ST 516
Experimental Statistics for Engineers II
Blocking in the 2k Design
Blocking may be required because:
we cannot perform all required runs under homogeneous
conditions; e.g. raw material comes in limited batches;
we want to carry out runs under a variety of conditions; e.g. to
use material from different batches.
1 / 23
Blocking in the 2k Design
Introduction
ST 516
Experimental Statistics for Engineers II
Blocking a Replicated Design
If blocks are large enough for 2k runs, we can carry out each replicate
in a single block.
E.g. 2 × 2 yield example, in 3 blocks each of 4 runs (yield.txt):
summary(aov(Yield ~ Rep + A * B, yield))
Output
Df Sum Sq Mean Sq
Rep
2
6.500
3.250
A
1 208.333 208.333
B
1 75.000 75.000
A:B
1
8.333
8.333
Residuals
6 24.833
4.139
--Signif. codes: 0 *** 0.001 **
2 / 23
F value
0.7852
50.3356
18.1208
2.0134
Pr(>F)
0.4978348
0.0003937 ***
0.0053397 **
0.2057101
0.01 * 0.05 . 0.1
Blocking in the 2k Design
1
Blocking a Replicated Design
ST 516
Experimental Statistics for Engineers II
Analysis as a replicated design
This data set was previously analyzed as a replicated design, not
blocked:
summary(aov(Yield ~ A * B, yield))
Output
Df Sum Sq Mean Sq
A
1 208.333 208.333
B
1 75.000 75.000
A:B
1
8.333
8.333
Residuals
8 31.333
3.917
--Signif. codes: 0 *** 0.001 **
3 / 23
F value
Pr(>F)
53.1915 8.444e-05 ***
19.1489 0.002362 **
2.1277 0.182776
0.01 * 0.05 . 0.1
Blocking in the 2k Design
1
Blocking a Replicated Design
ST 516
Experimental Statistics for Engineers II
Differences
2(= n − 1) degrees of freedom are broken out from “Residuals” into
“Rep” (= blocks). Residual mean square changed, and also F-ratios
and P-values.
4 / 23
Blocking in the 2k Design
Blocking a Replicated Design
ST 516
Experimental Statistics for Engineers II
Confounding
If blocks are not large enough for 2k runs, we must use an incomplete
block design.
Simplest case: 2 × 2 design in 2 blocks of size 2.
Treatment
Effect
Combination
(1)
a
b
ab
5 / 23
I
A
B
AB
Block
+ - + + + - +
+ + +
+
+
2
1
1
2
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
Note that each block has both levels of A, and also both levels of B,
so main effects can be estimated from within-block differences.
But the AB interaction is the difference between block averages, and
is confounded with blocks.
We could also use either the A or B column to assign runs to blocks,
but in this case a main effect would be confounded; we usually
choose to confound the interaction.
6 / 23
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
The general 2k design can be carried out in 2 blocks each of 2k−1
runs in the same way: use the signs in the column for the
highest-order interaction.
Note: runs are sometimes assigned to blocks using a defining contrast
L = α1 x1 + α2 x2 + · · · + αk xk ,
where:
each αi is 1 if factor i is in the interaction to be confounded,
and 0 otherwise;
xi is 0 for the low level of factor i and 1 for the high level;
L is evaluated modulo 2.
7 / 23
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
Example: a 23 design with ABC confounded with blocks
Block assignments:
Treatment
Effect
Block
Combination
I
A
B
AB
C
AC
BC
ABC
(1)
a
b
ab
c
ac
bc
abc
+
+
+
+
+
+
+
+
- + - +
+ +
- + - +
+ +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
8 / 23
Blocking in the 2k Design
Confounding
1
2
2
1
2
1
1
2
ST 516
Experimental Statistics for Engineers II
Terminology: the block containing (1) is the principal block.
In this case, the principal block is (1), ab, ac, and bc.
These form a group: the product of any pair of elements is another
element in the principal block (recall that e.g. a2 = (1)).
You can form the other block by multiplying these by any run not in
the principal block, e.g. a or abc.
9 / 23
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
Example: filtration rate data
filtration.txt, 4 factors in 2 blocks:
# factors have already been converted from "-","+" to -1, +1 coding
filtration$Block <- filtration$A * filtration$B *
filtration$C * filtration$D
summary(lm(Rate ~ Block + A * B * C * D, filtration))
Note that ABCD cannot be estimated, because it is confounded with
blocks.
Output
Call:
lm(formula = Rate ~ Block + A * B * C * D, data = filtration)
Residuals:
ALL 16 residuals are 0: no residual degrees of freedom!
