STAT 512
2-Level Factorial Experiments: Regular Fractions
TWO-LEVEL FACTORIAL EXPERIMENTS (Fractional Factorials)
• Bottom Line: A regular fractional factorial design consists of the
treatments in one block of a blocked full 2f experiment
• Example: f = 5, s = 2
– 25 (32 possible treatments)
– 2−2 fraction containing 25−2 treatments satisfying:
I = +ABC = −ADE (= −BCDE)
• Now just “+” or “−’ for each “word”’, not “±”, because we are
talking about only one block
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STAT 512
2-Level Factorial Experiments: Regular Fractions
ABC = + →
A
B
C
D
E
+
+
+
+
−
+
+
+
−
+
−
−
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+
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−
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+
+
−
+
−
−
−
+
−
−
+
−
+
−
−
−
+
← given A, ADE = −
• We know already that we’ve “lost” information on 3 effects (ABC,
ADE, BCDE)
• BUT, there are 25 − 1(mean) − 3(confounded effects) = 28 MORE
factorial effects, and only 8 observations (in an unreplicated
experiment) ... something else is missing
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2-Level Factorial Experiments: Regular Fractions
• Look, for example, at the BC interaction:
A
B
C
D
E
BC
+
+
+
+
−
+
+
+
+
−
+
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−
+
+
• A = BC ... But neither is always +1 or −1 and so would not be
confounded with blocks in a full blocked design
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2-Level Factorial Experiments: Regular Fractions
• This is not a problem when we have all 4 blocks because:
I = +ABC → +A = +BC in two blocks
I = −ABC → +A = −BC in two blocks
• In fact, ALL “estimable” factorial effects are now aliased in groups
... ABC, ADE, and BCDE with I, and all other effects in other
groups of size 4
• The “defining relation” (or “generating relation”, “identifying
relation”) for this design is:
I = +ABC = −ADE = −BCDE
i.e. the relationship between the effects intentionally aliased with
the intercept (and confounded with blocks in a full 2f ).
• Recall that these are “words” or “generators” that stand for
columns in the model matrix. We can use element-wise
multiplication of these columns to identify the groups of aliased
effects.
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STAT 512
2-Level Factorial Experiments: Regular Fractions
• Continued Example: Main effect A
I = +ABC = −ADE = −BCDE
(2s − 1 words)
A = +AABC = −AADE = −ABCDE
A = +BC = −DE = −ABCDE
The A main effect is aliased with 2s − 1 other words
Note: Underlines are added to highlight the lowest-order effect aliased
with A.
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2-Level Factorial Experiments: Regular Fractions
A
B
C
D
E
BC
DE
ABCDE
+
+
+
+
−
+
−
−
+
+
+
−
+
+
−
−
−
−
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STAT 512
2-Level Factorial Experiments: Regular Fractions
• Also from I = +ABC = −ADE = −BCDE:
B = +AC = −ABDE = −CDE
C = +AB = −ACDE = −BDE
D = +ABCD = −AE = −BCE
E = +ABCE = −AD = −BCD
• For an unreplicated experiment, N = 8 = uncorrected total d.f.:
– 1 for I and its aliases (correction for mean)
– 5 estimable “strings” containing main effects
– So, 2 more estimable strings ... generate these by using any
two effects that are not in the first 5 sets:
BD = +ACD = −ABE = −CE
BE = +ACE = −ABD = −CD
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STAT 512
2-Level Factorial Experiments: Regular Fractions
Analysis
• The result is 8 estimable strings of effects, 7 of which don’t
include I (or µ)
• The estimate of α is really an estimate of a “string” of effects:
E[α̂] = α + (βγ) − (δ) − (αβγδ)
• Similarly for other main effects, but their alias strings each include
only 1 two-factor interaction
• Given significance (or apparent significance via normal plot) of
some collection of these “strings”, effects that are most likely
“real” must be identified by other information – expert knowledge,
hierarchy or heredity rules, further experiments ...
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2-Level Factorial Experiments: Regular Fractions
Comparison of Fractions: Resolution
• Design resolution focuses on the shortest-length word in the
defining relation.
• Suppose: I = + AB (lowest-order effect) = + ...
Then: A = + B = ... i.e. can’t resolve main effects
• Suppose: I = + ABC (lowest-order effect) = + ...
Then: A = + BC = ... i.e. o.k. for estimating all main effects
cleanly if there are no two-factor interactions
• Suppose: I = + ABCD (lowest-order effect) = + ...
