.
With 3 treatments only two are free to vary.
Thus, 3 – 1 = 2df

16. SSres df The mean for each subject is fixed. Thus, (k-1)
Treatment
Subject 1 2 3 4 5 1 2 3 T 15 5 33 11 27 9 45 21 7 T 41 59
8.2
11.8
GM=9.4
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
SSbs
SSws
SSTreat
SSres(error)

18. SSws = SSTotal – SSbs or SSTreat + SSerror
Treatments
Subjects 1 x2 2 3 T 4 16 7 49 15 10
100 9 81 14
196 33 8 64 12
144 27 13
149 18
324 45 5 6 36 21 41 59
141 =
SSTotal =
= 227.6 =
SSbs =
= 177.6 =
SSTreat =
= 43.2
SS(res)error = =
= 6.8
SSws = SSTotal – SSbs or SSTreat + SSerror