Factorial Within Subjects
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Transcript of Factorial Within Subjects
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Factorial Within Subjects
Psy 420
Ainsworth
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Factorial WS Designs
AnalysisFactorial – deviation and computational
Power, relative efficiency and sample size Effect size Missing data Specific Comparisons
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Example for Deviation Approach
b1: Science Fiction b2: Mystery Case
Means a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3
S1 1 3 6 5 4 1 S1 = 3.333 S2 1 4 8 8 8 4 S2 = 5.500 S3 3 3 6 4 5 3 S3 = 4.000 S4 5 5 7 3 2 0 S4 = 3.667
S5 2 4 5 5 6 3 S5 = 4.167
Treatment Means a1b1 = 2.4 a2b1 = 3.8 a3b1 = 6.4 a1b2 = 5 a2b2 = 5 a3b2 = 2.2 GM = 4.133
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Analysis – Deviation
What effects do we have?ABABSASBSABST
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Analysis – Deviation
DFsDFA = 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
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Sums of Squares - Deviation
2 2 2
... . . ... .. ...
2 2 2
. ... . . ... .. ...
22 2
.. ... . . . .. ...
2
. . . .. ..
Total A B AB S AS BS ABS
ijk j j k k
jk jk j j k k
i ij j i
i k j i
SS SS SS SS SS SS SS SS
Y Y n Y Y n Y Y
n Y Y n Y Y n Y Y
jk Y Y k Y Y jk Y Y
j Y Y jk Y Y
2
.
2 2 22
. .. ... . . . . . .ijk jk i ij j i k jY Y jk Y Y k Y Y j Y Y
The total variability can be partitioned into A, B, AB, Subjects and a Separate Error Variability for Each Effect
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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
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For the A and AS effects
Remake data, averaging over B
a1: Month 1 a2: Month 2 a3: Month 3 Case Means
S1 3 3.5 3.5 S1 = 3.333 S2 4.5 6 6 S2 = 5.500 S3 3.5 4 4.5 S3 = 4.000 S4 4 3.5 3.5 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
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2 2 2 2
. . ...
2 2 2 2
.. ...
2 2
2 2
. . . .. ...
2
. . .
10*[ 3.7 4.133 4.4 4.133 4.3 4.133 ] 2.867
(3*2)*[ 3.333 4.133 5.5 4.133 4 4.133
3.667 4.133 4.167 4.133 16.467
2*[
A j j
S i
AS ij j i
ij j
SS n Y Y
SS jk Y Y
SS k Y Y jk Y Y
k Y Y
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
3 3.7 4.5 3.7 3.5 3.7 4 3.7 3.5 3.7
3.5 4.4 6 4.4 4 4.4 3.5 4.4 5 4.4
3.5 4.3 6 4.3 4.5 4.3 3.5 4.3 4 4.3 ] 20.6
20.6 16.467 4.133ASSS
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For B and BS effects Remake data, averaging over A
b1: Science Fiction
b2: Mystery Case Means
S1 3.333 3.333 S1 = 3.333 S2 4.333 6.667 S2 = 5.500 S3 4.000 4.000 S3 = 4.000 S4 5.667 1.667 S4 = 3.667 S5 3.667 4.667 S5 = 4.167
Treatment Means b1 = 4.200 b2 = 4.067 GM = 4.133
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2 2 2
.. ...
2 2
. .. .. ...
2 2 2 2
. ..
2 2 2 2
2 2
15*[ 4.2 4.133 4.067 4.133 ] .133
3*[ 3.333 4.2 4.333 4.2 4 4.2
5.667 4.2 3.667 4.2 3.333 4.067 6.667 4.067
4 4.067 1.667 4.067
B k k
BS i k k i
i k k
SS n Y Y
SS j Y Y jk Y Y
j Y Y
2
2
.. ...
4.667 4.067 ] 50
16.467
50 16.467 33.533
S i
BS
SS jk Y Y
SS
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For AB and ABS effects
Use the data in its original form
b1: Science Fiction b2: Mystery Case
Means a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3
S1 1 3 6 5 4 1 S1 = 3.333 S2 1 4 8 8 8 4 S2 = 5.500 S3 3 3 6 4 5 3 S3 = 4.000 S4 5 5 7 3 2 0 S4 = 3.667
S5 2 4 5 5 6 3 S5 = 4.167
Treatment Means a1b1 = 2.4 a2b1 = 3.8 a3b1 = 6.4 a1b2 = 5 a2b2 = 5 a3b2 = 2.2 GM = 4.133
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2 2 2
. ... . . ... .. ...
2 2 2 2
. ...
2 2 2
2
. . ...
2
.. ...
5*[ 2.4 4.133 3.8 4.133 6.4 4.133
5 4.133 5 4.133 2.2 4.133 ] 67.467
2.867
.133
67.467 2.867 .1
AB jk jk j j k k
jk jk
j j A
k k B
AB
SS n Y Y n Y Y n Y Y
n Y Y
n Y Y SS
n Y Y SS
SS
33 64.467
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2 2 22
. .. ... . . . . . .
2 2 2 2
.
