Factorial Within
Subjects
Psy 420
Ainsworth
Factorial WS Designs

Analysis

Factorial – deviation and computational
Power, relative efficiency and sample size
 Effect size
 Missing data
 Specific Comparisons

Example for Deviation
Approach
b2: Mystery
b1: Science Fiction
a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3
Case
Means
S1
S2
S3
S4
1
1
3
5
3
4
3
5
6
8
6
7
5
8
4
3
4
8
5
2
1
4
3
0
S1 = 3.333
S2 = 5.500
S3 = 4.000
S4 = 3.667
S5
2
4
5
5
6
3
S5 = 4.167
a2b1 = 3.8
a3b1 = 6.4
a1b2 = 5
a2b2 = 5
Treatment
a1b1 = 2.4
Means
a3b2 = 2.2 GM = 4.133
Analysis – Deviation

What effects do we have?
A
B
 AB
S
 AS
 BS
 ABS
T

Analysis – Deviation

DFs
DFA = a – 1
 DFB = b – 1
 DFAB = (a – 1)(b – 1)
 DFS = (s – 1)
 DFAS = (a – 1)(s – 1)
 DFBS = (b – 1)(s – 1)
 DFABS = (a – 1)(b – 1)(s – 1)
 DFT = abs - 1

Sums of Squares - Deviation

The total variability can be partitioned
into A, B, AB, Subjects and a Separate
Error Variability for Each Effect
SSTotal  SS A  SS B  SS AB  SS S  SS AS  SS BS  SS ABS
  Y  Y    n Y  Y    n Y  Y  
   n Y  Y    n Y  Y    n Y  Y  


2
ijk
2
...
j
. j.
2
...
k
2
jk
. jk
...
..k
...
2
j
. j.
...
2
k
..k
...
2
2

 jk  Yi..  Y...    k  Yij .  Y. j .   jk  Yi..  Y...   


2
2

  j  Yi.k  Y. j.   jk  Yi..  Y...   


2
2
2
2
2

   Yijk  Y. jk   jk  Yi..  Y...   k  Yij .  Y. j .   j  Yi.k  Y. j .  


Analysis – Deviation
Before we can calculate the sums of
squares we need to rearrange the data
 When analyzing the effects of A (Month) you
need to AVERAGE over the effects of B
(Novel) and vice versa
 The data in its original form is only useful for
calculating the AB and ABS interactions

For the A and AS effects

Remake data, averaging over B
a1: Month 1 a2: Month 2 a3: Month 3 Case Means
3
4.5
3.5
6
3.5
6
S1 = 3.333
S2 = 5.500
S4
3.5
4
4
3.5
4.5
3.5
S3 = 4.000
S4 = 3.667
S5
3.5
5
4
S5 = 4.167
Treatment
Means
a1 = 3.7
a2 = 4.4
a3 = 4.3
GM = 4.133
S1
S2
S3
SS A   n j Y. j .  Y...   10*[ 3.7  4.133   4.4  4.133   4.3  4.133  ]  2.867
2
2
2
2
SS S  jk  Yi..  Y...   (3*2)*[ 3.333  4.133   5.5  4.133    4  4.133  
2
2
2
2
  3.667  4.133   4.167  4.133   16.467
2
2
SS AS  k  Yij .  Y. j .   jk  Yi..  Y...  
2
2
k  Yij .  Y. j .   2*[ 3  3.7    4.5  3.7    3.5  3.7    4  3.7    3.5  3.7  
2
2
2
2
2
  3.5  4.4    6  4.4    4  4.4    3.5  4.4    5  4.4  
2
2
2
2
2
  3.5  4.3   6  4.3   4.5  4.3   3.5  4.3   4  4.3 ]  20.6
2
2
SS AS  20.6  16.467  4.133
2
2
2
2
For B and BS effects

