Applied Statistics and Probability for Engineers

544 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

This design is shown in Fig. 14-24. Notice that block 1 contains the treatment combina-

tions (1) and aband that block 2 contains aand b. The contrasts for estimating the main

effects of factors Aand Bare

Note that these contrasts are unaffected by blocking since in each contrast there is one plus

and one minus treatment combination from each block. That is, any difference between block

1 and block 2 that increases the readings in one block by an additive constant cancels out. The

contrast for the ABinteraction is

Since the two treatment combinations with the plus signs, aband (1), are in block 1 and the

two with the minus signs, aand b, are in block 2, the block effect and the ABinteraction are

identical. That is, the ABinteraction is confounded with blocks.

The reason for this is apparent from the table of plus and minus signs for the 2^2 design

shown in Table 14-12. From the table we see that all treatment combinations that have a plus

on ABare assigned to block 1, whereas all treatment combinations that have a minus sign on

ABare assigned to block 2.

This scheme can be used to confound any 2kdesign in two blocks. As a second example,

consider a 2^3 design, run in two blocks. From the table of plus and minus signs, shown in Table

14-15, we assign the treatment combinations that are minus in the ABCcolumn to block 1 and

those that are plus in the ABCcolumn to block 2. The resulting design is shown in Fig. 14-25.

ContrastABab 112 ab

ContrastBabba 112

ContrastAabab 112

b

+

(1)

a

ab

A

Geometric view

(a)

Assignment of the four

runs to two blocks

(b)

(1)

Block 1

ab

a

Block 2

b

= Run in block 1

= Run in block 2

Figure 14-24 A 2^2

design in two blocks.

(a) Geometric view. (b)

Assignment of the four

runs to two blocks.

= Run in block 1

= Run in block 2

A

C

B

bc abc

c

b

ac

a

ab

(1)

(a)

Geometric view

Assignment of the eight

runs to two blocks

(b)

(1)

Block 1

ab

a

Block 2

b

ac

bc

c

abc

Figure 14-25 The 2^3

design in two blocks

with ABC confounded.

(a) Geometric View.

(b) Assignment of the

eight runs to two

blocks.

c 14 .qxd 5/9/02 7:54 PM Page 544 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH112 FIN L: