Summarize the content for 10 marks & include the examples in content - BLOCKING AND CONFOUNDING
SYSTEM FOR TWO-LEVEL FACTORIALS
13.1.1. Introduction
Under homogeneous conditions, there are many conditions in which it is not possible to
perform all runs in a 2
k
factorial experiment.
For example, a single group of raw materials may not be sufficient to make other
requirements in process.
There are many case where it may be desirable to vary the experimental conditions to
ensure that treatments are equally effective in many situations, likely to come into
practice.
For example, an engineer may run a pilot plant experiment with multiple batches of raw
materials as he knows quality grades of raw materials that are likely to be used in the
process.
Bloacking is a design tehcnique that can be used in these situations. It is a method of
dealing controllable nuisance variables.
13.1.2. Blocking a Replicated 2
k
Factorial Design
Suppose the 2
k
factorial design is repeated n times. In case, there exist n replicas, then
every non-homogeneous conditions defines a block, and each replication is run in a
block. The replicates (or runs) in each block will be created randomly. Design analysis is
similar to any blocked factorial experiment.
Example 1: Let us suppose that the chemical process experiment. Consider that only four
experimental tests can be conducted from a single batch of raw materials. Thus, three
batches would be required to run all three replicates.
Table 13.1 illustrates the design, where every batch of raw materials matches to one
block.
Table 13.1:Chemical Process Experiment in Three Blocks
Block 1 Block 2 Block 3
(1)
= 28
a
= 36
b
= 18
(D = 25
a
= 32
b = 19
(1)
= 27
a
= 32
b
= 23
ab
= 31 ab = 30 ab = 29
Block Totals B|
= 113 B2 = 106 B3= lll
For this blocking design, the analysis of variance is shown in table 13.2.
268 Biostatistics & Research Methodology
Table 13.2: Analysis of Variance for the Chemical Process
Experiment in Three Blocks
Source of
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Square
F. PValue
Blocks 6.50 2 3.25
A 208.33 208.33 1 50.32 0.0004
(Concentration)
B (catalyst) 75.00 1 75.00 18.12 0.0053
AB 8.33 1 8.33 2.01 0.2060
Error 24.84 6 4.14
Total 323.00 11
All sums of squares are calculated similar to a standard unblocked 2
k
design. From the
block sum, the sum of squares for a block is calculated. B|
, B2 and B^
shows block totals
as shown in table 13.1:
3 r
2 ..2
=
y
y ...
s
' 4 12 i=l ^
(113)
2
+(106)
2
+(111)
2
(330)
2
SSBlock
= 6.50
4 12
These three blocks have two degrees of freedom. Table 13.2 shows that the summary
from this analysis, the design was run in blocks and the block effect is relatively small.
13.1.3. Confounding in the 2
k
Factorial Design
There are many situations where it is not possible to perform a complete replication of a
factorial design in a block. Confounding is a design technique in which complete factorial
experiment is arranged in blocks and the block size is smaller than the number of
treatment combinations in one replica. The technique causes information about certain
treatment effects to be confounded with block.
For 2
k
factorial design, we typically focus on confounding systems. Sometimes the
designs shown are incomplete block designs as every block does not include all
treatments combinations. The particular structure of a 2
k
factorial system permits a
simple method of analysis. We generally focus on design and analysis of 2
k
factorial
design in 2
P
incomplete blocks having condition, p < k. These designs can be generally
run with two blocks, four blocks, eight blocks and so on.
13.1.4. Confounding the 2
k
Factorial Design in Two Blocks
Assume that someone wants to run a single replica of 22 designs. Each of the 22 = 4
treatment combinations needs the same amount of raw material. For example, each batch
of raw materials is sufficient to test two treatment combinations.
Therefore, two batches of raw materials are needed. In case, batches of raw materials are
taken as blocks, we must specify two of four treatment combinations in every block. For
this problem,figure 13.1 shows a possible design.
Figure 13.1a shows the geometric view and indicates that the treatment combination is
assigned to different blocks on opposing diagonals. Figure 13.1b shows that that Block 1
contains the treatment combination ( 1 ) and ab and Block 2 contains q and b. The order is
Blocking and Confounding System (Chapter 13) 269
randomly defined for running treatment combinations within a block. The block which to
run first is decided by random manner. We estimate the main effects of A and B by
assuming that there is no blocking. We get:
A=-
^
[ab+a -b-(l)]
B=-
^
[ab+ b-a-(l)]
We see that both A and B are not affected by blocking as each estimate has a plus and a
minus treatment combination from every block. This means that there is no difference
between Block 1 and Block 2. Let us assume that we have AB interaction:
^
AB = [ab+(l)-a -b]
The two treatment with the minus sign (A and B) are in block 2 and the two with the plus
sign [Ab and (1)] are in block 1, the block effect and AB interaction are the identical.
This shows that AB is confounded with blocks.
The reason is shown in the table of plus and minus signs for 22 designs. It is illustrated in
table 13.3. In table, we see that all treatment combinations with a plus sign on AB are
assigned to block 1. Similarly, all treatment combinations with a minus sign on AB are
placed in block 2.
This method can be used to confound any effect A, B, or AB with blocks. For example,
in case, (1) and b are assigned to block 1 and a and ab to block 2, then the main effect A
will be confounded with block.
+<>
Block 1 Block 2
( 1 ) a
B ab b
A +
b) Assignment of the Four
•= Run in block 1 Runs to Two Blocks
o — Run in block 2
a) Geometric View
Figure 13.1: A 21 Design in Two Blocks
Table 13.3: Table of Plus and Minus Signs for the 21 Design
Treatnient
Combination
Factorial Effect
1 A B AB
( 1 ) +
a + +
b + +
ab + + + +
This method can be very useful to confound any 2
k
design in two blocks. For example,
let us assume that a 23 design run in two blocks. Let we confound the three-factor
interaction ABC with blocks. Table 13.4 shows plus and minus signs.
270 Biostatistics & Research Methodology
We assign treatment combinations of minus to block 1 on ABC and plus to block on
ABC.
Table 13.4: Table of Plus and Minus Signs for the 23 Design
Treatment
Combination
Factorial Effect
I A B AB C AC BC ABC
( 1) + +
a + + + +
b + + + +
ab + + + +
c + + +
ac + + +
be + + +
abc + + + + + + + +
Figure 13.2 illustrates the resultant design. The treatments within a block are again run
randomly.
Block 1 Block 2
(1) a
# = Run in block 1
ab b
O =
Run in block 2
ac c
,0
be abc
a) Geometric View
b) Assignment of the Eight Runs to Two Blocks
Figure 13.2: 23 Design in Two Blocks with ABC Confounded
Other Methods for Constructing the Blocks
These designs can be constructed by using another method. The method applies linear
combination:
L = aiXi + (X2X2
+ + akxk, . . (1)
Where, Xj
=
The level of i
,h
factor in a specific treatment combination
a
=
Exponent appearing on the ilh factor in the effect to be confounded.
Note: For the 2
k
system, we have a. = 0 or 1 and x, = 0 (low level) or x4 = 1 (high level).
Equation 1 is known as defining contrast. Treatment combinations that give the similar
value of L (mod 2) will be assigned in the similar block. Since 0 and 1 are the only
possible values of L (mod 2), it will specify 2k treatment combinations in exactly two
segments.
To see the approach, assume 23 designs with ABC, confounded by blocks.
Where Xi corresponds to A, x2 to B, X3 to C, and oil = 012 = 013 = 1 . Therefore, the contrast
corresponding to ABC is as follows:
L = X|+ X2 + X3
The treatment combination ( 1 ) is written 000 in the (0, 1 ) notation, thus we get:
L = 1 (0) + 1 (0) + 1(0) = 0
= 0 (mod 2)
Blocking and Confounding System (Chapter 13) 271
Similarly, the treatment combination is 100, obtaining:
L = 1(1) + 1(0) + 1 (0)
= 1 = 1 (mod 2)
Thus, (1) and 'a' will be run in different blocks. For the remaining treatment
combinations, we get:
b: L = 1(0) + 1(1) + 1(0)
= 1 = 1 (mod 2)
ab: L = 1(1) + 1(1) + 1(0)
= 2
= 0 (mod 2)
c: L = 1(0) + 1(0) + 1(1)
= 1 = 1 (mod 2)
ac: L = 1 (1) + 1(0) + 1(1 )
= 2 = 0 (mod 2)
be: L = 1(0) + 1(1) + 1(1)
= 2 = 0 (mod 2)
abc: L = 1(1) + 1(1) + 1(1 )
= 3
= 1 (mod 2)
Thus (1), ab, ac, and be are run in blocks 1 and a, b, c. abc are run in block 2.This is the same
design shown in Figure 13.3, which originated from the table of plus and minussigns.
Additional methods can be used to make these designs. The block containing the
treatment combination (1) is known as the principal block. This block has a useful
group-theoretic property in treatment combinations. That is, they form a group against the
multiplication modulus. This implies that if good quality material was used then any
element except the original (1 ) would be principal. Figure 13.3b shows the resulting
responses. These can be found by subtracting the block effect from the original
observations. This shows that the original response to treatment combination ( 1) was 45,
and it was calculated (1)
= 25 (
= 45 - 20) in figure 13.3b. Other responses can also be
obtained in this block. After being tested in block 1, block 2 performed eight trials.
D + Block 1 Block 2
a = 71
b = 48
c = 68
d = 43
abc = 65
bed = 86
abd = 86
abd =
( 1 ) = 25
ab = 45
ac = 40
be = 60
ad = 80
bd = 25
cd = 55
abed =
76 104
# = Runs in block 1 C
b) Assignment of the 16 Runs to Two O = Blocks Runs in block 2
a) Geometric View
Figure 13J: 24 Design in Two Blocks
Table 13.5 shows the effect estimates for this “modified” version. The estimates of 4
main effects, 6 two-factor interactions, and 4 three-factor interactions are similar to affect
estimates where there was no block effect. Similar to the original experiment, the factors
A, C, D and AC and AD interactions emerge as significant effects when a normal
probability of effect estimates is constructed.
In the original experiment, this effect was estimated as ABCD = 1.375. In this example,
the estimate of the ABCD = -18.625. Since ABCD is confounded with blocks, the
ABCD interaction estimates the original interaction effect (1.375) with addition to the
block effect (-20), hence ABCD = 1.375 + (-20)
=-18.625.
272 Biostatistics & Research Methodology
Table 13.5: Effect Estimates for the Blocked 24 Design
Regression
Coefficient
Sum of
Squares
Per cent
Contribution Model Term Effect Estimate
21.625
3.125
9.875
14.625
0.125
-18.125
16.625
2.375
-0.375
-1.125
1.875
4.125
-1.625
-2.625
-18.625
1870.5625
39.0625
390.0625
855.5625
0.0625
1314.0625
1105.5625
22.5625
0.5625
5.0625
14.0625
68.0625
10.5625
27.5625
1387.5625
A 10.81 26.30
B 1.56 0.55
C 4.94 5.49
7.31 12.03
<0.01
18.48
15.55
D
0.062
-9.06
AB
AC
AD 8.31
BC 1.19 0.32
BD -0.19
-0.56
<0.01
CD 0.07
ABC 0.94 0.20
ABD 2.06 0.96
ACD -0.81
-1.31
0.15
BCD 0.39
Blocks (ABCD) 19.51
The block effect can be calculated between two blocks as follows:
Block Effect -y Block!
-y Block2
406 555 -149
8
~~
8
”
8
=-18.625
These effects are estimates as Block + ABCD.
Table 13.6 gives the summary of the analysis of variance. Effects with large estimates
are included in the model, and the block sum of squares is as follows:
(406 )
2
+(555 )
2
(961 )
2
SS =1387.5625 Blocks 8 16
Table 13.6: Analysis of Variance
Source of
Variation
Degrees of
Freedom Sum of Squares Mean Square F„ P-Value
Blocks
(ABCD)
1387.5625 1
1870.5625
390.0625
855.5625
1314.0625
1105.5625
187.5625
7111.4375
1870.5625
390.625
855.5625
1314.0625
1105.5625
20.8403
1 89.76
18.72
41.05
63.05
53.05
<0.0001
0.0019
0.0001
<0.0001
<0.0001
A
c 1
D 1
AC 1
AD 1
Error 9
Total 15
The findings of this experiment match those exactly where no block effect was present.
