STAT 512
2-Level Factorial Experiments: Irregular Fractions
IRREGULAR TWO-LEVEL FRACTIONAL FACTORIALS
• A major practical weakness of “regular” fractional factorial
designs is that N must be a power of 2:
8
16
32
64
128
(large gaps)
• A broader class of 2-level designs for which low-order effects
are orthogonally estimable when higher-order effects are
assumed zero:
1
STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Orthogonal Array of strength t, OA(t): Design for which
every subset of t factors (ignoring the rest) forms a full 2t
factorial design, possibly replicated
– less rigid structure requirement than for regular ff’s – we
don’t require that every pair of higher-order effects be
either orthogonal or completely confounded
– allows for more flexibility in the value of N , e.g. an
OA(2) could be of size 12 (every pair of factors a 22 × 3
reps ...)
– generally not so easy to construct as regular fractional
factorials
• A useful class of OA’s of strength 2:
2
STAT 512
2-Level Factorial Experiments: Irregular Fractions
3
PLACKETT-BURMAN DESIGNS (Biometrika 1946)
• Resolution III designs (estimable main effects in the absence
of interactions)

 = 0 mod[4]
• Available for N
 ≥ f +1
• Characterized by first row of design matrix:
N
first row
8
+
+
+
−
+
−
−
12
+
+
−
+
+
+
−
16
.
.
.
(through N ≈ 100 in the PB paper)
−
−
+
−
STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Construction of N -run design matrix D:
1. use given row for first run
2. use cyclic permutations of the given row for runs 2
through N − 1
3. use (− − −...−) for N th run
4. can (and usually should) randomize:
– columns in the resulting design matrix to the physical
experimental factors.
– order of run execution
4
STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Example, N = 8, f = 7:









M = (1, D) = 








+
+ +
+ − +
+ + −
−

+ −

















−
+
− +
+
− − + + + − +
+
+ −
− + +
+ −
+
− +
− − +
+
+
+
+ −
+ − − +
+
+
+ + − + − − +
+
− −
−
5
− − −
−
STAT 512
2-Level Factorial Experiments: Irregular Fractions
• For f = 4, 5, 6, use a subset of these columns
−s
• For N = 2c , these are equivalent to 2fIII
regular fractions
• For N 6= 2c , these are nongeometric, still providing
orthogonal estimates of main effects, but alias structure is
more complex
6
STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Review of Bias in Linear Models
Suppose y = X1 θ 1 + X2 θ 2 + , usual assumptions ...
But we fit parameters as though model is y = X1 θ 1 + θ̂ 1 = (X01 X1 )−1 X01 y
E[θ̂ 1 ] = (X01 X1 )−1 X01 (X1 θ 1 + X 2 θ2 )
= θ 1 + (X01 X1 )−1 X01 X2 θ 2 = θ 1 + Aθ 2
• When:
– θ 1 contains the intercept and main effects
– θ 2 contains two-factor interactions
– columns of X1 are +1/-1 and orthogonal
• A=
1
0X
X
N 1 2
• Look at this for P-B designs in 8 and 12 runs:
7
STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Example, N = 8, elements of A0
I
AB
AC
AD
AE
AF
AG
BC
BD
BE
BF
BG
CD
CE
CF
CG
DE
DF
DG
EF
EG
FG
A
B
C
D
E
F
-1
G
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
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STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Look at column for A ... A = -BF = -CD = -EG
• ... or, I = -ABF = -ACD = -AEG
• But that’s not all ...
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STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Continuing this idea:
A = -BF = -CD = -EG
I = -ABF = -ACD = -AEG
B = -AF = -CG = -DE
I = -ABF = -BCG = -BDE
C = -AD = -BG = -EF
I = -ACD = -BCG = -CEF
D = -AC = -BE = -FG
I = -ACD = -BDE = -DFG
E = -AG = -BD = -CF
I = -AEG = -BDE = -CEF
F = -AB = -CE = -DG
I = -ABF = -CEF = -DFG
G = -AE = -BC = -DF
I = -AEG = -BCG = -DFG
• Generalized interactions of these are:
+BCDF, +BEFG, +ACFG, +ADEF, +ABCE, +CDEG, +ABDG,
-ABCDEFG
• 15 words = 24 − 1 ... this is a 27−4 regular fractional factorial
10
STAT 512
2-Level Factorial Experiments: Irregular Fractions
11
• Example, N = 12, f = 11, elements of 3A0
I
A
B
AB
C
D
E
F
G
H
J
K
L
-
-
-
+
-
-
+
+
-
+
-
-
+
-
+
-
-
+
+
-
-
-
-
-
-
-
-
-
+
+
-
+
-
-
-
+
-
+
-
-
-
+
-
+
AC
-
AD
-
+
AE
-
-
+
AF
+
-
+
-
AG
-
+
-
-
-
AH
-
-
-
-
+
+
AJ
+
+
-
-
-
-
-
AK
+
-
-
+
-
+
-
-
AL
-
-
-
+
-
-
+
+
-
-
-
-
+
-
-
+
+
+
-
-
+
-
+
-
+
+
-
-
-
-
-
-
-
-
+
BC
-
BD
-
-
BE
-
-
+
BF
+
-
-
+
...
-
STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Each main effect is partially aliased with all two-factor
interactions of other factors (only)
• The weight of each aliased term is 1/3 (rather than 1)
• So, for exmple:
E(β̂) = β + 13 [−(αγ) − (αδ)...]
• There are more two-factor interactions that could bias any
main effect, but the bias associated with any one of them is
reduced by a factor of 13 .
12
STAT 512
2-Level Factorial Experiments: Irregular Fractions
IRREGULAR RESOLUTION IV DESIGNS:
• P-B designs are Resolution III because main effects, while
orthogonally estimable, would be biased by 2-factor
interactions:
A=
1
0X
X
N 1 2
6= 0
• Think about the (complete) foldover of a P-B design (e.g.
all runs repeated with all factors reversed). Model matrices
for the “doubled” design:

