Applied Statistics and Probability for Engineers

factors requires 2kfactorial runs, 2kaxial runs, and at least one center point (three to five cen-

ter points are typically used). Designs for k2 and k3 factors are shown in Fig. S14-14.

The central composite design may be made rotatableby proper choice of the axial

spacing in Fig. S14-14. If the design is rotatable, the standard deviation of predicted

response is constant at all points that are the same distance from the center of the design. For

rotatability, choose (F)^1 ^4 , where Fis the number of points in the factorial part of the design

(usually F 2 k). For the case of k2 factors, (2^2 )^1 ^4 1.414, as was used in the

design in Example S14-4. Figure S14-15 presents a contour plot and a surface plot of the stan-

dard deviation of prediction for the quadratic model used for the yield response. Notice that

the contours are concentric circles, implying that yield is predicted with equal precision for all

points that are the same distance from the center of the design. Also, as one would expect, the

precision decreases with increasing distance from the design center.

yˆ

14-19

48.00

158.0

50.33 52.67 55.00 57.33 59.67 62.00

160.3

162.7

165.0

167.3

169.7

1 172.0

• 1

x^2

(temperature)

x 1 (time)

• 1+1 0

0

Contour plot

(a)

62.00

59.20

(b)

56.40

53.60

50.80

158.0 48.00

160.8

163.6

166.4

169.2

172.0

27.35

33.83

40.32

46.81

x 2 (temperature) x 1 (time)

Viscosity

+1

0

0

+1

– (^1) – 1
Surface plot
46.00
44.00
42.00
40.00
38.00
36.00
34.00
32.00
30.00
Figure S14-12
Response surface
plots for the
viscosity response,
Example S14-4.
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