Response Surface Methodology

Abstract

Experiments for fitting a predictive model involving several continuous variables are known as response surface experiments. The objectives of response surface methodology include the determination of variable settings for which the mean response is optimized and the estimation of the response surface in the vicinity of this good location. The first part this chapter discusses first-order designs and first-order models, including lack of fit and the path of steepest ascent to locate the optimum. The second part of the chapter introduces second-order designs and models for exploring the vicinity of the optimum location. The application of response surface methodology is demonstrated through a real experiment. The concepts introduced in this chapter are illustrated through the use of SAS and R software.

Exercises

• 1. Paint experiment, continued

  The paint experiment of Eibl et al. (1992) was discussed in Example 16.2.1 (p. 569), where the first-order model was fitted to the data. For the fitted first-order model, do the following.

  • (a) Plot the residuals versus run order, and use the plot to check the independence assumption. (The order of the observations was not randomized in this experiment. Rather, the observations were collected in the order they are shown row by row in Table 16.1, p. 570.)

  • (b) Plot the residuals versus the predicted values, and use the plot to check the assumption of equal variance.

  • (c) Plot the residuals versus their normal scores, and use the plot to check the normality assumption.

  • (d) Verify that the design is orthogonal.

• 2. Paint followup experiment

  The data of the second paint experiment described by Eibl et al. (1992) are given in Table 16.23. This experiment involves factors A–D, as these had significant effects in the first experiment (Example 16.2.1). The factors are

  $$\begin{aligned} \begin{array}{ll} A{:}\,\text {belt speed} &{} B{:}\,\text {tube width}\\ C{:}\,\text {pump pressure} &{} D{:}\,\text {paint viscosity} \end{array} \end{aligned}$$

  All four factors are at lower levels than in the first experiment. Lowering the levels of factors B–D was indicated by the analysis of the first experiment. Lowering the level of factor A was based on a conjecture of the experimenters.  

  Table 16.23 Paint thickness $y_{\mathbf{z}t}$ for the paint followup experiment

   

  • (a) The experiment consists of two replicates of a half-fraction. Find the defining relation for the half-fraction.

  • (b) Fit the first-order model, recoding the factor levels as $\pm 1$.

  • (c) Test for lack of fit of the first-order model.

  • (d) What would you recommend the experimenters do next?

• 3. Fractionation experiment

  Sosada (1993) studied the effects of extraction time, solvent volume, ethanol concentration, and temperature on the yield and phosphatidylcholine enrichment (PCE) of deoiled rapeseed lecithin when fractionated with ethanol. Initially, a single-replicate $2^4$ experiment was conducted, augmented by three center points.

  • (a) The results for the 16 factorial points are shown as the first 16 runs in Table 16.24. Fit the first-order model for the response variable “PCE” and conduct the corresponding analysis of variance.

  • (b) The design also included $n_0=3$ center-point observations of PCE. The sample variance of these three observations was $s^2_0 = 1.120$. Test the first-order model for lack of fit, using a 5% level of significance. (Hint: Since the factorial points include no replication, ${{\textit{msPE}}} = s_0^2$, and ssE based on all 19 runs is equal to ssE from the factorial portion of the design plus $(n_0-1)s_0^2$.)

  • (c) Based on the results of parts (a) and (b), what subsequent experimentation would you recommend?

• 4. Fractionation experiment, continued

  The fractionation experiment was described in Exercise 3, where the response PCE was used. Consider here, instead, the analysis of “Yield”.

  • (a) Fit the first-order model for the response variable “Yield” based on the initial first-order $2^4$ factorial design, shown as the first 16 runs in Table 16.24. Conduct the corresponding analysis of variance.

  • (b) At the design center point, three additional observations were collected, for which the sample variance was $s^2_0 = 0.090$. Test the first-order model for lack of fit, using a 5% level of significance. (Hint: Since the factorial points include no replication, ${{\textit{msPE}}} = s_0^2$, and ssE based on all 19 runs is equal to ssE from the factorial portion of the design plus $(n_0-1)s_0^2$.)

  • (c) Based on the results of parts (a) and (b), what subsequent experimentation would you recommend?

  Table 16.24 Purified lecithin yield and phosphatidylcholine enrichment (PCE), given extraction time ($z_1$), solvent volume ($z_2$), ethanol concentration ($z_3$), and temperature ($z_4$); fractionation experiment

• 5. Fractionation experiment, continued

  The fractionation experiment was described in Exercise 3, and analysis of the first-order model for “Yield” was considered in Exercise 4. Based on the analysis of the first-order design, the experimenter chose to augment the 16 factorial points of the first-order design into a 25-run central composite design, the yields from which are shown in Table 16.24.

  • (a) Determine whether the central composite design used is rotatable or orthogonal.

  • (b) Fit the second-order response surface model and determine which effects are significantly different from zero.

  • (c) Conduct a canonical analysis and discuss the results with respect to the following items. What is the nature of the critical point? Noting that the objective is to increase yield, in what direction should one move in subsequent experimentation?

