STAT 512
2-Level Factorial Experiments: Blocking
TWO-LEVEL FACTORIAL EXPERIMENTS (Blocking)
Some “traditional” notation:
• Upper-case letters are associated with factors, or “regressors” of
factorial effects, e.g.
– ABC ≡ x1 x2 x3
• The treatment combination associated with µ122112...1 is
sometimes designated by listing lower-case letters associated with
factors set to level 2, e.g.
– “ac” ↔ factors A and C at level 2, others at level 1
– “abcd” ↔ factors A-D at level 2, others at level 1
– “(1)” ↔ all factors at level 1
• So upper-case letters are associated with columns, and lower-case
letters with rows, of X.
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STAT 512
2-Level Factorial Experiments: Blocking
Blocks of size 2f
• Put one full unreplicated factorial experiment in each block
• e.g. CBD for f = 2:
(1)
(1)
(1)
a
a
b
b
b
ab
ab
ab
...
—– r blocks —–
a
← all 2f treatments in each block
2
STAT 512
2-Level Factorial Experiments: Blocking
3
• Model:
(
ymijl... =
µ + ρm + (αi + ...) + mijl...
µ + ρm + (±α ± ...) + mijl...
(overparameterized)
(full rank)
• i = 1, 2, et cetera; m = 1, 2, 3, ...r
• Assumes no block-by-treatment interaction
source
df
blocks
r−1
treatments
2f − 1
residual
(2f − 1)(r − 1)
corr’d total
r2f − 1
sum-of-squares
P f
2
2
(ȳ
−
ȳ
)
m···
···
m
P
2
r(ȳ
−
ȳ
)
·ijl...
···
ijl...
–difference–
P
2
mijl... (ymijl... − ȳ··· )
STAT 512
2-Level Factorial Experiments: Blocking
• “residual” would be block-by-treatment interaction if that had
been included in the model
• “treatments” can be decomposed into 1-df components for each
effect, e.g. N α̂2
– just as in unblocked case
– leave out µ ... this is taken out as “correction factor”
2
2
2
d , df=3
– e.g. f = 2, SST = N α̂ + N β̂ + N (αβ)
4
STAT 512
2-Level Factorial Experiments: Blocking
Blocks of size 2f −1
• Put one “half-replicate” – one-half of all treatments – in each
block
• Start with a single, full, unreplicated design ... divide into two
blocks of size 2f −1 so that each treatment appears exactly once
• This leads to:
source
df
blocks
1
(two blocks)
treatments
2f − 2
(???)
residual
0
(no replication)
c.t.
2f − 1
(N − 1)
• Doesn’t accommodate 2f − 1 treatment d.f., even without any
residual d.f.
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STAT 512
2-Level Factorial Experiments: Blocking
• Need to give up one of the “treatment” degrees of freedom ... do
this by intentionally confounding one of the factorial effects with
blocks
• Arrange pairs of blocks so that, for a selected effect (−),
– (−) is always + in one block
– (−) is always − in the other block
• Can’t estimate (−) now because it is confounded with the block
difference, so we generally want to select an effect that is:
– least likely to be important or interesting
– most likely to be zero
• The highest-order interaction is often used
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STAT 512
2-Level Factorial Experiments: Blocking
7
• Example: 23 , confounding (αβγ) with blocks
treatment
I
A
B
C
AB
AC
BC
ABC
block
(1)
+
−
−
−
+
+
+
−
1
a
+
+
−
−
−
−
+
+
2
b
+
−
+
−
−
+
−
+
2
c
+
−
−
+
+
−
−
+
2
ab
+
+
+
−
+
−
−
−
1
ac
+
+
−
+
−
+
−
−
1
bc
+
−
+
+
−
−
+
−
1
abc
+
+
+
+
+
+
+
+
2
STAT 512
2-Level Factorial Experiments: Blocking
Notes:
•
(1)
a
ab
b
ac
c
bc
abc
→
not a BIBD (or any other
design we’ve studied so far)
• COULD have used another effect to “split” ... e.g. AB rather
than ABC, if this had made sense
• Over the entire design, all factorial effects are orthogonal, so all
except ABC are orthogonal to “ABC+block”
– Other treatment estimates and SS’s are unchanged
• Without replication, there is no “pure error”
– e.g. use normal plots omitting µ̂ AND block d
+ (αβγ)
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STAT 512
2-Level Factorial Experiments: Blocking
• Now, suppose we can afford to apply each treatment r > 1 times,
but must still use blocks of size 2f −1
(1)
a
ab
b
...
ac
c
bc
abc
– r blocks –
...
– r blocks –
ABC confounded with difference
• Let m index block ... not replicate ... so there are 2r blocks:
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STAT 512
2-Level Factorial Experiments: Blocking
source
df
blk’s
2r − 1
trt’s
2f − 2∗
resid
(2f − 2)(r − 1)
c.t.
r2f − 1
sum-of-squares
P f −1
2
2
(ȳ
−
ȳ
)
m···
···
m
P d2
d ∗
N (−) (omit (αβγ))
–difference–
P
2
(y
−
ȳ
)
mijl...
