Applied Statistics and Probability for Engineers

14-9

the test statistics for main effects and interactions have been constructed properly using

Equations S14-9 through S14-11.

14-7.4 Addition of Center Points to a 2kDesign (CD Only)

A potential concern in the use of two-level factorial designs is the assumption of linearity in the

factor effects. Of course, perfect linearity is unnecessary, and the 2ksystem will work quite well

even when the linearity assumption holds only approximately. However, there is a method of

replicating certain points in the 2kfactorial that will provide protection against curvature as well

as allow an independent estimate of error to be obtained. The method consists of adding center

pointsto the 2kdesign. These consist of nCreplicates run at the point xi0 (i1, 2,... , k). One

important reason for adding the replicate runs at the design center is that center points do not

affect the usual effects estimates in a 2kdesign. We assume that the kfactors are quantitative.

To illustrate the approach, consider a 2^2 design with one observation at each of the

factorial points ( , ), (, ), ( ,), and (,) and nCobservations at the center points

(0, 0). Figure S14-3 illustrates the situation. Let yFbe the average of the four runs at the four

Table S14-5 Minitab Analysis of Variance for the Breaking Strength Data in Table S14-1 Where

Operators Are Fixed and Machines Are Random (Mixed Model)

Analysis of Variance (Balanced Designs)

Factor Type Levels Values

Operator fixed 3 A B C

Machine random 4 1234

Analysis of Variance for Strength

Source DF SS MS F P

Operator 2 26.333 13.167 2.21 0.190

Machine 3 24.333 8.111 2.16 0.145

Operator*Machine 6 35.667 5.944 1.59 0.234

Error 12 45.000 3.750

Total 23 131.333

Figure S14-3 A 2^2

design with center

points.

y

B

A

–1

0

+1

+1

0

–1

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