Applied Statistics and Probability for Engineers

14-9 FRACTIONAL REPLICATION OF THE 2kDESIGN 561

EXAMPLE 14-9 To illustrate the use of Table 14-29, suppose that we have seven factors and that we are

interested in estimating the seven main effects and obtaining some insight regarding

the two-factor interactions. We are willing to assume that three-factor and higher interactions

are negligible. This information suggests that a resolution IV design would be appropriate.

Table 14-29 shows that two resolution IV fractions are available: the with 32 runs

and the with 16 runs. The aliases involving main effects and two- and three-factor inter-

actions for the 16-run design are presented in Table 14-30. Notice that all seven main effects

are aliased with three-factor interactions. All the two-factor interactions are aliased in groups

of three. Therefore, this design will satisfy our objectives; that is, it will allow the estimation

of the main effects, and it will give some insight regarding two-factor interactions. It is not

necessary to run the design, which would require 32 runs. The construction of the

design is shown in Table 14-31. Notice that it was constructed by starting with the 16-run

24 design in A, B, C, and Das the basic design and then adding the three columns E ABC,

F BCD, and G ACDas suggested in Table 14-29. Thus, the generators for this design

areI ABCE, I BCDF, and I ACDG. The complete defining relation is I ABCE

BCDF ADEF ACDG BDEG CEFG ABFG. This defining relation was used to

produce the aliases in Table 14-30. For example, the alias relationship of Ais

which, if we ignore interactions higher than three factors, agrees with Table 14-30.

For seven factors, we can reduce the number of runs even further. The 2^7 ^4 design is an

eight-run experiment accommodating seven variables. This is a 116th fraction and is ob-

tained by first writing down a 2^3 design as the basic design in the factors A, B, and C, and then

forming the four new columns from I ABD, I ACE, I BCF, and I ABCG, as sug-

gested in Table 14-29. The design is shown in Table 14-32.

The complete defining relation is found by multiplying the generators together two, three,

and finally four at a time, producing

The alias of any main effect is found by multiplying that effect through each term in the

BEGAFGDEFADEGCEFGBDFGABCDEFG

IABDACEBCFABCGBCDEACDFCDGABEF

ABCEABCDFDEFCDGABDEGACEFGBFG

(^2) IV^7 ^22 IV^7 ^3
(^2) IV^7 ^3
(^2) IV^7 ^2
Table 14-30 Generators, Defining Relation, and Aliases for the 2IV^7 ^3
Fractional Factorial Design
Generators and Defining Relation
EABC, FBCD, GACD
IABCEBCDFADEFACDGBDEGABFGCEFG
Aliases
ABCEDEFCDGBFG ABCEFG
BACECDFDEGAFG ACBEDG
CABEBDFADGEFG ADEFCG
DBCFAEFACGBEG AEBCDF
EABCADFBDGCFG AFDEBG
FBCDADEABGCEG AGCDBF
GACDBDEABFCEF BDCFEG
ABDCDEACFBEFBCGAEGDFG
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