Remake data, averaging over A
b1: Science
b2: Mystery Case Means
Fiction
S4
3.333
4.333
4.000
5.667
3.333
6.667
4.000
1.667
S1 = 3.333
S2 = 5.500
S3 = 4.000
S4 = 3.667
S5
3.667
4.667
S5 = 4.167
Treatment
Means
b1 = 4.200
b2 = 4.067
GM = 4.133
S1
S2
S3
SS B   nk Y..k  Y...   15*[ 4.2  4.133   4.067  4.133  ]  .133
2
2
2
SS BS  j  Yi.k  Y..k   jk  Yi..  Y...  
2
2
j  Yi.k  Y..k   3*[ 3.333  4.2    4.333  4.2    4  4.2  
2
2
2
2
  5.667  4.2    3.667  4.2    3.333  4.067    6.667  4.067  
2
2
2
2
  4  4.067   1.667  4.067    4.667  4.067  ]  50
2
2
SS S  jk  Yi..  Y...   16.467
2
SS BS  50  16.467  33.533
2
For AB and ABS effects

Use the data in its original form
b2: Mystery
b1: Science Fiction
a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3
Case
Means
S1
S2
1
1
3
4
6
8
5
8
4
8
1
4
S1 = 3.333
S2 = 5.500
S3
S4
3
5
3
5
6
7
4
3
5
2
3
0
S3 = 4.000
S4 = 3.667
S5
2
4
5
5
6
3
S5 = 4.167
a2b1 = 3.8
a3b1 = 6.4
a1b2 = 5
a2b2 = 5
Treatment
a1b1 = 2.4
Means
a3b2 = 2.2 GM = 4.133
SS AB   n jk Y. jk  Y...    n j Y. j .  Y...    nk Y..k  Y...  
2
 n Y
jk
2
 Y...   5*[ 2.4  4.133   3.8  4.133   6.4  4.133
2
. jk
2
2
2
  5  4.133   5  4.133   2.2  4.133 ]  67.467
2
2
 n Y
 Y...   SS A  2.867
 n Y
 Y...   SS B  .133
j
k
. j.
..k
2
2
SS AB  67.467  2.867  .133  64.467
2
2
2
2
2
2
SS ABS    Yijk  Y. jk   jk  Yi ..  Y...   k  Yij .  Y. j .   j  Yi.k  Y. j .   


 Yijk  Y. jk   1  2.4   1  2.4    3  2.4  
2
2
2
2
  5  2.4    2  2.4    3  3.8    4  3.8  
2
2
2
2
  3  3.8    5  3.8    4  3.8    6  6.4  
2
2
2
2
  8  6.4    6  6.4    7  6.4    5  6.4  
2
2
2
2
  6  5  8  5   4  5   3  5 
2
2
2
2
  5  5   4  5  8  5   5  5 
2
2
2
2
  2  5    6  5   1  2.2    4  2.2  
2
2
2
2
  3  2.2    0  2.2    3  2.2   64
2
2
2
SS ABS  64  16.467  20.6  50  9.867
SSTotal  1  4.133  1  4.133   3  4.133 
2
2
2
  5  4.133   2  4.133   3  4.133   4  4.133 
2
2
2
2
  3  4.133   5  4.133   4  4.133   6  4.133  
2
2
2
2
  8  4.133   6  4.133   7  4.133   5  4.133  
2
2
2
2
  6  4.133   8  4.133   4  4.133   3  4.133  
2
2
2
2
  5  4.133   4  4.133   8  4.133   5  4.133 
2
2
2
2
  2  4.133   6  4.133  1  4.133   4  4.133 
2
2
2
  3  4.133   0  4.133   3  4.133  131.467
2
2
2
2
Analysis – Deviation

DFs
DFA = a – 1 = 3 – 1 = 2
 DFB = b – 1 = 2 – 1 = 1
 DFAB = (a – 1)(b – 1) = 2 * 1 = 2
 DFS = (s – 1) = 5 – 1 = 4
 DFAS = (a – 1)(s – 1) = 2 * 4 = 8
 DFBS = (b – 1)(s – 1) = 1 * 4 = 4
 DFABS = (a – 1)(b – 1)(s – 1) = 2 * 1 * 4 = 8
 DFT = abs – 1 = [3*(2)*(5)] – 1 = 30 – 1 = 29

Source Table - Deviation
Source
A
AS
B
BS
AB
ABS
S
Total
SS
2.867
4.133
0.133
33.533
64.467
9.867
16.467
131.467
df
2
8
1
4
2
8
4
29
MS
1.433
0.517
0.133
8.383
32.233
1.233
4.117
F
2.774
0.016
26.135
Analysis – Computational