Then: A = + BCD = ... i.e. o.k. for estimating all main effects
cleanly if there are no three-factor interactions
Then: AB = + CD = ... i.e. can’t resolve two-factor interactions
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STAT 512
2-Level Factorial Experiments: Regular Fractions
• Suppose: I = + ABCDE (lowest-order effect) = + ...
Then: A = + BCDE = ... i.e. o.k. for estimating all main
effects cleanly if there are no four-factor interactions
Then: AB = + CDE = ... i.e. o.k. for estimating all two-factor
interactions if there are no three-factor interaction
• The worst case of aliasing lower-order effects with higher-order
effects is determined by the lowest-order effect aliased with I, i.e.
the “shortest word” in the defining relation
• The length of this shortest word (i.e. number of letters involved)
is called the resolution of the design, often denoted by a roman
numeral
• The bigger, the better.
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STAT 512
2-Level Factorial Experiments: Regular Fractions
• Summary: In a design of resolution R, no O-order effect is aliased
with any effect of order less than R−O
– Res. III: main effects aren’t aliased with main effects
– Res. IV: main effects aren’t aliased with other main effects or
two-factor interactions; but two-factor interactions are aliased
with other two-factor interaction
– Res. V: main effects aren’t aliased with other main effects,
two-factor interactions, or three-factor interactions; two-factor
interactions aren’t aliased with other two-factor interactions
• ... these are the classes of regular fractional factorials that are
most commonly used in practice
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STAT 512
2-Level Factorial Experiments: Regular Fractions
• Examples: 25
I = +ABCDE is a 25−1
V
I = +ABCD is a 25−1
IV , would usually be considered worse
(5−2)
I = +ABC = −BDE ( = −ACDE) is a 2III , of less resolution,
but also a smaller design
Note: ()’s are used to emphaize that ACDE is actually implied by ABC
and BDE ... a more compact notation could omit this with no loss of
information.
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STAT 512
2-Level Factorial Experiments: Regular Fractions
Comparing Fractions of Equal Resolution: Aberration
• Example: 27−2
IV
I = +ABCD = +DEFG = +ABCEFG
I = +ABCD = +CDEFG = +ABEFG
The second design has fewer pairs of aliased 2-factor
interactions, less “aberration”
• Goal is to find a design of:
1. maximum resolution (maximum length of shortest word), and
among these
2. minimum aberration (minimum number of shortest words)
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2-Level Factorial Experiments: Regular Fractions
14
• These two criteria can be combined by looking at a list of word
lengths, the word length pattern, for each candidate design:
design
length of words
1
2
3
4
5
6
7
I=ABCD=DEFG=ABCEFG
(0
0
0
2
0
1
0)
Res IV
I=ABCD=CDEFG=ABEFG
(0
0
0
1
2
0
0)
Res IV, min ab.
I=ABC=DEFG=ABCDEFG
(0
0
1
1
0
0
1)
Res III
STAT 512
2-Level Factorial Experiments: Regular Fractions
Blocking Regular Fractional Factorial Designs
• As with full factorial experiments, but now realizing that the
effects we chose to estimate or confound with blocks are really
“strings” of aliased effects
• Previous 25−2 example, 8 estimable strings:
Defining Relation: I = +ABC = −ADE (= −BCDE)
“I” = I +ABC −ADE −BCDE
“D” = D +ABCD −AE −BCE
“A” = A +BC −DE −ABCDE
“E” = E +ABCE −AD −BCD
“B” = B +AC −ABDE −CDE
“BD” = BD +ACD −ABE −CE
“C” = C +AB −ACDE −BDE
“BE” = BE +ACE −ABD −CD
• Without blocking, the last 7 of these are associated with the 7 d.f.
that would be available for treatments
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2-Level Factorial Experiments: Regular Fractions
16
• Original fractional factorial:
ad
ae
b
bde
c
cde
abcd
abce
• To divide into 2 blocks (of size 4), we must confound one of the
“effect strings” with blocks
• Pick, e.g., “BD” (split using BD column in design matrix ...)
source
df
ae
bde
ad
b
blk
1
c
abcd
cde
abce
trt
6
c.t.
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STAT 512
2-Level Factorial Experiments: Regular Fractions
17
• We COULD split a second time by confounding a second effect
string, say “BE”
• BUT, this would involve the generalized interaction also: BD ×
BE = DE ...