10 / 23
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
Output, continued
Coefficients: (1 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 60.0625
NA
NA
NA
Block
0.6875
NA
NA
NA
A
10.8125
NA
NA
NA
B
1.5625
NA
NA
NA
C
4.9375
NA
NA
NA
D
7.3125
NA
NA
NA
A:B
0.0625
NA
NA
NA
A:C
-9.0625
NA
NA
NA
B:C
1.1875
NA
NA
NA
A:D
8.3125
NA
NA
NA
B:D
-0.1875
NA
NA
NA
C:D
-0.5625
NA
NA
NA
A:B:C
0.9375
NA
NA
NA
A:B:D
2.0625
NA
NA
NA
A:C:D
-0.8125
NA
NA
NA
B:C:D
-1.3125
NA
NA
NA
A:B:C:D
NA
NA
NA
NA
11 / 23
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
Reduced model
summary(aov(Rate ~ Block + A + C + D + A * C + A * D, filtration));
Output
Df Sum Sq Mean Sq F value
Pr(>F)
Block
1
7.56
7.56 0.3629 0.5617799
A
1 1870.56 1870.56 89.757 5.600e-06
C
1 390.06 390.06 18.717 0.0019155
D
1 855.56 855.56 41.053 0.0001242
A:C
1 1314.06 1314.06 63.054 2.349e-05
A:D
1 1105.56 1105.56 53.049 4.646e-05
Residuals
9 187.56
20.84
--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1
***
**
***
***
***
1
Note that all effects in the reduced model can be estimated and
tested.
12 / 23
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
Confounding in More Than 2 Blocks
Suppose that blocks hold only 2k−p runs ⇒ we need 2p blocks.
Choose p effects to be confounded with blocks, where no effect is the
product of others.
Use the combination of signs in those p columns (or p defining
contrasts) to assign runs to blocks.
13 / 23
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
E.g. a 25 design in 4 blocks: we decide to confound ADE and BCE ;
Block assignments:
Run
ADE
BCE
Block
(0)
a
b
..
.
+
..
.
+
..
.
Block−−
Block+−
Block−+
..
.
abcde
+
+
Block++
and so on; the four combinations of + and − are used to make four
block labels (distinct, but otherwise arbitrary; could be Larry, Moe,
Curly, and Shemp).
14 / 23
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
The 4 blocks are:
Block−−: (1), ad, bc, abcd, abe, ace, cde, bde;
Block+−: a, d, abc, bcd, be, abde, ce, acde;
Block−+: b, abd, c, acd, ae, de, abce, bcde;
Block++: e, ade, bce, abcde, ab, bd, ac, cd.
Note: the 4 blocks will remove 3 degrees of freedom; in addition to
ADE and BCE , one other effect must be confounded. It is their
product ADE × BCE = ABCDE 2 = ABCD.
Note that I , ADE , BCE , and ABCD also form a group.
15 / 23
Blocking in the 2k Design
Confounding
ST 516
Experimental Statistics for Engineers II
Replication and Partial Confounding
Suppose that a design is replicated and confounded by blocking.
If the same confounding structure is used in each replicate, the
confounded effects are not estimable; they are said to be completely
confounded.
If different effects are confounded in each replicate, the design gives
some information about all effects; they are said to be partially
confounded.
16 / 23
Blocking in the 2k Design
Partial Confounding
ST 516
Experimental Statistics for Engineers II
E.g. 23 in 2 replicates each in 2 blocks, with ABC confounded with
blocks in Rep I, and AB confounded in Rep II (plasma etching tool
data):
plasmaLongRep1 <- plasmaLong[plasmaLong$Rep == 1,]
A <- coded(plasmaLongRep1$A)
B <- coded(plasmaLongRep1$B)
C <- coded(plasmaLongRep1$C)
plasmaLongRep1$Block <- ifelse(A * B * C < 0, 1, 2)
plasmaLongRep1 <- plasmaLongRep1[order(plasmaLongRep1$Block),]
plasmaLongRep2 <- plasmaLong[plasmaLong$Rep == 2,]
A <- coded(plasmaLongRep2$A)
B <- coded(plasmaLongRep2$B)
plasmaLongRep2$Block <- ifelse(A * B > 0, 1, 2)
plasmaLongRep2 <- plasmaLongRep2[order(plasmaLongRep2$Block),]
partialConfounding <- rbind(plasmaLongRep1, plasmaLongRep2)
partialConfounding$Rep
<- factor(partialConfounding$Rep)
partialConfounding$Block <- factor(partialConfounding$Block)
17 / 23
Blocking in the 2k Design
Partial Confounding
ST 516
Experimental Statistics for Engineers II
partialConfounding
1.1
4.1
6.1
7.1
2.1
3.1
5.1
8.1
1.2
4.2
5.2
8.2
2.2
3.2
6.2
7.2
A
+
+
+
+
+
+
+
+
-
B
+
+
+
+
+
+
+
+
C Rep Rate id Block
1 550 1
1
1 642 4
1
+
1 749 6
1
+
1 1075 7
1
1 669 2
2
1 633 3
2
+
1 1037 5
2
+
1 729 8
2
2 604 1
1
2 635 4
1
+
2 1052 5
1
+
2 860 8
1
2 650 2
2
2 601 3
2
+
2 868 6
2
+
2 1063 7
2
18 / 23
Blocking in the 2k Design
Partial Confounding
ST 516
Experimental Statistics for Engineers II
Note
The Blocks are labeled 1 and 2 in both Reps, but Block 1 in Rep I is
not the same as Block 1 in Rep II.