Example data re-calculated with totals
b1: Science Fiction
b2: Mystery
Case
a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3 Totals
S1
1
3
6
5
4
1
S1=20
S2
1
4
8
8
8
4
S2=33
S3
3
3
6
4
5
3
S3=24
S4
5
5
7
3
2
0
S4=22
S5
Treatment
Totals
2
4
5
5
6
3
S5=25
a1b1 = 12
a2b1 = 19
a3b1 = 32
a1b2 = 25
a2b2 = 25
a3b2 = 11 T = 124
Analysis – Computational
Before we can calculate the sums of
squares we need to rearrange the data
 When analyzing the effects of A (Month) you
need to SUM over the effects of B (Novel)
and vice versa
 The data in its original form is only useful for
calculating the AB and ABS interactions

Analysis – Computational

For the Effect of A (Month) and A x S
Month
Case
a1: Month 1 a2: Month 2 a3: Month 3 Totals
S1
6
7
7
S1=20
S2
9
12
12
S2=33
S3
7
8
9
S3=24
S4
8
7
7
S4=22
S5
Treatment
Totals
7
10
8
S5=25
37
44
43
T = 124
a 




2
T 2 37 2  442  432
1242
SS A




bs
abs
2(5)
2(3)(5)
5154 15376
SS A 

 515.4  512.533  2.867
10
30
j
SS S 
   si 
2
T 2 202  332  242  222  252


 512.533 
abs
2(3)
ab
3174
SS S 
 512.533  529  512.533  16.467
6
   a j si 
2
  a j 
2
   si 
2
T2
SS AS 




b
bs
ab
abs
62  92  7 2  82  7 2  7 2  122  82  7 2  102  7 2  12 2  9 2  7 2  82
SS AS 

2
1072
515.4  529  512.533 
 515.4  529  512.533 
2
SS AS  536  515.4  529  512.533  4.133
Analysis – Traditional

For B (Novel) and B x S
Type of Novel
b1: Science Fiction
b2: Mystery
Case
Totals
S1
10
10
S1=20
S2
13
20
S2=33
S3
12
12
S3=24
S4
17
5
S4=22
S5
Treatment
Totals
11
14
S5=25
63
61
T = 124
SS B
b 




2
k
as
T 2 632  612


 512.533 
abs
3(5)
7690
SS B
 512.533  512.667  512.533  .133
15
b s    b    s 






2
2
2
T2
SS BS


a
as
ab
abs
102  132  122  17 2  112  102  202  122  52  142
SS BS 

3
512.667  529  512.533 
1688
SS BS 
 512.667  529  512.533
3
SS BS  562.667  512.667  529  512.533  33.533
k i
k
i
Analysis – Computational

Example data re-calculated with totals
b1: Science Fiction
b2: Mystery
Case
a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3 Totals
S1
1
3
6
5
4
1
S1=20
S2
1
4
8
8
8
4
S2=33
S3
3
3
6
4
5
3
S3=24
S4
5
5
7
3
2
0
S4=22
S5
Treatment
Totals
2
4
5
5
6
3
S5=25
a1b1 = 12
a2b1 = 19
a3b1 = 32
a1b2 = 25
a2b2 = 25
a3b2 = 11 T = 124
a b    a    b 






2
SS AB
SS AB
SS AB
SS AB
j k
j
k
2
T2


abs
s
bs
as
122  192  322  252  252  112

 515.4  512.667  512.533 
5
2900

 515.4  512.667  512.533 
5
 580  515.4  512.667  512.533  64.467
SS ABS   Y 2 

2
  a j 
SS ABS
SSTotal
2
   a j bk 
s
2

   a j si 
b
b    s 





2
2

2
   bk si 
a
2

T2


bs
as
ab
abs
 644  580  536  562.667  515.4  512.667  529  512.533  9.867
k
i
T2
 Y 
 644  512.533  131.467
abs
2
Analysis – Computational
DFs are the same

