Applied Statistics and Probability for Engineers

design (with a single replicate of each factorial run) augmented with five center points. The

design and the yield data are shown in Fig. S14-4.

Table S14-6 summarizes the analysis of variance for this experiment. The mean square

error is calculated from the center points as follows:

The average of the points in the factorial portion of the design is , and the average of

the points at the center is. The difference 40.425 40.46  

0.035

appears to be small. The curvature sum of squares in the analysis of variance table is computed

from Equation S14-12 as follows:

The analysis of variance indicates that both factors exhibit significant main effects, that there

is no interaction, and that there is no evidence of curvature in the response over the region of

exploration. That is, the null hypothesis cannot be rejected.

14-10 RESPONSE SURFACE METHODS AND DESIGNS (CD ONLY)

Response surface methodology, or RSM, is a collection of mathematical and statistical tech-

niques that are useful for modeling and analysis in applications where a response of interest is

influenced by several variables and the objective is to optimize this response.For example,

suppose that a chemical engineer wishes to find the levels of temperature (x 1 ) and feed con-

centration (x 2 ) that maximize the yield (y) of a process. The process yield is a function of the

levels of temperature and feed concentration, say,

Yf 1 x 1 , x 22 

H 0 : gkj 1 jj 0



1421521

 0.035 22

4  5

0.0027

SSCurvature

nFnC 1 yF yC 22

nFnC

yC40.46 yF yC

yF40.425



a

5

i 1

1 yi 40.46 22

4



0.1720

4

0.0430

MSE

SSE

nC 

1



a

Center points

1 yi yC 22

nC 

1

14-11

Table S14-6 Analysis of Variance for Example S14-2

Source of Sum of Degrees of Mean

Variation Squares Freedom Square f 0 P-Value

A(Time) 2.4025 1 2.4025 55.87 0.0017

B(Temperature) 0.4225 1 0.4225 9.83 0.0350

AB 0.0025 1 0.0025 0.06 0.8237

Curvature 0.0027 1 0.0027 0.06 0.8163

Error 0.1720 4 0.0430

Total 3.0022 8

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