“A” = A +BC −DE −ABCDE
• SO, A isn’t estimable, even with aliases. Still, we COULD ...
source
df
bde
ae
b
ad
blk
3
c
abcd
cde
abce
trt
4∗
c.t.
7
• * associated with “B”, “C”, “D”, and “E”
STAT 512
2-Level Factorial Experiments: Regular Fractions
Recombining Fractions
• Suppose a 25−2 has been completed:
– I = +ABC = −ADE (= −BCDE)
– no blocking
– 8 estimable strings: “I”, “A” through “E”, “BD”, “BE”
• Results were interesting, but indicate more (or more complex)
effects of factors than was expected ... now we want to expand
the study – augment the design
• Can convert this 1/4 fraction to a 1/2 fraction by adding any one
of the other 1/4 fractions based on the same defining relation (but
different signs). For example:
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2-Level Factorial Experiments: Regular Fractions
1st 1/4 fraction:
I=
+ABC =
−ADE =
(−BCDE)
2nd 1/4 fraction:
I=
−ABC =
−ADE =
(+BCDE)
1/2 fraction :
I=
−ADE
• Note: Estimable strings are now half as long (i.e. aliased groups
are now half the size) as before – and there are twice as many of
them (counting “I”)
• Note: Selecting a different 2nd 1/4 fraction results in a different
augmented design. For example:
1st 1/4 fraction:
I=
+ABC =
−ADE =
(−BCDE)
2nd 1/4 fraction:
I=
−ABC =
+ADE =
(−BCDE)
1/2 fraction :
I=
−BCDE
• This one is of greater resolution, and so would ordinarily be
preferred
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STAT 512
2-Level Factorial Experiments: Regular Fractions
Generally:
• Start with 2f −s : I = W1 = W2 = ... Ws ( = W1 W2 ... )
– 2s − 1 words (other than I) in all
• Add 2f −s : I = −W1 = W2 = ... −Ws ( = −W1 W2 ... )
– -’s on any combination of independent generators
– 2s−1 of all words will have −’s ... how would you show this?
• Together, 2f −s+1 : I = all independent words and G.I.s for which
sign didn’t change
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STAT 512
2-Level Factorial Experiments: Regular Fractions
Example: 26−3
I
= +ABC
= +CDE
( = +ABDE)
= −ADF
( = −BCDF
= −ACEF
= −BEF)
• Would be good to eliminate all words of length 3 here ... this
would increase resolution to IV
– W1 = +ABC
W2 = +CDE
W3 = −ADF
– W1 W2 W3 = −BEF – if signs on first 3 are changed, this one
will be also ... so add:
I
= −ABC
= −CDE
( = +ABDE)
= +ADF
( = −BCDF
= −ACEF
= +BEF)
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STAT 512
2-Level Factorial Experiments: Regular Fractions
• Result: 26−2
I = +ABDE = −BCDF (= −ACEF)
• Could subsequently expand this 1/4 fraction to a 1/2 fraction the
same way, e.g. add
I = −ABDE = −BCDF (= +ACEF)
• Result: 26−1
I = −BCDF
• But note: we COULD have had a resolution VI 1/2 fraction if we
had begun by selecting the “best” 26−1 , e.g.:
I = +ABCDEF
6−1
• Can you find another 26−3
or 26−1
III that yields a 2V
V I when
doubled twice?
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STAT 512
2-Level Factorial Experiments: Regular Fractions
Fold-Over Designs
• Recall from the discussion of blocked factorial designs: Given 1
block of runs, you can construct another by REVERSING SIGNS
on a selected set of factors
• This leads to two practical techniques for augmenting a Resolution
III fractional factorial design, based on the analysis of the data
Example:
• 26−3
III
• I = +ABC = +CDE ( = +ABDE ) = −ADF ( = −BCDF =
−ACEF = −BEF )
• A = +BC ..., AB = +C ..., et cetera
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STAT 512
2-Level Factorial Experiments: Regular Fractions
Suppose analysis suggests that factor A is potentially important, and
we want more information on this factor. REVERSE THE SIGN FOR
ONLY FACTOR A in the augmenting fraction:
• I = −ABC = +CDE ( = −ABDE ) = +ADF ( = −BCDF =
+ACEF = −BEF )
Together:
• I = +CDE = −BCDF ( = −BEF ) ← no A’s
• A = +ACDE = −ABCDF = −ABEF ← each ≥ 4 letters
• AB = +ABCDE = −ACDF = −AEF ← each ≥ 3 letters
• The A main effect and all two-factor interactions involving A are
estimable if there are no interactions of order 3 or more (“Res V”
for A only)
• Still Res III for all other factors
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STAT 512
2-Level Factorial Experiments: Regular Fractions
Suppose analysis suggests that ALL factors are potentially interesting,
and we want more information on the entire system. REVERSE THE
SIGNS ON ALL LETTERS in the augmenting fraction:
• I = −ABC = −CDE ( = +ABDE ) = +ADF ( = −BCDF =
−ACEF = +BEF )
Together:
• I = +ABDE = −BCDE ( = −ACEF )
← no odd-length (3, esp.) words
• A = +BDE = −ABCDE = −CEF ← each ≥ 3 letters
• All main effects are estimable if there are no interactions of order
3 or more (Res IV for all factors)
What happens in intermediate cases, where signs are changed on a
SUBSET of factors?
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STAT 512
2-Level Factorial Experiments: Regular Fractions
Practical Reality of Experimenting in Stages ...
• operating conditions or “raw material” may change
• ... 2s fractions may need to be treated as blocks
Example (again):
• 25−2 : I = +ABC = −ADE ( = −BCDE) ← block 1
– 8 estimable strings of 4 effects each
• 25−2 : I = −ABC = +ADE ( = −BCDE) ← block 2
– same 8 strings of effects, but different signs within strings
• Together: I = −BCDE
– 16 strings of 2 effects each
– one of them is “ABC” = ABC − ADE
← THE WORDS THAT CHANGED SIGNS
– this is the contrast “sacrificed to” (confounded with) blocks
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STAT 512
2-Level Factorial Experiments: Regular Fractions
Example:
• Begin with 26−3 (block 1):
I
= +ABC
= +CDE
( = +ABDE )
= −ADF
( = −BCDF
= −ACEF
= −BEF )
• Estimable strings contain 8 effects at this point.
• Add 26−3 (block 2):
I
= −ABC
= −CDE
( = +ABDE )
= +ADF
( = −BCDF
= −ACEF
= +BEF )
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STAT 512
2-Level Factorial Experiments: Regular Fractions
• Result 26−2 :
I = +ABDE = −BCDF ( = −ACEF )
ABC + CDE − ADF − BEF is confounded with blocks
• Now estimable strings contain 4 effects each.
• Add 26−2 (blocks 3 and 4):
I
= +ABDE
= +BCDF
( = +ACEF )
• Result 26−1 :
I = +ABDE
−BCDF − ACEF is confounded with blocks
+ABC + CDE is confounded with blocks
−ADF − BEF is confounded with blocks
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STAT 512
2-Level Factorial Experiments: Regular Fractions
• Add 26−1 (blocks 5 through 8):
I = −ABDE
• Result 26 :
+ABDE is confounded with blocks
−BCDF is confounded with blocks
−ACEF is confounded with blocks
...
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STAT 512
2-Level Factorial Experiments: Regular Fractions
30
Summary of Example:
ABC
CDE
(ABDE)
ADF
(BCDF
ACEF
BEF)
block 1
+
+
+
−
−
−
−
8 runs
block 2
−
−
+
+
−
−
+
8 runs
−
−
accumulated
+
16 runs
block 3
+
+
+
+
+
+
+
8 runs
block 4
−
−
+
−
+
+
−
8 runs
accumulated
+
32 runs
block 5
+
−
−
−
−
+
+
8 runs
block 6
+
−
−
+
+
−
−
8 runs
block 7
−
+
−
−
+
−
+
8 runs
block 8
−
+
−
+
−
+
−
8 runs
accumulated
64 runs
STAT 512
2-Level Factorial Experiments: Regular Fractions
31
Alternatively, it may make more sense to think of this as four blocks of
increasing size, with a new block added at each “doubling”:
ABC
CDE
(ABDE)
ADF
(BCDF
ACEF
BEF)
block 1
+
+
+
−
−
−
−
8 runs
block 2
−
−
+
+
−
−
+
8 runs
+
+
block 3
+
block 4
−
1
1
2
16 runs
32 runs
1
3
3
1
• 1: aliased, but estimable from data in blocks 3 and 4
• 2: not estimable ... confounded throughout with blocks
• 3: aliased, but estimable from data in block 4
Note: In most cases, this isn’t done. Blocks of equal size:
• make operation simpler
• are generally more consistend with an assumption of equal control
(and “noise”) within blocks