Transcript Factorial Design

Factorial Design
One Between-Subject Variable
One Within-Subject Variable
SSTotal
SSbetween subjects
Groups
Subjects
within groups
Differences Between
Subjects
SSwithin subjects
Treatments
by Groups
Treatments
Treatments by
Subjects within
groups
Differences Within
Subjects
Groups – differences between groups of subjects
SS w/in Groups – differences between subjects w/in a group
Treatment – differences between subject’s scores across treatments
Treat x Groups – interaction between Treatments and Groups
Treats x Ss w/in Groups – interaction between Subjects and
Treatments hold Groups factor constant
SStotal   ( yij  GM )2
SSbsub  k  ( yi  GM ) 2
SS wsub  SStotal  SSbsub
SScells  n ( yij  GM ) 2
SS grps  kn ( ygrp  GM ) 2
SS wgrps  SSbsub  SS grps
SStreat  gn ( y· j  GM ) 2
SS treat x grps  SS cells  SStreat  SS grps
SS treat x sub(wgrp)  SS w / sub  SStreat  SS treat x grp
  ( yij yi·  y· j  GM ) 2
Example
Speed (Repeated Measure)
G
r
o
u
p
1
G
r
o
u
p
2
Sub #
5
15
25
35
y
1
7
7
4
3
5.25
2
8
8
5
6
6.75
3
9
7
4
3
5.75
4
8
6
3
3
5.00
Trc
32
28
16
15
y rc
8.0
7.0
4.0
3.75
5.69
1
10
5
2
1
4.5
2
10
6
3
2
5.25
3
9
5
4
2
5.00
4
11
6
3
2
5.5
Trc
40
22
12
7
y rc
10
5.5
3
1.75
5.06
Tc
72
50
28
22
172
yc
9
6.25
3.5
2.75
5.38
=GT
=GM
Calculate MS
Divide SS by appropriate df
SSbs by #Ss - 1
SSgrp by #Grps - 1
SSss w/in grps by (#Singrp-1) x (# of grps)
SSws by #Ss (# Treatments – 1)
SStreat by # Treatments - 1
SSTxG by (#grp – 1) (#Treats -1)
SSTxS w/in grpsby (#Treats -1) x (n-1) x (# of grps)
7
Prepare Summary Table
Source
Btwn S
SS
Df
12.5
7
MS
Grp
3.125
1
3.125
Ss w/in
Grp
9.375
6
1.563
Within S
223
194.5
3
TxG
19.375
3
6.458
TxS
w/in grp
9.125
18
.507
235.531
Interpolation?
2
n.s.
64.833 127.89 < 0.01
What are the appropriate error terms?
(the denominators for the Fratios)
8
P
24
Treat
Total
F
12.74 < 0.01
Repeated Measures Assumptions
1)
normality
2)
homogeneity of variance
3)
compound symmetry
- constant variances on diagonal
- constant covariances off diagonal
A variance / covariance matrix for each group and overall
( x  x )( y  y )

Cov 
n 1
Cov 
Cov 
4)

x y

xy 
N 1
 TiT j
N
T T


i
n 1
j
n
T X Ss interactions are constant across groups
- test with Fmax
Example
No STRAT
5
15
25
35
Var/Covar Matrix
Speed
5
15
25
35
0.66
0
0
0
0.66
0.66
1.0
0.66
1.0
2.25
The assumption of compound symmetry is usually replaced
by the assumption of sphericity
 y2Ti  yT j =

2
y1 y2
a constant across all pairs of conditions



2
1
N1
2
T5 T15


2
2
N2
.66 .66


4
4

2
T5 T35
= .574
.66 2.25
= .853


4
4
Simple Effects
One B-S variable
One W-S variable
Factorial Design
The W-S variable
- Separate One-Way ANOVAs (repeated measures)
∙ Error terms pooled = MS T X Ss w/in groups
∙ Or, use the MST X Ss for each separate analysis
No STRAT
STRAT
SSTotal = 67.44
SSTotal = 168.04
SSbs
SSbs
=
7.14
=
5.29
SSTreat = 54.69
SSTreat = 159.19
SSerror =
SSerror =
5.56
3.56
No STRAT
SSTotal
67.44
STRAT
+
168.04
=
235.48
SSTotal
(overall)
SSbs
7.19
+
SSTreat
54.69
+
df  3
5.29
=
159.19
=
12.45
SSbs
(overall)
213.88
SSTreat
df  3
+
SST X G
(overall)
Why?
SSerror
5.56
df  9
+
3.56
df  9
=
9.12
SST X S w/in group
(overall)
Between-Subjects Simple
Effects
We could do a separate analysis of each level
- unnecessary loss of df
SSgrp at 5
=
=
8.0
SSgrp at 15
=
=
4.5
SSgrp at 25
=
=
2.0
SSgrp at 35
=
=
8.0
MS all 1 df
SSerror term = SSw/cells = SSSs w/in grp + SS T X Ss w/in grps
Why?
MSerror
=
SS w/in cells
dfSw/in grp  df T X Sw/in grps
SS error  18 .5
df  24
MSerror  .77