Fractional Factorial Experiments

15.6 Fractional Factorial Experiments

The 2 k factorial experiment can become quite demanding, in terms of the number of experimental units required, when k is large. One of the real advantages of this experimental plan is that it allows a degree of freedom for each interaction. However, in many experimental situations, it is known that certain interactions are negligible, and thus it would be a waste of experimental effort to use the complete factorial experiment. In fact, the experimenter may have an economic constraint that disallows taking observations at all of the 2 k treatment combinations. When k is large, we can often make use of a fractional factorial experiment where per-

15.6 Fractional Factorial Experiments 627 haps one-half, one-fourth, or even one-eighth of the total factorial plan is actually

carried out.

Construction of 1

2 Fraction

The construction of the half-replicate design is identical to the allocation of the

2 k factorial experiment into two blocks. We begin by selecting a defining contrast that is to be completely sacrificed. We then construct the two blocks accordingly and choose either of them as the experimental plan.

A 1 fraction of a 2 2 k factorial is often referred to as a 2 k−1 design, the latter indicating the number of design points. Our first illustration of a 2 k−1 will be a 1 2 of 2 3 , or a 2 3−1 , design. In other words, the scientist or engineer cannot use the full complement (i.e., the full 2 3 with 8 design points) and hence must settle for a design with only four design points. The question is, of the design points (1), a,

b, ab, ac, c, bc, and abc, which four design points would result in the most useful design? The answer, along with the important concepts involved, appears in the table of + and − signs displaying contrasts for the full 2 3 . Consider Table 15.9.

Table 15.9: Contrasts for the Seven Available Effects for a 2 3 Factorial Experiment

Treatment

Effects

+ + − Note that the two 1 2 fractions are {a, b, c, abc} and {ab, ac, bc, (1)}. Note also

from Table 15.9 that in both designs ABC has no contrast but all other effects do have contrasts. In one of the fractions we have ABC containing all + signs, and in the other fraction the ABC effect contains all − signs. As a result, we say that the top design in the table is described by ABC = I and the bottom design by ABC = −I. The interaction ABC is called the design generator, and ABC = I (or ABC = −I for the second design) is called the defining relation.

Aliases in the 2 3−1

If we focus on the ABC = I design (the upper 2 3−1 ), it becomes apparent that six effects contain contrasts. This produces the initial appearance that all effects can be studied apart from ABC. However, the reader can certainly recall that with only four design points, even if points are replicated, the degrees of freedom available (apart from experimental error) are

628 Chapter 15 2 k Factorial Experiments and Fractions Regression model terms 3

Intercept

A closer look suggests that the seven effects are not orthogonal, and each contrast is represented in another effect. In fact, using ≡ to signify identical contrasts, we have

As a result, within a pair an effect cannot be estimated independently of its alias “partner.” The effects

2 2 will produce the same numerical result and thus contain the same information.

In fact, it is often said that they share a degree of freedom. In truth, the estimated effect actually estimates the sum, namely A + BC. We say that A and BC are aliases, B and AC are aliases, and C and AB are aliases.

For the ABC = −I fraction we can observe that the aliases are the same as those for the ABC = I fraction, apart from sign. Thus, we have

C ≡ −AB. The two fractions appear on corners of the cubes in Figures 15.15(a) and

A ≡ −BC;

B ≡ −AC;

(a ) The A B C = I fraction (b ) The A B C = − I fraction

Figure 15.15: The 1 2 fractions of the 2 3 factorial.

How Aliases Are Determined in General

In general, for a 2 k−1 , each effect, apart from that defined by the generator, will have a single alias partner. The effect defined by the generator will not be aliased

15.6 Fractional Factorial Experiments 629 by another effect but rather will be aliased with the mean since the least squares

estimator will be the mean. To determine the alias for each effect, one merely begins with the defining relation, say ABC = I for the 2 3−1 . Then to find, say, the alias for effect A, multiply A by both sides of the equation ABC = I and reduce any exponent by modulo 2. For example,

A · ABC = A,

thus, BC ≡ A.

In a similar fashion,

B ≡ B · ABC ≡ AB 2

C ≡ AC,

and, of course,

C ≡ C · ABC ≡ ABC 2 ≡ AB.

Now for the second fraction (i.e., defined by the relation ABC = −I),

C ≡ −AB. As a result, the numerical value of effect A is actually estimating A−BC. Similarly,

A ≡ −BC;

B ≡ −AC;

the value of B estimates B − AC, and the value of C estimates C − AB.

Formal Construction of the 2 k−1

A clear understanding of the concept of aliasing makes it very simple to understand the construction of the 2 k−1 . We begin with investigation of the 2 3−1 . There are three factors and four design points required. The procedure begins with a full factorial in k − 1 = 2 factors A and B. Then a third factor is added according to the desired alias structures. For example, with ABC as the generator, clearly

C = ±AB. Thus, C = AB or C = −AB is found to supplement the full factorial in A and B. Table 15.10 illustrates what is a very simple procedure.

Table 15.10: Construction of the Two 2 3−1 Designs

Basic 2 2 2 3−1 ; ABC = I

− Note that we saw earlier that ABC = I gives the design points a, b, c, and abc

while ABC = −I gives (1), ac, bc, and ab. Earlier we were able to construct the same designs using the table of contrasts in Table 15.9. However, as the design becomes more complicated with higher fractions, these contrast tables become more difficult to deal with.

