Applied Statistics and Probability for Engineers

14-7 2kFACTORIAL DESIGNS 537

Projection of 2kDesigns

Any 2kdesign will collapse or project into another 2kdesign in fewer variables if one or more of

the original factors are dropped. Sometimes this can provide additional insight into the

remaining factors. For example, consider the surface roughness experiment. Since factor Cand

all its interactions are negligible, we could eliminate factor Cfrom the design. The result is to

collapse the cube in Fig. 14-18 into a square in the ABplane; therefore, each of the four runs

in the new design has four replicates. In general, if we delete hfactors so that rkhfactors

remain, the original 2kdesign with nreplicates will project into a 2rdesign with n 2 hreplicates.

14-7.3 Single Replicate of the 2kDesign

As the number of factors in a factorial experiment grows, the number of effects that can be

estimated also grows. For example, a 2^4 experiment has 4 main effects, 6 two-factor interac-

tions, 4 three-factor interactions, and 1 four-factor interaction, while a 2^6 experiment has 6

main effects, 15 two-factor interactions, 20 three-factor interactions, 15 four-factor interac-

tions, 6 five-factor interactions, and 1 six-factor interaction. In most situations the sparsity of

effects principleapplies; that is, the system is usually dominated by the main effects and low-

order interactions. The three-factor and higher order interactions are usually negligible.

Therefore, when the number of factors is moderately large, say, k 4 or 5, a common prac-

tice is to run only a single replicate of the 2kdesign and then pool or combine the higher order

interactions as an estimate of error. Sometimes a single replicate of a 2kdesign is called an

unreplicated 2 kfactorial design.

When analyzing data from unreplicated factorial designs, occasionally real high-order

interactions occur. The use of an error mean square obtained by pooling high-order interactions

is inappropriate in these cases. A simple method of analysis can be used to overcome this prob-

lem. Construct a plot of the estimates of the effects on a normal probability scale. The effects

that are negligible are normally distributed, with mean zero and variance 

2 and will tend to fall

along a straight line on this plot, whereas significant effects will have nonzero means and will

not lie along the straight line. We will illustrate this method in the next example.

• 2.250

1

8

10

30

20

60

70

80

90

95

• 1.653 –0.917 –0.250 0.417 1.083 1.750

99

Normal probability

Residual

Figure 14-20

Normal probability

plot of residuals from

the surface roughness

experiment.

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