Applied Statistics and Probability for Engineers

530 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

for each of the three factors. This table is sometimes called the design matrix.This design al-

lows three main effects to be estimated (A, B, and C) along with three two-factor interactions

(AB, AC, and BC) and a three-factor interaction (ABC).

The main effects can easily be estimated. Remember that the lowercase letters (1), a, b,

ab, c, ac, bc, and abcrepresent the total of all nreplicates at each of the eight runs in the de-

sign. As seen in Fig. 14-19(a), the main effect of Acan be estimated by averaging the four

treatment combinations on the right-hand side of the cube, where Ais at the high level, and by

C

a

B

c

bc

abc

ab

A +

+

(1)

ac

+

b

(a) Geometric view (b) The 2

(^3) design matrix
Run
1 2 3 4 5 6 7 8
A – + – + – + – +
B – – + + – – + +
C – – – – + + + +
Figure 14-18 The 2^3
design.
ABC
AB AC BC
(a)
(b)
ABC
(c)
A
C
B

+

+ – + – – +

+

+

= + runs

= – runs

Main effects

Two-factor interactions

Three-factor interaction

+

Figure 14-19

Geometric presenta-

tion of contrasts corre-

sponding to the main

effects and interaction

in the 2^3 design. (a)

Main effects. (b) Two-

factor interactions.

(c) Three-factor

interaction.

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