DFs
DFA = a – 1 = 3 – 1 = 2
 DFB = b – 1 = 2 – 1 = 1
 DFAB = (a – 1)(b – 1) = 2 * 1 = 2
 DFS = (s – 1) = 5 – 1 = 4
 DFAS = (a – 1)(s – 1) = 2 * 4 = 8
 DFBS = (b – 1)(s – 1) = 1 * 4 = 4
 DFABS = (a – 1)(b – 1)(s – 1) = 2 * 1 * 4 = 8
 DFT = abs – 1 = [3*(2)*(5)] – 1 = 30 – 1 = 29

Source Table – Computational
Is also the same
Source
A
AS
B
BS
AB
ABS
S
Total
SS
2.867
4.133
0.133
33.533
64.467
9.867
16.467
131.467
df
2
8
1
4
2
8
4
29
MS
1.433
0.517
0.133
8.383
32.233
1.233
4.117
F
2.774
0.016
26.135
Relative Efficiency
One way of conceptualizing the power of a
within subjects design is to calculate how
efficient (uses less subjects) the design is
compared to a between subjects design
 One way of increasing power is to minimize
error, which is what WS designs do,
compared to BG designs
 WS designs are more powerful, require less
subjects, therefore are more efficient

Relative Efficiency
Efficiency is not an absolute value in terms
of how it’s calculated
 In short, efficiency is a measure of how
much smaller the WS error term is
compared to the BG error term
 Remember that in a one-way WS design:
SSAS=SSS/A - SSS

Relative Efficiency
MSS / A  df AS  1  df S / A  3 
Relative Efficiency =



MS AS  df AS  3  df S / A  1 

MSS/A can be found by either re-running the
analysis as a BG design or for one-way:
SSS + SSAS = SSS/A and
dfS + dfAS = dfS/A
MSS / A
s  1 MSS   a  1 s  1 MS AS


as  1
Relative Efficiency

From our one-way example:
Source
A
S
AxS
Total
SS
41.20
9.70
9.50
60.40
df
2
4
8
14
MS
20.60
2.43
1.19
F
17.35
2.04
SS S / A  SS S  SS AS  9.7  9.5  19.2
df S / A  df S  df AS  4  8  12
MS S / A  19.2 /12  1.6
Relative Efficiency
1.6  8  1  12  3 
Relative Efficiency =


  1.26
1.2  8  3  12  1 
From the example, the within subjects design
is roughly 26% more efficient than a between
subjects design (controlling for degrees of
freedom)
 For # of BG subjects (n) to match WS (s)
efficiency:

n  s(relative efficiency)  5(1.26)  6.3  7
Power and Sample Size
Estimating sample size is the same from before
 For a one-way within subjects you have and AS
interaction but you only estimate sample size
based on the main effect
 For factorial designs you need to estimate
sample size based on each effect and use the
largest estimate
 Realize that if it (PC-Size, formula) estimates 5
subjects, that total, not per cell as with BG
designs

Effects Size

Because of the risk of a true AS interaction
we need to estimate lower and upper bound
estimates
SS A
SS A
  

SSTotal SS A  SS S  SS AS
2
2
L
SS A
SS A
partial  

SS A  SSerror SS A  SS AS
2
df A ( MS A  MS AS )
ˆ 
df A ( MS A  MS AS )  as ( MS AS )  s ( MS S )
2
L
partial ˆ 2 
df A ( MS A  MS AS )
df A ( MS A  MS AS )  as ( MS AS )
Missing Values
In repeated measures designs it is often the
case that the subjects may not have scores
at all levels (e.g. inadmissible score, dropout, etc.)
 The default in most programs is to delete the
case if it doesn’t have complete scores at all
levels
 If you have a lot of data that’s fine
 If you only have a limited cases…

Missing Values

Estimating values for missing cases in WS designs
(one method: takes mean of Aj, mean for the case
and the grand mean)
Yij* 
sSi'  aA'j  T '
(a  1)( s  1)
where :
Yij*  predicted score to replace missing score
s  number of cases
Si'  sum of known values for that case
a  number of levels of A
A'j  sum of known values for A
T '  sum of all known values
Specific Comparisons

F-tests for comparisons are exactly the
same as for BG
n(  w jY j ) /  w
2
F
2
j
MSerror
Except that MSerror is not just MSAS, it is
going to be different for every comparison
 It is recommended to just calculate this
through SPSS