2 2 2 2
2 2 2 2
2 2 2 2
2 2 2
1 2.4 1 2.4 3 2.4
5 2.4 2 2.4 3 3.8 4 3.8
3 3.8 5 3.8 4 3.8 6 6.4
8 6.4 6 6.4 7 6.4 5 6.4
6 5 8 5 4 5
ABS ijk jk i ij j i k j
ijk jk
SS Y Y jk Y Y k Y Y j Y Y
Y Y
2
2 2 2 2
2 2 2 2
2 2 2
3 5
5 5 4 5 8 5 5 5
2 5 6 5 1 2.2 4 2.2
3 2.2 0 2.2 3 2.2 64
64 16.467 20.6 50 9.867ABSSS
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2 2 2
2 2 2 2
2 2 2 2
2 2 2 2
2 2 2 2
2
1 4.133 1 4.133 3 4.133
5 4.133 2 4.133 3 4.133 4 4.133
3 4.133 5 4.133 4 4.133 6 4.133
8 4.133 6 4.133 7 4.133 5 4.133
6 4.133 8 4.133 4 4.133 3 4.133
5 4.133 4 4.133
TotalSS
2 2 2
2 2 2 2
2 2 2
8 4.133 5 4.133
2 4.133 6 4.133 1 4.133 4 4.133
3 4.133 0 4.133 3 4.133 131.467
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Analysis – Deviation
DFsDFA = 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
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Source SS df MS FA 2.867 2 1.433 2.774AS 4.133 8 0.517B 0.133 1 0.133 0.016
BS 33.533 4 8.383AB 64.467 2 32.233 26.135ABS 9.867 8 1.233
S 16.467 4 4.117Total 131.467 29
Source Table - Deviation
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Analysis – Computational
b1: Science Fiction b2: Mystery Case
Totals a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3
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 2 4 5 5 6 3 S5=25
Treatment Totals
a1b1 = 12 a2b1 = 19 a3b1 = 32 a1b2 = 25 a2b2 = 25 a3b2 = 11 T = 124
Example data re-calculated with totals
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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
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Analysis – Computational
For the Effect of A (Month) and A x S
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22 2 2 2 2
22 2 2 2 2 2
2 2 2
37 44 43 124
2(5) 2(3)(5)
5154 15376515.4 512.533 2.867
10 30
20 33 24 22 25512.533
2(3)
3174512.533 529 512.533 16.467
6
j
A
A
i
S
S
j i j i
AS
a TSS
bs abs
SS
s TSS
ab abs
SS
a s a sSS
b bs ab
2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 26 9 7 8 7 7 12 8 7 10 7 12 9 7 8
21072
515.4 529 512.533 515.4 529 512.5332
536 515.4 529 512.533 4.133
AS
AS
T
abs
SS
SS
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Analysis – Traditional
For B (Novel) and B x S
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22 2 2
2 2 22
2 2 2 2 2 2 2 2 2 2
63 61512.533
3(5)
7690512.533 512.667 512.533 .133
15
10 13 12 17 11 10 20 12 5 14
3512.667 529 512.533
1688512.667 529
3
k
B
B
k i k i
BS
BS
BS
b TSS
as abs
SS
b s b s TSS
a as ab abs
SS
SS
512.533
562.667 512.667 529 512.533 33.533BSSS
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Analysis – Computational
b1: Science Fiction b2: Mystery Case
Totals a1: Month 1 a2: Month 2 a3: Month 3 a1: Month 1 a2: Month 2 a3: Month 3
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 2 4 5 5 6 3 S5=25
Treatment Totals
a1b1 = 12 a2b1 = 19 a3b1 = 32 a1b2 = 25 a2b2 = 25 a3b2 = 11 T = 124
Example data re-calculated with totals
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2 2 22
2 2 2 2 2 2
2 2 2
2
2
12 19 32 25 25 11515.4 512.667 512.533
52900
515.4 512.667 512.5335
580 515.4 512.667 512.533 64.467
j k j k
AB
AB
AB
AB
j k j i k i
ABS
j
a b a b TSS
s bs as abs
SS
SS
SS
a b a s b sSS Y
s b a
a
2 22
22
644 580 536 562.667 515.4 512.667 529 512.533 9.867
644 512.533 131.467
k i
ABS
Total
b s T
bs as ab absSS
TSS Y
abs
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Analysis – ComputationalDFs are the same
DFsDFA = 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
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Source Table – ComputationalIs also the same
Source SS df MS FA 2.867 2 1.433 2.774AS 4.133 8 0.517B 0.133 1 0.133 0.016
BS 33.533 4 8.383AB 64.467 2 32.233 26.135ABS 9.867 8 1.233
S 16.467 4 4.117Total 131.467 29
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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
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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
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Relative Efficiency
/ /
/
1 3Relative Efficiency =
3 1S A AS S A
AS AS S A
MS df df
MS df df
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
/
1 1 1
1S AS
S A
s MS a s MSMS
as
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Relative Efficiency
From our one-way example:
/
/
/
9.7 9.5 19.2
4 8 12
19.2 /12 1.6
S A S AS
S A S AS
S A
SS SS SS
df df df
MS
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Relative Efficiency
1.6 8 1 12 3Relative 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:
(relative efficiency) 5(1.26) 6.3 7n s
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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
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Effects Size
2 2
2
2
2
( )ˆ
( ) ( ) ( )
( )ˆ
( ) ( )
A AL
Total A S AS
A A
A error A AS
A A ASL
A A AS AS S
A A AS
A A AS AS
SS SS
SS SS SS SS
SS SSpartial
SS SS SS SS
df MS MS
df MS MS as MS s MS
df MS MSpartial
df MS MS as MS
Because of the risk of a true AS interaction we need to estimate lower and upper bound estimates
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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, drop-out, 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…
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Missing Values Estimating values for missing cases in WS designs
(one method: takes mean of Aj, mean for the case and the grand mean)
' ' '*
*
'
'
'
( 1)( 1)
:
predicted score to replace missing score
number of cases
sum of known values for that case
number of levels of A
sum of known values for A
sum of all kn
i jij
ij
i
j
sS aA TY
a s
where
Y
s
S
a
A
T
own values
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Specific Comparisons
F-tests for comparisons are exactly the same as for BG
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
2 2( ) /j j j
error
n w Y wF
MS