···
mijl...
• In the last sum, not all possible combinations of index values
appear since not all treatments (i, j, ...) appear in each block (m)
• For r = 1, there are no d.f. for residual under the full model (of
course), but an error estimate could come from d.f. in treatments
corresponding to terms omitted from the model (and not
confounded with blocks)
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STAT 512
2-Level Factorial Experiments: Blocking
• If blocks can be regarded as random, this can be analyzed as a
split-plot design, with levels of ABC compared between blocks,
and other effects compared within blocks:
stratum
source
df
sum-of-squares
ABC
1
d
N (αβγ)
blk’s|ABC
2(r − 1)
resid. from prev. blk SS
c.t.
2r − 1
prev. blk SS
2
w.p.
s.p.
other trt’s
2 −2
P c2
d
N
(−) (omit (αβγ))
resid
(2f − 2)(r − 1)
–difference–
c.t.
f
f
r2 − 1
2
(y
−
ȳ
)
···
mijl...
mijl...
P
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STAT 512
2-Level Factorial Experiments: Blocking
• Last 3 lines of the ANOVA table are the same as with CRD.
• In this case, there is NO information about ABC | blocks:
P
f −1
2
– blocks SS = 2
(ȳ
−
ȳ
)
m...
....
m
– r blocks including (1),ab,ac,bc ... ABC=−
– r blocks including a,b,c,abc ... ABC=+
– as with split-plot design assigning r dams to diet 1, r dams to
diet 2 ...
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STAT 512
2-Level Factorial Experiments: Blocking
13
• But what if you:
– don’t want to assume random blocks
– need to make inferences about all factorial effects
– are required to use blocks of size 2f −1
• Partial confounding:
(1)
a
(1)
b
...
...
ab
b
bc
c
...
...
ac
c
a
ab
...
...
bc
abc
abc
ac
...
...
ABC conf.
BC conf.
(-) conf.
→r
STAT 512
2-Level Factorial Experiments: Blocking
• Estimates and SS’s for non-confounded effects are computed as in
the unblocked case, e.g. N α̂2
• Estimates and SS’s for confounded effects are computed from
2
f
d
replicates in which they aren’t confounded, e.g. 2 (r − 1)(αβγ) ,
with estimate computed from data in replicates 2-through-r only
source
df
blocks
2r − 1
unconfounded effects
2f − 1 − r
confounded effects
r
resid
(2f − 2)(r − 1) − 1∗
c.t.
2f r − 1
• *: −1 did not appear before because the single confounded effect
was not recovered ... now all effects are estimated, some using
partial data
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STAT 512
2-Level Factorial Experiments: Blocking
Blocks of size 2f −2
• Put one “quarter replicate” – 1/4 of all possible treatments – in
each block
• Accomplish this by further splitting of 2f −1 -size blocks, by
selecting a second factorial effect to confound with blocks
• Continuing previous f = 3 example where ABC was chosen to
generate the first split, now add BC for the second. (Now we
won’t be able to estimate the factorial effect (βγ) associated with
this contrast either.)
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STAT 512
2-Level Factorial Experiments: Blocking
16
treatment
I
A
B
C
ABC
BC
block
(−)
+
−
−
−
−
+
1
a
+
+
−
−
+
+
2
b
+
−
+
−
+
−
3
c
+
−
−
+
+
−
3
ab
+
+
+
−
−
−
4
ac
+
+
−
+
−
−
4
bc
+
−
+
+
−
+
1
abc
+
+
+
+
+
+
2
• Or,
(-)
a
b
ab
bc
abc
c
ac
STAT 512
2-Level Factorial Experiments: Blocking
• So,
– 8 observations
– 7=8-1 d.f. after “correction for the mean”
– 4=7-3 d.f. after accounting for 4 blocks
• But, there are 5 factorial effects not used in “splitting”:
A, B, C, AB, AC ... but not BC, ABC
– What else can’t be estimated within blocks?
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STAT 512
2-Level Factorial Experiments: Blocking
• Within each block:
– ABC = x1 x2 x3 is constant
– BC = x2 x3 is constant
– But x1 x2 x3 × x2 x3 = x1
– So, A = x1 = ABC×BC is constant within each block
• Symbolically, within:
block 1,
I=
−ABC =
+BC =
−A
block 2,
I=
+ABC =
+BC =
+A
block 3,
I=
+ABC =
−BC =
−A
block 4,
I=
−ABC =
−BC =
+A
• By choosing to confound ABC and BC with blocks, we also
confound their generalized interaction, A. (This is a bad choice for
most purposes since this is a main effect.)
