Applied Statistics and Probability for Engineers

14-9 FRACTIONAL REPLICATION OF THE 2kDESIGN 549

14-9 FRACTIONAL REPLICATION OF THE 2kDESIGN

As the number of factors in a 2kfactorial design increases, the number of runs required

increases rapidly. For example, a 2^5 requires 32 runs. In this design, only 5 degrees of free-

dom correspond to main effects, and 10 degrees of freedom correspond to two-factor

interactions. Sixteen of the 31 degrees of freedom are used to estimate high-order interac-

tions—that is, three-factor and higher order interactions. Often there is little interest in

these high-order interactions, particularly when we first begin to study a process or system.

If we can assume that certain high-order interactions are negligible, a fractional factorial

designinvolving fewer than the complete set of 2kruns can be used to obtain information

on the main effects and low-order interactions. In this section, we will introduce fractional

replications of the 2kdesign.

A major use of fractional factorials is in screening experiments.These are experiments

in which many factors are considered with the purpose of identifying those factors (if any) that

have large effects. Screening experiments are usually performed in the early stages of a

project when it is likely that many of the factors initially considered have little or no effect

on the response. The factors that are identified as important are then investigated more thor-

oughly in subsequent experiments.

14-9.1 One-Half Fraction of the 2kDesign

A one-half fraction of the 2kdesign contains 2k^1 runs and is often called a 2k^1 fractional fac-

torial design. As an example, consider the 2^3 ^1 design—that is, a one-half fraction of the 2^3.

This design has only four runs, in contrast to the full factorial that would require eight runs.

The table of plus and minus signs for the 2^3 design is shown in Table 14-24. Suppose we se-

lect the four treatment combinations a, b, c, and abc, as our one-half fraction. These treatment

combinations are shown in the top half of Table 14-24 and in Fig. 14-28(a).

Notice that the 2^3 ^1 design is formed by selecting only those treatment combinations that

yield a plus on the ABCeffect. Thus, ABCis called the generatorof this particular fraction.

Factorial Effect

I A B C AB AC BC ABC

a 

b 

c 

abc 

ab 

ac 

bc 

112 

Table 14-24 Plus and Minus Signs for the 2^3 Factorial Design

Treatment

Combination

(a) Analyze the data from this experiment.

(b) Analyze the residuals and comment on model ade-

quacy.

(c) Comment on the efficiency of this design. Note that we

have replicated the experiment twice, yet we have no

information on the ABCinteraction.

(d) Suggest a better design, specifically, one that would

provide some information on allinteractions.

14-30. Consider the 2^6 factorial design. Set up a design to

be run in four blocks of 16 runs each. Show that a design that

confounds three of the four-factor interactions with blocks is

the best possible blocking arrangement.

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