Page 478 - Design for Six Sigma a Roadmap for Product Development

Fundamentals of Experimental Design 437

TABLE 12.11 Number of Runs for a
k
2 Full Factorial
Number of factors Number of runs
2 4
3 8
4 16
5 32
6 64
7 128

y β 0 β 1 x 1 β 2 x 2 β 3 x 3 β 13 x 1 x 3 β 23 x 2 x 3 ε (12.13)

From MINITAB, this is

Y 19.75 1.90x 1 2.75x 2 0.60x 3 0.45x 1 x 3 2.30x 2 x 3

12.3.5 Full factorial designs in two levels
If there are k factors, each at two levels, a full factorial design has
2 runs.
k
As shown in Table 12.11, when the number of factors is five or
greater, a full factorial design requires a large number of runs and is
not very efficient. A fractional factorial design or a Plackett-Burman
design is a better choice for five or more factors.

12.4 Fractional Two-Level Factorial Design
As the number of factors k increase, the number of runs specified
for a full factorial can quickly become very large. For example, when
k 6, then 2 64. However, in this six-factor experiment, there
6
are six main effects, say, A,B,C,D,E,F, 15 two-factor interactions,
AB,AC,AD,AE,AF,BC,BD,BE,BF,CD,CE,CF,DE,DF,EF, 20 three-factors
interactions, ABC,ABD,…, 15 four-factor interactions, ABCD,…, 6 five-
factor interactions, and one six-factor interaction.
During many years of applications of factorial design, people have
found that higher-order interaction effects (i.e., interaction effects
involving three or more factors) are very seldom significant. In most
experimental case studies, only some main effects and two-factor
6
interactions are significant. However, in the 2 experiment above, out
of 63 main effects and interactions, 42 of them are higher-order inter-
actions, and only 21 of them are main-effects and two-factor interac-
tions. As k increases, the overwhelming proportion of effects in the full
factorials will be higher-order interactions. Since those effects are
most likely to be insignificant, a lot of information in full factorial