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STAT 512
2-Level Factorial Experiments: Blocking
• Usually better:
I = ± AB = ± BC → = ± AC
(-)
c
a
b
abc
ab
bc
ac
• Now, replicate each of these blocks r times for r full replicates of
the design in 4r blocks of size 2
• For fixed block effects:
source
d.f.
blocks
4r − 1
treatments
2f − 4
residual
(2f − 4)(r − 1)
c.t.
2f r − 1
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STAT 512
2-Level Factorial Experiments: Blocking
20
• For random block effects:
stratum
source
d.f.
sums-of-squares
AB,AC,BC
3
d + ...
N (αβ)
blocks | AB,AC,BC
4r − 4
c.t.
4r − 1
(fixed ”blocks”)
other fact. effects
2f − 4
(as before)
residual
(2f − 4)(r − 1)
(as before)
c.t.
2f r − 1
2
w.p.
s.p.
STAT 512
2-Level Factorial Experiments: Blocking
• Partial confounding also works here
• Example: 24−2 , 2 replicates
– Rep 1: I = ± ABC = ± BCD ( = ± AD )
– Rep 2: I = ± ABCD = ± BC ( = ± AD )
– ABC and BCD estimates and SS from Rep 2 only
– ABCD and BC estimates and SS from Rep 1 only
– AD confounded with blocks in both replicates; no information
available
– df(resid) = 31 (c.t.) − 7 (blocks) − 4 (ABC,BCD,ABCD,BC)
− 10 ( others except for AD) = 10
• If confounded effects had been entirely different in each rep:
– df(resid) = 31 (c.t.) − 7 (blocks) − 6 (all confounded effects)
− 9 (others) = 9
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STAT 512
2-Level Factorial Experiments: Blocking
Blocks of size 2f −s , s < f
• Have already taken care of s = 1, 2
• Ideas can be made more general by sequentially re-splitting blocks
(s times), and realizing that new generalized interactions are also
confounded
• Example 26−4 :
– 26 = 64 treatments
– 24 = 16 blocks
– 26−4 = 4 observations per block
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STAT 512
2-Level Factorial Experiments: Blocking
23
• Think in terms of the effects we select to confound with blocks:
I
= ± ABF
1st split
= ± ACF
(= ± BC )
= ± BDF
(= ± AD
= ± ABCD
= ±CDF )
3rd split
= ± DEF
(= ± ABDE
= ± ACDE
= ± BCDEF
4th split
= ± BE
= ± AEF
= ± ABCEF
2nd split
= ± CE)
• Generalized interactions added at each stage are between the new
“independent” effect, and all previously implicated effects
(independent or GI’s)
• 2s − 1 effects or “words”, in all, are confounded with blocks (all
possible combinations of s independent effects)
STAT 512
2-Level Factorial Experiments: Blocking
• Note: Cannot pick a previously identified GI as a new independent
effect. Why?
– because these are ALREADY constant-within-blocks
• What do these blocks look like? The block containing treatment
(+++...+) is:
I = +ABF = +ACF = +BDF = +DEF
A
B
F
C
D
E
+
+
+
+
+
+
−
−
+
−
−
−
−
+
−
+
−
+
+
−
−
−
+
−
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STAT 512
2-Level Factorial Experiments: Blocking
• All other blocks can be formed by reversing entire columns from
this set, e.g. A and B:
A
B
F
C
D
E
−
−
+
+
+
+
+
+
+
−
−
−
+
−
−
+
−
+
−
+
−
−
+
−
→ I = +ABF = −ACF = −BDF = +DEF
• Signs on ACF and BDF are reversed because they each have one
of A or B
• Signs on ABF and DEF are not reversed because they each have
both of A and B ... likewise with generalized interactions
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STAT 512
2-Level Factorial Experiments: Blocking
• “Principal” block: Reverse all, so that run set at the low level for
each factor will be included
A
B
F
C
D
E
−
−
−
−
−
−
+
+
−
+
+
+
+
−
+
−
+
−
−
+
+
+
−
+
→ I = −ABF = −ACF = −BDF = −DEF
• In general, the generating relation for the principal block assigns
−/+ to words of odd/even length
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STAT 512
2-Level Factorial Experiments: Blocking
• For fixed block effects:
source
d.f.
blocks
2s r − 1
treatments
2f − 2s
residual
difference
c.t.
2f r − 1
sum of squares
P f −s
2
(ȳblock − ȳ)2
P d2
N (−) except confounded
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STAT 512
2-Level Factorial Experiments: Blocking
• For random block effects:
stratum
source
d.f.
w.p.
confounded effects
2s − 1
blocks | conf effects
2s (r − 1)
c.t.
2s r − 1
(split-plot ”blocks”)
other fact. effects
2f − 2s
(as before)
residual
difference
(as before)
c.t.
2f r − 1
s.p.
• As with 2f −2 , partial confounding can also be used
• For example: r = 5, ABC in confounding pattern for replicates 1
and 2, estimate from replicates 3-5 only ... some effects may be
confounded in more replicates than other effects
28