A factor (here Block) whose levels are labeled the same, but with
different meanings, across levels of another factor (here Rep), is said
to be nested within that factor: “Blocks are nested within Reps.”
The main effect of a nested factor has no meaning, and should be
left out of the analysis; only the interaction of the nested factor with
the outer factor has any meaning.
19 / 23
Blocking in the 2k Design
Partial Confounding
ST 516
Experimental Statistics for Engineers II
summary(aov(Rate ~ Block:Rep + A * B * C, partialConfounding))
Output
Df Sum Sq Mean Sq F value
Pr(>F)
A
1 41311
41311 16.1941 0.010079 *
B
1
218
218
0.0853 0.781987
C
1 374850 374850 146.9446 6.75e-05 ***
Block:Rep
3
4333
1444
0.5662 0.660744
A:B
1
3528
3528
1.3830 0.292529
A:C
1 94403
94403 37.0066 0.001736 **
B:C
1
18
18
0.0071 0.936205
A:B:C
1
6
6
0.0024 0.962816
Residuals
5 12755
2551
--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1
1
20 / 23
Blocking in the 2k Design
Partial Confounding
ST 516
Experimental Statistics for Engineers II
If blocks are given unique labels, the analysis is simpler:
partialConfounding$Blocks <- factor(paste(partialConfounding$Rep,
partialConfounding$Block,
sep = "-"))
Output
partialConfounding
Rep Block A B
1
I
1 - 2
I
1 + +
3
I
1 + 4
I
1 - +
5
I
2 + 6
I
2 - +
7
I
2 - 8
I
2 + +
9
II
1 - 10 II
1 - . . .
16 II
2 - +
21 / 23
C
+
+
+
+
+
Rate Blocks
550
I-1
642
I-1
749
I-1
1075
I-1
669
I-2
633
I-2
1037
I-2
729
I-2
604
II-1
1052
II-1
+ 1063
II-2
Blocking in the 2k Design
Partial Confounding
ST 516
Experimental Statistics for Engineers II
summary(aov(Rate ~ Blocks + A * B * C, partialConfounding))
Output
Df Sum Sq Mean Sq F value
Pr(>F)
Blocks
3
4333
1444
0.5662 0.660744
A
1 41311
41311 16.1941 0.010079 *
B
1
218
218
0.0853 0.781987
C
1 374850 374850 146.9446 6.75e-05 ***
A:B
1
3528
3528
1.3830 0.292529
A:C
1 94403
94403 37.0066 0.001736 **
B:C
1
18
18
0.0071 0.936205
A:B:C
1
6
6
0.0024 0.962816
Residuals
5 12755
2551
--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1
1
22 / 23
Blocking in the 2k Design
Partial Confounding
ST 516
Experimental Statistics for Engineers II
Note
√ larger standard errors for partially confounded effects (factor
of 2):
summary(lm(Rate ~ Blocks + coded(A) * coded(B) * coded(C),
partialConfounding))
Coefficients:
(Intercept)
BlocksI-2
BlocksII-1
BlocksII-2
coded(A)
coded(B)
coded(C)
coded(A):coded(B)
coded(A):coded(C)
coded(B):coded(C)
coded(A):coded(B):coded(C)
--Signif. codes: 0 *** 0.001
23 / 23
Estimate Std. Error t value Pr(>|t|)
753.125
30.929 24.350 2.18e-06 ***
14.750
50.507
0.292 0.78199
55.625
43.740
1.272 0.25942
21.375
43.740
0.489 0.64575
-50.812
12.627 -4.024 0.01008 *
3.688
12.627
0.292 0.78199
153.062
12.627 12.122 6.75e-05 ***
-21.000
17.857 -1.176 0.29253
-76.812
12.627 -6.083 0.00174 **
-1.062
12.627 -0.084 0.93621
-0.875
17.857 -0.049 0.96282
** 0.01 * 0.05 . 0.1
Blocking in the 2k Design
1
Partial Confounding