Applied Statistics and Probability for Engineers

14-9 FRACTIONAL REPLICATION OF THE 2kDESIGN 551

since and. The aliases of Band Care

and

Now suppose that we had chosen the other one-half fraction, that is, the treatment com-

binations in Table 14-24 associated with minus on ABC. These four runs are shown in the

lower half of Table 14-24 and in Fig. 14-28(b). The defining relation for this design is

I ABC. The aliases are A BC, B AC, and C AB. Thus, estimates of A, B, and

Cthat result from this fraction really estimate A BC, B AC, and C AB. In practice, it

usually does not matter which one-half fraction we select. The fraction with the plus sign in

the defining relation is usually called the principal fraction,and the other fraction is usually

called the alternate fraction.

Note that if we had chosen ABas the generator for the fractional factorial,

and the two main effects of Aand Bwould be aliased. This typically loses important information.

Sometimes we use sequencesof fractional factorial designs to estimate effects. For

example, suppose we had run the principal fraction of the 2^3 ^1 design with generator ABC.

From this design we have the following effect estimates:

Suppose that we are willing to assume at this point that the two-factor interactions are negli-

gible. If they are, the 2^3 ^1 design has produced estimates of the three main effects A, B, and C.

However, if after running the principal fraction we are uncertain about the interactions, it is

possible to estimate them by running the alternatefraction. The alternate fraction produces

the following effect estimates:

We may now obtain de-aliased estimates of the main effects and two-factor interactions

by adding and subtracting the linear combinations of effects estimated in the two individual

fractions. For example, suppose we want to de-alias Afrom the two-factor interaction BC.

Since and , we can combine these effect estimates as follows:

and

1

2

1 /A/¿A 2 

1

2

1 ABCABC 2 BC

1

2

1 /A/¿A 2 

1

2

1 ABCABC 2 A

/AABC /A¿ABC

/¿CCAB

/¿BBAC

/¿AABC

/CCAB

/BBAC

/AABC

AAABB

CCABCABC^2 AB

BBABCAB^2 CAC

AIA A^2 I

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