Note that in case, the experiment was not run in blocks, and in case the magnitude (-20)
had affected the first eight trials, then the results vary.
Blocking and Confounding System (Chapter 13)
13.1.5. Confounding the 2
k
Factorial Design in Four Blocks
A 2
k
factorial design confounded in four sections of 2
k‘2
observations can be constructed
easily. These designs are useful when the number of factors is moderately large (k > 4),
and the block sizes are very small.
For example, assume 25 designs. In case, each block would have only eight runs, then
four blocks would be useful. This design can be constructed in straightforward. Assume
two effects confounded with blocks, according to ADE and BCE. Such effects have two
defining contrasts associated with them.
L, = x, + x4 + x5
Li = x2 + x3 + x5
273
Every treatment combination will give a particular values of Li (mod 2) and L2
(mod 2)
like (L|, L2
)
= (0, 0), (0, 1 ) ( 1 , 0), or ( 1 , 1 ). The treatment combinations having the same
values are assigned to a single block. In this example we get:
Li = 0, L2 = 0 For ( 1 ), ad, be, abed, abe, ace, ede, bde
L|
= 1 , Li = 0 For a, d, abc, bed, be, abde, ce, aede
Li = 0, L2= 1 For b, abd, c, acd, ae, de, abce, bede
L, = 1 , L2 = 1 For e, ade, bee, abede, ab, bd, ac, cd
The above combinations will be assigned to different blocks. Figure 13.4 shows the
complete design.
Block 1
L, = 1
Lz = l
Block 1
L, = 0
L2 = 0
Block 1
L,
= 1
Lj= 0
Block 1
L, = 0
L2 = 1
(1 ) abc be e abede a b abce
ad ace d abde ade bd abd ae
be ede bee ac abc ce c bede
abed bde bed aede ab cd acd de
Figure 13.4: 25 Design in Four Blocks with ADE, BCE and ABCD Confounded
In addition to ADE and BCE, another effect must be confounded with the block. Another
effect of one degree of freedom must be confounded because there are four blocks of
three degrees of freedom between them and because ADE and BCE each have only one
degree of freedom. This effect is the generalised form of ADE and BCE and can be
determined by product of ADE and BCE modulus 2.
Therefore generalised interaction (ADE) (BCE)
=
ABCDE2
=
ABCD is also confounded
with blocks. This can be easily verified for 25 designs, like a table of plus and minus
signs. The treatment combination assigned to blocks is shown in the table below:
Treatment Combinations In Sign on ADE Sign on BCE Sign on ABCD
Block 1
Block 2
Block 3
Block 4
+
+
+
+ + +
The product of the signs of any two effects for a block (e.g., ADE and BCE) gives
additional effects for that block (ABCD). Therefore, ADE, BCE and ABCD are all
confounded with blocks.
274 Biostatistics & Research Methodology
The group-theory properties of principal block also hold. For example, the product of
principal block of the two treatment combination gives another element of the principal
block. This gives:
ad • be = abed and abe • bde
= ab2
=de2
= ad
and so on. To design another block, we have to select that treatment combination that is
not in the principal block such as b and multiply b by all the treatment combinations in
the principal block. It provides:
b (1)
= b b • ad
= abd b • be
= b
2
c
= c b • abed = ab2
cd
= acd
And so on, that is, it would produce eight treatment combinations in block 3. The
principal blocks can be derived from the defined contrasts and group-theoretic property.
The remaining blocks can be determined from these treatment combinations by the
method given above.
The typical process of building a 2
k
design divided into four segments is to select two
effects to make blocks, automatically confounding a third effect which is a generalized
form of the first two. Then, the design is constructed using the group-theory properties of
the principal block and two defined contrasts (L|
, Li). To select confounded effects with
blocks, precaution must be done to obtain a design that does not confound effects that
may be of interest.
For example, in 25 designs, we may select ABCDE and ABD, which automatically
confounds CE, an effect of interest. A best option is to confound ADE and BCE, which
confounds ABCD automatically. It is preferable to focus on three-factor interactions
ADE and BCE rather than two-factor interaction CE.