  Table 16.25 Apparent viscosity $y_{\mathbf{z}t}$ of film-forming solution, for combinations of levels of protein concentration (g/100 g solution), pH, and temperature ($^{\circ }$C)

• 6. Film viscosity experiment

  Cuq et al. (1995, Journal of Food Science) used a central composite design to study the effects of protein concentration (g/100 g solution), pH, and temperature ($^\circ$C), denoted by P, H, and T, respectively, on the apparent viscosity Y (mPa) of film-forming solution, in the development of edible packaging films based on fish myofibrillar proteins. The data are shown in Table 16.25.

  • (a) Is this central composite design rotatable or orthogonal?

  • (b) Fit the second-order model to the data using the coded factor levels, and check the model assumptions. Would you recommend that a transformation of the data be taken?

  • (c) Fit the second-order model to the natural log of the data, $\ln (y)$, using the coded factor levels.

  • (d) Conduct the test for lack of fit of the second-order model for $\ln (y)$.

  • (e) Check the model assumptions for $\ln (y)$.

  • (f) Conduct the canonical analysis for $\ln (y)$.

  • (g) Conduct the analysis of variance for $\ln (y)$.

  • (h) Compute the coefficient of multiple determination $R^2$ for the second-order model for $\ln (y)$.

  • (i) Assess the results of the experiment, based on the model for $\ln (y)$.

• 7. Flour production experiment, continued

  Consider again the flour production experiment of Sect. 16.5. The data were given in Table 16.11 (p. 590), along with the statistics $\overline{y}_{.\mathbf{z}}$ and $100\log _{10}(s_\mathbf{z})$ computed for the observations at each design-factor combination z.

  • (a) Plot $\log _{10}(s_\mathbf{z})$ versus $\log _{10}(\overline{y}_{.\mathbf{z}})$, and use the methods of Sect. 5.6.2 to determine an appropriate variance-stabilizing transformation for these data. (Use of $\log _{10}$ is equivalent to use of $\ln$ for choosing a transformation.)

  • (b) Repeat the first analysis of variance of Sect. 16.5, for which the response variable was $\overline{y}_{.\mathbf{z}}$, after applying the transformation determined in part (a) to the observations $y_{h\mathbf{z}}$. Compare your conclusions with those reached in Sect. 16.5.

  • (c) Repeat the second analysis of variance of Sect. 16.5, for which the response variable was $100\log _{10}(s_\mathbf{z})$, after applying the transformation determined in part (a) to the observations $y_{h\mathbf{z}}$. Compare your conclusions to those reached in Sect. 16.5.

• 8. Central composite design

  Consider using a central composite design for three factors, to include eight factorial points and six axial points.

  • (a) Determine the value of $\alpha$ to make the design rotatable.

  • (b) Investigate how $\alpha$ and the number of center points should be chosen to make the design both rotatable and orthogonal, if possible. If this is not possible, how can the design be made rotatable and nearly orthogonal?

  • (c) Investigate whether the design can be rotatable with orthogonal blocking. If not, then investigate whether orthogonal blocking is possible. If so, how many blocks could be used? Investigate whether orthogonal blocking and near rotatability is possible.

• 9. Central composite design

  Repeat Exercise 8 for a central composite design for four factors, to include 16 factorial points and eight axial points.

  Table 16.26 Resin impurity content $y_{\mathbf{z}t}$ (ppm)

• 10. Resin impurity experiment

  An experiment was conducted using a design close to a central composite design to study the effects of drying time (hours) and temperature ($^\circ$C) on the content y (ppm) of undesirable compounds in a resin. The data are shown in Table 16.26.

  • (a) Determine the coded levels of time and temperature, as well as the values of $n_f$, $n_a$, $n_0$. What values of $\alpha$ for each factor were selected by the experimenters for the axial points? Why is the design not quite a central composite design?

  • (b) Fit the second-order model, using coded factor levels.

  • (c) Test for model lack of fit.

  • (d) Check the equal variance and normality assumptions of the model using residual plots.

  • (e) Conduct the canonical analysis.

  • (f) Conduct the analysis of variance.

  • (g) Summarize the results.

  Table 16.27 Resin degradation (ppm) for the resin moisture experiment

• 11. Resin moisture experiment

  A Box–Behnken design was used to determine whether specific drying conditions for a process could yield a resin that is sufficiently devoid of moisture and low-molecular-weight components. The three factors T, H, and P under study were temperature (150, 185, 220$^\circ$C), relative humidity (0, 50, 100%), and air pressure (1, 5, 9 torr). The response variable y was a measure of product degradation (ppm). The design and data are shown in Table 16.27.

  • (a) Fit the second-order model, using coded factor levels.

  • (b) Test for model lack of fit.

  • (c) Check the equal-variance and normality model assumptions using residual plots.

  • (d) Conduct the canonical analysis.

  • (e) Conduct the analysis of variance.

  • (f) Summarize the results.

• 12. Box–Behnken design

  • (a) Construct a Box–Behnken design for three factors based on the balanced incomplete block design for three treatments in three blocks of size two and the $2^2$ factorial design.

  • (b) Determine whether the design constructed in part (a) is rotatable.

  • (c) For the design constructed in part (a), determine whether orthogonal blocking is possible.

• 13. Box–Behnken design

  • (a) Construct a Box–Behnken design for five factors based on the balanced incomplete block design for five treatments in 10 blocks of size two.

  • (b) Determine whether the design constructed in part (a) is rotatable.

  • (c) For the design constructed in part (a), determine whether orthogonal blocking is possible.