Introduction to Fractional Factorial Designs
(1) FACTORS (display location, display height)
have LEVELS (end of aisle EOA, not end NE;
lower shelf L, upper shelf U)
(2) LOCATION Effect on SALES (=Y)
SALES for 1 week, 8 stores
EOA: 50, 46, 67, 69
mean 58, SSq = 64+144+81+121 = 410 (3 df)
NE: 47, 43, 50, 52
mean 48 SSq 1+25+4+16=46 (3 df)
Error sum of squares 410+46=456
Error degrees of freedom 3+3=6
Mean Squared Error (MSE) = 456/6 = 76.
(3) t test H0: No effect of location on sales.
Y1  Y2  58  48  10
Standard error of this difference
MSE / n1  MSE / n2  76 / 4  76 / 4  38  6.1644
t test (ratio) = 10/6.1644=1.62 with 6 degrees
of freedom. Need REPLICATION for this.
(4) Link to regression:
Plot EOA data at 1, NE at -1 and regress
DEMO IAA_DESIGN1.SAS
Slope is 5=(1/2)(10), standard error is also half
of that in PROC TEST so t ratio the same!
(5) Two factors – interaction.
Factor L: Location XL=-1, 1
Factor S: Shelf Height XS=-1, 1
45
48
Low S: mean 46.5
51
68
High S: mean 59.5
Low L
High L
Mean 48 Mean 58
Simple effect of L at low S: 48-45=3
Simple effect of L at high S: 68-51=17
Main effect of L = 58-48=10=(17+3)/2
Main effect of S: 13, simple effects 6 and 20
Interaction ½ difference of simple effects
14/2=7
XL*XS measures interaction in regression:
(6) Matrix formulation
1

1
1

1
X 
1

1
1

1
 1 1
1
 47 

 
 1 1 1 
 43 
 50 
1 1 1

 
1 1 1
52
Y  
 50 
1 1 1

 
1 1 1
 46 
 67 
1 1 1

 
1 1 1 
 69 
8 0

0 8
1

b  ( X ' X ) ( X 'Y ) 
0 0

0 0
 53   overall mean 
  

5
(
L
main
effect
)
/
2
 

 6.5   ( S main effect ) / 2 
  

3.5
(
SxL
)
/
2
  

1
0 0   424 
 

0 0   232  192 




8 0
238  186
 

0 8   226  198 
Run demo IAA_Design2.sas
Factors A, B, C: Interactions AxB, AxC, BxC,
AxBxC (failure of AxB pattern to be the same at
each level of C)
(7) Fractional Factorials:
(EOA, Shelf Ht., Sign, Free Sample, On Sale)
Factors A,B,C,D,E: 5 main effects, 10 2-way
interactions, 10 3-way, 5 4-way and 1 5-way.
25=32 too many runs.
X variables -1, 1 as before.
Take only runs with (XA)(XB)(XC)=1 and
(XC)(XD)(XE)=1 (we say ABC=CDE=1). Then
(XA)(XB)(XC)2(XD)(XE)= (XA)(XB)(XD)(XE)=1 .
NO REPLICATION
I=ABC=CDE=ABDE is “defining contrast”
Note this implies that XB=(XA)(XB)2(XC)=
(XB)(XC)(XD)(XE)=(XA)(XB)2(XD)(XE)
which we write as B=AC=BCDE=ADE.
B is “aliased” with AC, BCDE, and ADE. Columns
for these 4 are all the same – cannot estimate
separately!
I=ABC=CDE=ABCE  B=AC=BCDE=ACE
Likewise D = ABCD=CE=ABCDE
Shortest “word” (ABC) in defining contrast is
resolution. Resolution 3 main effects are
aliased with 3-1=2 way interactions at worst.
Resolution 2 designs are useless. Need
resolution 5 to separate main effects from 2way interactions.
Run IAA_Design3.sas
(8) Confounding: Run all combinations but do
so in blocks. Drop X notation for simplicity.
Block 1 (store 1, 8 periods) : only runs with
ABC=-1 and CDE=-1
Block 2 ABC = -1 and CDE = 1
Block 3 (ABC=1,CDE=-1) Block 4 (ABC=1,CDE=1)
Loose ABC, CDE, and “generalized interaction”
ABDE because they can’t be separated from
(are “confounded with”) block effects.
Blocks increase accuracy (e.g. comparisons of
periods within stores, not store to store)
(9) Self study(?): Daniels’ normal plot.
Suppose we want to study 7 effects with just 8
runs. Example: unreplicated 23 plan.
1

1
1

1
X 
1

1
1

1
1 1 1 1
1
1
1 1 1 1
1 1
1 1
1 1 1
1
1 1 1
1 1 1
1
1 1 1 1 1
1
1
1
1 1
1 1
1
1
1 1
1 1
1 1 1 1 1
1

1 
1 

1
1 

1
1

1 
8

0
0

0
X 'X 
0

0
0

0
0 0 0 0 0 0 0

8 0 0 0 0 0 0
0 8 0 0 0 0 0

0 0 8 0 0 0 0
 8I
0 0 0 8 0 0 0

0 0 0 0 8 0 0
0 0 0 0 0 8 0

0 0 0 0 0 0 8 
Daniels’ idea: Vector of regression estimates b
H0: no effects  E(each bj)=0
Theory b has variance matrix (X’X)-12 so each
bj~N(0,2/8) under H0.
Under H0, normal plot should be a straight line.
Anything off the line is an active effect. Here’s
the one for demo IAA_Design3.sas where CE
(alias is D) is the only active effect.
Daniels normal plot
CE (=D) is only active effect
Plot of COL1*rbetas$_NAME_.
C
o
e
f
f
i
c
i
e
n
t
Symbol points to label.
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6 +
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> D
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4 +
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2 +
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> AD
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0 +
> A > B
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> BD > C
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> E
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-2 +
Š-+-------------+-------------+-------------+-------------+--2
-1
0
1
2
Z score