Applied Statistics and Probability for Engineers

550 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

Furthermore, the identity element Iis also plus for the four runs, so we call

the defining relationfor the design.

The treatment combinations in the 2^3 ^1 design yields three degrees of freedom associated

with the main effects. From the upper half of Table 14-24, we obtain the estimates of the main

effects as linear combinations of the observations, say,

It is also easy to verify that the estimates of the two-factor interactions should be the follow-

ing linear combinations of the observations:

Thus, the linear combination of observations in column A, , estimates both the main effect

of Aand the BCinteraction. That is, the linear combination estimates the sum of these two

effects A BC. Similarly, estimates B AC, and estimates CAB. Two or more ef-

fects that have this property are called aliases.In our 2^3 ^1 design, Aand BCare aliases, Band

ACare aliases, and Cand ABare aliases. Aliasing is the direct result of fractional replication.

In many practical situations, it will be possible to select the fraction so that the main effects

and low-order interactions that are of interest will be aliased only with high-order interactions

(which are probably negligible).

The alias structure for this design is found by using the defining relation I ABC.

Multiplying any effect by the defining relation yields the aliases for that effect. In our

example, the alias of Ais

AAABCA^2 BCBC

/B /C

/A

/A

AB^1  23 a bcabc 4

AC ^1  23 a bcabc 4

BC^1  23 abcabc 4

C^1  23 a bcabc 4

B^1  23 a bcabc 4

A^1  23 abcabc 4

IABC

A

C

B

abc

c

b

a

(a)

The principal fraction, I = +ABC

bc

ac

ab

(1)

(b)

The alternate fraction, I = –ABC

Figure 14-28 The

one-half fractions of

the 2^3 design. (a) The

principal fraction,

I ABC. (b) The

alternate fraction,

I ABC.

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