Question 14.4: Table 14.5 lists the uncoded factor levels, coded factor lev...
Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2^{3} factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10.
R=\beta_0+\beta_a A+\beta_b B+\beta_c C+\beta_{a b} A B+\beta_{a c} A C+\beta_{b c} B C+\beta_{a b c} A B C (14.10)
Table 14.5 Uncoded and Coded Factor Levels and Responses for the 2³ Factorial Design of Example 14.4
| Run | A | B | C | A* | B* | C* | Response |
| 1 | 15 | 30 | 45 | +1 | +1 | +1 | 137.25 |
| 2 | 15 | 30 | 15 | +1 | +1 | –1 | 54.75 |
| 3 | 15 | 10 | 45 | +1 | –1 | +1 | 73.75 |
| 4 | 15 | 10 | 15 | +1 | –1 | –1 | 30.25 |
| 5 | 5 | 30 | 45 | –1 | +1 | +1 | 61.75 |
| 6 | 5 | 30 | 15 | –1 | +1 | –1 | 30.25 |
| 7 | 5 | 10 | 45 | –1 | –1 | +1 | 41.25 |
| 8 | 5 | 10 | 15 | –1 | –1 | –1 | 18.75 |
We begin by calculating the estimated parameters using equations 14.6-14.9 and 14.11-14.14.
\begin{aligned} &\beta_a \approx b_a=\frac{1}{n} \sum A_i^* R_i & (14.6)\\ &\beta_b \approx b_b=\frac{1}{n} \sum B_i^* R_i & (14.7) \end{aligned}
\left.\beta_{a b}\,\approx\,b_{a b}\,=\,\frac{1}{n}\sum A_{i}^{\ast}B_{i}^{\ast}R_{i}\right. (14.8)
R=15.0+2.0A^{\star}+5.0B^{\star}+0.5A^{\star}B^{\star} (14.9)
\begin{gathered} \beta_c \approx b_c=\frac{1}{n} \sum C_i^* R_i & (14.11)\\ \beta_{a c} \approx b_{a c}=\frac{1}{n} \sum A_i^* C_i^* R_i & (14.12)\\ \beta_{b c} \approx b_{b c}=\frac{1}{n} \sum B_i^* C_i^* R_i & (14.13)\\ \beta_{a b c} \approx b_{a b c}=\frac{1}{n} \sum A_i^* B_i^* C_i^* R_i & (14.14) \end{gathered}
\begin{aligned} & b_{0}=\frac{1}{8}(137.25+54.75+73.75+30.25+61.75+30.25+41.25+18.75)=56.0 \\ & b_{a}=\frac{1}{8}(137.25+54.75+73.75+30.25-61.75-30.25-41.25-18.75)=18.0 \\ & b_{b}=\frac{1}{8}(137.25+54.75-73.75-30.25+61.75+30.25-41.25-18.75)=15.0 \end{aligned}
\begin{aligned} & b_{c}=\frac{1}{8}(137.25-54.75+73.75-30.25+61.75-30.25+41.25-18.75)=22.5 \\ & b_{a b}=\frac{1}{8}(137.25+54.75-73.75-30.25-61.75-30.25+41.25+18.75)=7.0 \\ & b_{a c}=\frac{1}{8}(137.25-54.75+73.75-30.25-61.75+30.25-41.25+18.75)=9.0 \\ & b_{b c}=\frac{1}{8}(137.25-54.75-73.75+30.25+61.75-30.25-41.25+18.75)=6.0 \\ & b_{a b c}=\frac{1}{8}(137.25-54.75-73.75+30.25-61.75+30.25+41.25-18.75)=3.75 \end{aligned}
The coded empirical model, therefore, is
R=56+18 A^{\star}+15 B^{\star}+22.5 C^{\star}+7 A^{\star} B^{\star}+9 A^{\star} C^{\star}+6 B^{\star} C^{\star}+3.75 A^{\star} B^{\star} C^{\star}
To check the result we substitute the coded factor levels for the first run into the coded empirical model, giving
\begin{aligned} R= & 56+(18)(+1)+(15)(+1)+(22.5)(+1)+(7)(+1)(+1)+(9)(+1)(+1) \\ & +(6)(+1)(+1)+(3.75)(+1)(+1)(+1)=137.25 \end{aligned}
which agrees with the measured response.
To transform the coded empirical model into its uncoded form, it is necessary to replace A^{\star}, B^{\star}, and C^{\star} with the following relationships
A^{\star}=0.2 A-2 \quad B^{\star}=0.1 B-2 \quad C^{\star}=\frac{C}{15}-2
the derivations of which are left as an exercise. Substituting these relationships into the coded empirical model and simplifying (which also is left as an exercise) gives the following result for the uncoded empirical model
R=3+0.2 A+0.4 B+0.5 C-0.01 A B+0.02 A C-0.01 B C+0.005 A B C