Consider now a 2 4−1 (i.e., a 1 2 of a 2 4 factorial design) involving factors A, B,

C, and D. As in the case of the 2 3−1 , the highest-order interaction, in this case

630 Chapter 15 2 k Factorial Experiments and Fractions ABCD, is used as the generator. We must keep in mind that ABCD = I; the

defining relation suggests that the information on ABCD is sacrificed. Here we begin with the full 2 3 in A, B, and C and form D = ±ABC to generate the two

2 4−1 designs. Table 15.11 illustrates the construction of both designs.

Table 15.11: Construction of the Two 2 4−1 Designs

Basic 2 3 2 4−1 ; ABCD = I

2 4−1 ; ABCD = −I

A B C A B C D = ABC

A B C D = −ABC

+ + + − Here, using the notation a, b, c, and so on, we have the following designs:

ABCD = I, (1), ad, bd, ab, cd, ac, bc, abcd ABCD = −I, d, a, b, abd, c, acd, bcd, abc.

The aliases in the case of the 2 4−1 are found as illustrated earlier for the 2 3−1 . Each effect has a single alias partner and is found by multiplication via the use of the defining relation. For example, the alias for A for the ABCD = I design is given by

A = A · ABCD = A 2 BCD = BCD.

The alias for AB is given by

AB = AB · ABCD = A 2 B 2 CD = CD.

As we can observe easily, main effects are aliased with three-factor interactions and two-factor interactions are aliased with other two-factor interactions. A complete listing is given by

D = ABC.

Construction of the 1

4 Fraction

In the case of the 1 4 fraction, two interactions are selected to be sacrificed rather than one, and the third results from finding the generalized interaction of the

15.6 Fractional Factorial Experiments 631 selected two. Note that this is very much like the construction of four blocks.

The fraction used is simply one of the blocks. A simple example aids a great deal in seeing the connection to the construction of the 1 fraction. Consider the construction of 1 of a 2 5 4 2 factorial (i.e., a 2 5−2 ), with factors A, B, C, D, and E. One procedure that avoids the confounding of two main effects is the choice of ABD and ACE as the interactions that correspond to the two generators, giving ABD = I and ACE = I as the defining relations. The third interaction sacrificed

would then be (ABD)(ACE) = A 2 BCDE = BCDE. For the construction of the design, we begin with a 2 5−2 =2 3 factorial in A, B, and C. We use the interactions ABD and ACE to supply the generators, so the 2 3 factorial in A, B, and C is supplemented by factor D = ±AB and E = ±AC. Thus, one of the fractions is given by

abcde The other three fractions are found by using the generators {D = −AB, E = AC},

{D = AB, E = −AC}, and {D = −AB, E = −AC}. Consider an analysis of the above 2 5−2 design. It contains eight design points to study five factors. The aliases for main effects are given by

A(ABD) ≡ BD

A(ACE) ≡ CE

A(BCDE) ≡ ABCDE

C ≡ ABCD

≡ BCD Aliases for other effects can be found in the same fashion. The breakdown of

E ≡ ABDE

≡ AC

degrees of freedom is given by (apart from replication)

Main effects 5 Lack of fit

2 (CD = BE, BC = DE)

Total

We list interactions only through degree 2 in the lack of fit. Consider now the case of a 2 6−2 , which allows 16 design points to study six factors. Once again two design generators are chosen. A pragmatic choice to supplement a 2 6−2 =2 4 full factorial in A, B, C, and D is to use E = ±ABC and

F = ±BCD. The construction is given in Table 15.12. Obviously, with eight more design points than in the 2 5−2 , the aliases for main effects will not present as difficult a problem. In fact, note that with defining relations ABCE = ±I, BCDF = ±I, and (ABCE)(BCDF ) = ADEF = ±I,

632 Chapter 15 2 k Factorial Experiments and Fractions

Table 15.12: A 2 6−2 Design

Treatment

A B C D E = ABC

F = BCD Combination

main effects will be aliased with interactions that are no less complex than those of third order. The alias structure for main effects is written

A ≡ BCE ≡ ABCDF ≡ DEF,

D ≡ ABCDE ≡ BCF ≡ AEF,

B ≡ ACE ≡ CDF ≡ ABDEF,

E ≡ ABC ≡ BCDEF ≡ ADF,

F ≡ ABCEF ≡ BCD ≡ ADE, each with a single degree of freedom. For the two-factor interactions,

C ≡ ABE ≡ BDF ≡ ACDEF,

AB ≡ CE ≡ ACDF ≡ BDEF,

AF ≡ BCEF ≡ ABCD ≡ DE,

AC ≡ BE ≡ ABDF ≡ CDEF,

BD ≡ ACDE ≡ CF ≡ ABEF,

AD ≡ BCDE ≡ ABCF ≡ EF,

BF ≡ ACEF ≡ CD ≡ ABDE,

AE ≡ BC ≡ ABCDEF ≡ DF. Here, of course, there is some aliasing among the two-factor interactions. The

remaining 2 degrees of freedom are accounted for by the following groups:

ACD ≡ BDE ≡ ABF ≡ CEF. It becomes evident that we should always be aware of what the alias structure

ABD ≡ CDE ≡ ACF ≡ BEF,

is for a fractional experiment before we finally recommend the experimental plan. Proper choice in defining contrasts is important, since it dictates the alias structure.