14-8 BLOCKING AND CONFOUNDING IN THE 2kDESIGN 545There is a more general method of constructing the blocks. The method employs a
defining contrast,say(14-24)where xiis the level of the ith factor appearing in a treatment combination and (^) iis the expo-
nent appearing on the ith factor in the effect that is to be confounded with blocks. For the 2k
system, we have either (^) i 0 or 1, and either xi 0 (low level) or xi 1 (high level).
Treatment combinations that produce the same value of L(modulus 2) will be placed in the
same block. Since the only possible values of L(mod 2) are 0 and 1, this will assign the 2k
treatment combinations to exactly two blocks.
As an example, consider the 2^3 design with ABCconfounded with blocks. Here x 1 corre-
sponds to A, x 2 to B, x 3 to C, and 1 2 3 1. Thus, the defining contrast that would be
used to confound ABCwith blocks is
To assign the treatment combinations to the two blocks, we substitute the treatment combina-
tions into the defining contrast as follows:
1 mod 2 2
1 mod 2 2
1 mod 2 2
1 mod 2 2
1 mod 2 2
1 mod 2 2
1 mod 2 2
1 mod 2 2
Thus (1), ab, ac, and bcare run in block 1, and a, b, c, and abcare run in block 2. This same
design is shown in Fig. 14-25.
A shortcut method is useful in constructing these designs. The block containing the treat-
ment combination (1) is called the principal block.Any element [except (1)] in the principal
block may be generated by multiplying two other elements in the principal block modulus 2
on the exponents. For example, consider the principal block of the 2^3 design with ABC
confounded, shown in Fig. 14-25. Note that
Treatment combinations in the other block (or blocks) may be generated by multiplying
one element in the new block by each element in the principal block modulus 2 on the
exponents. For the 2^3 with ABCconfounded, since the principal block is (1), ab, ac, and bc,
acbc abc^2 ab
abbcab^2 c ac
abac a^2 bcbc
abc: L 1112 1112 1112 3 1
bc: L 1102 1112 1112 2 0
ac: L 1112 1102 1112 2 0
c: L 1102 1102 1112 1 1
ab: L 1112 1112 1102 2 0
b: L 1102 1112 1102 1 1
a: L 1112 1102 1102 1 1
112 : L 1102 1102 1102 0 0
Lx 1 x 2 x 3
L 1 x 1 2 x 2 kxk
c 14 .qxd 5/9/02 7:54 PM Page 545 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH112 FIN L: