552 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORSFor all three pairs of effect estimates, we would obtain the following results:
Effect,i from (lili) from (lili)
i A
i B
i C
Thus, by combining a sequence of two fractional factorial designs, we can isolate both the
main effects and the two-factor interactions. This property makes the fractional factorial de-
sign highly useful in experimental problems since we can run sequences of small, efficient ex-
periments, combine information across severalexperiments, and take advantage of learning
about the process we are experimenting with as we go along. This is an illustration of the con-
cept of sequential experimentation.
A 2k^1 design may be constructed by writing down the treatment combinations for a full
factorial with k 1 factors, called the basic design,and then adding the kth factor by identi-
fying its plus and minus levels with the plus and minus signs of the highest order interaction.
Therefore, a 2^3 ^1 fractional factorial is constructed by writing down the basic design as a full
22 factorial and then equating factor Cwith the ABinteraction. Thus, to construct the prin-
cipal fraction, we would use C ABas follows:(^1) 21 CABCAB 2 C (^1) 23 CAB 1 CAB 24 AB
(^1) 21 BACBAC 2 B (^1) 23 BAC 1 BAC 24 AC
(^1) 21 ABCABC 2 A (^1) 2 3 ABC 1 ABC 24 BC
(^1)
2
(^1)
2
Table 14-25 The 2^4 ^1 Design with Defining Relation I ABCD
Treatment Etch
ABC D ABC Combination Rate
550
ad 749
bd 1052
ab 650
cd 1075
ac 642
bc 601
abcd 729
112
Basic Design Fractional Design
Full 2^223 ^1 ,IABC
AB A BC AB
To obtain the alternate fraction we would equate the last column to C AB.
EXAMPLE 14-7 To illustrate the use of a one-half fraction, consider the plasma etch experiment described in
Example 14-5. Suppose that we decide to use a 2^4 ^1 design with I ABCDto investigate the
four factors gap (A), pressure (B), C 2 F 6 flow rate (C), and power setting (D). This design
would be constructed by writing down as the basic design a 2^3 in the factors A, B, and Cand
then setting the levels of the fourth factor D ABC. The design and the resulting etch rates
are shown in Table 14-25. The design is shown graphically in Fig. 14-29.
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