X∗1
=
1
D
1 −D



X∗2
=
X2
X2


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STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Alias matrix:

A∗
=
1
2N


1
2N

10

D0 −D0

10
X2
X2
210 X2

D0 X2 − D0 X2


=
– 10 X2 = 0 because the P-B design is an OA(2)
– D0 X2 − D0 X2 = 0
– therefore A∗ = 0
• So this produces irregular Resolution IV designs in N = the
smallest multiple of 8 ≥ 2(f + 1)
– e.g. for f = 9 and N = 24
– regular fraction would require 29−4
IV = 32 runs.
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STAT 512
2-Level Factorial Experiments: Irregular Fractions
EXERCISE
• Suppose you begin a study with an N -run OA(2)
main-effects design (not necessarily a regular fractional
factorial).
• You decide to augment this initial design with its complete
fold-over, e.g. N more runs selected by reversing all factors
in all the original runs.
• What are the statistical properties of the main effect
estimates (sampling variances and possible bias due to
two-factor interactions) if the two halves must be treated as
blocks?
15
STAT 512
2-Level Factorial Experiments: Irregular Fractions
16
ONE-FACTOR-AT-A-TIME (OAT) DESIGNS
• For any f , N = f + 1
• Version 1:

− − ... −


f + 1 f − 3 f − 3 ... f − 3

 + − ... − 
 f − 3 f + 1 f − 3 ... f − 3 








D =  − + ... −  , X01 X1 =  f − 3 f − 3 f + 1 ... f − 3 




 ... ... ... ... 
 ...
...
... ... ... 
− − ... +
f − 3 f − 3 f − 3 ... f + 1
• Version 2:




f +1
f −1
f −3
... −(f − 1)
 f −1
 + − ... − 



0



D=
 + + ... −  , X1 X1 =  f − 3
 ...
 ... ... ... ... 
f +1
f −1
f −1
f +1
... −(f − 3) 

...
...
− − ... −
+ + ... +
... −(f − 5) 

...
−(f − 1) −(f − 3) −(f − 5) ...
...
f +1

STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Regardless of the value of f , for version 1:
– {(X01 X1 )−1 }i,i =
– {(X01 X1 )−1 }i,j =
1
2
1
4
(comp. to
1
N
for orth. designs)
(comp. to 0 for orth. designs)
(other than the first row/column)
• For version 2:
– {(X01 X1 )−1 }i,i =
1
2
– {(X01 X1 )−1 }i,i+1 = − 41 , other {(X01 X1 )−1 }i,j = 0
• Why? In each case β̂i = half the difference of two data
values ...
• So, much less efficient than orthogonal Resolution III designs
• Perhaps reasonable only if σ 2 is known to be very small
relative to the size of important effects
17
STAT 512
2-Level Factorial Experiments: Irregular Fractions
SUPERSATURATED DESIGNS
• N < f + 1 ... full main effects model isn’t estimable
• Used in preliminary factor screening, or where operational
requirements demand a very small N
• Effect sparsity must be “strongly assumed”
• A popular example: Lin (Technometrics 1993)
– For f factors, begin by constructing a P-B design in
f + 1 factors.
– Select one column as the “branching column”, include
only the runs associated with one level of this factor (i.e.
half the number of runs in the P-B plan).
– Remove the branching column (which is now confounded
with the intercept).
18
STAT 512
2-Level Factorial Experiments: Irregular Fractions
• Example: f = 10
• Construct f = 11 P-B design with 12 runs; include only the runs
with + in the 11th column:


−++−+++−−−
− − − + − + + − + +
+ − − − + − + + − +

D=
+ + − − − + − + + −
− + + + − − − + − +
+−+++−−−+−
• X01 X1 has
– diagonal elements = 6
– other row/column 1 elements = 0
– all other elements = ±2
• Can fit models including the mean and any 4 main effects
• Stepwise regression is often used for analysis
19
STAT 512
2-Level Factorial Experiments: Irregular Fractions
Can you do this with regular fractions of Resolution III
(including P-B designs with N a power of 2)?
• Yes, but ...
• Suppose you start with I = ABC = ...
• Use A as a branching column. This is the same as
“splitting” again in the regular fractional factorial
framework, adding A to the list of factorial effects in the
generating relation.
• Then I = BC results, so B=C
• The column-orthogonality of these designs prevents
simultaneous fitting of any small subset of main effects
20
STAT 512
2-Level Factorial Experiments: Irregular Fractions
SOME OTHER IRREGULAR FRACTIONAL FACTORIALS:
• Full 2f plus a regular fraction
– All factorial effects are estimable, some are correlated
– Variance of effect estimates is < σ 2 /2f , but > σ 2 /N
– Replication without doubling everything
• 2f −1 plus an included smaller fraction (e.g. 2f −2 )
– Same sub-comments as above, but for “strings” and “2f −1 ”
• 3 distinct 2f −3 ’s from the same generating relation (“3/8 rep”)
– More like P-B designs; complex aliasing, more estimable
functions
21