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1 STA 536 – Chapter 10 Response Surface Methodology Chapter 10 Response Surface Methodology To study the relationship between the response and the input factors. To optimize the response or to understand the underlying mechanism. The input factors are quantitative.

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2 STA 536 – Chapter 10 Response Surface Methodology 10.1 A Ranitidine Separation Experiment studied important factors in the separation of ranitidine and related products by capillary electrophoresis. From screening experiments, the investigators identified three factors as important: pH of the buffer solution, the voltage used in electrophoresis and the concentration of-CD, a component of the buffer solution. The response, chromatographic exponential function (CEF), is a quality measure in terms of separation achieved and time of final separation. the goal is to minimize CEF.

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3 STA 536 – Chapter 10 Response Surface Methodology The experiment used a central composite design: Runs 1–8 form a 2 3 design. Because they are on the corners of the 2 3 cube, they are called cube points or corner points. Runs 9–14 form three pairs of points along the three coordinate axes and are therefore called axial points or star points. Runs 15–20 are at the center of the design region and are called center points. These 15 distinct points of the design are represented in the three dimensional space. This is a second-order design in the sense that it allows all the linear and quadratic components of the main effects and the linear-by-linear interactions to be estimated.

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4 STA 536 – Chapter 10 Response Surface Methodology Table 10.1 Factors and Levels, Ranitidine Experiment

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5 STA 536 – Chapter 10 Response Surface Methodology Table 10.2 Design Matrix and Response Data, Ranitidine Experiment cube points or corner points 2 3 axial points or star points center points

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6 STA 536 – Chapter 10 Response Surface Methodology Fig. 10.1 A Central Composite Design in Three Dimensions [cube point (dot), star point (cross), center point (circle)]

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7 STA 536 – Chapter 10 Response Surface Methodology 10.2 Sequential Nature of Response Surface Methodology input factors (also called input variables or process variables) X1,X2,...,Xk (in original scales) relationship between the response y and X1,X2,...,Xk y = f(X1,X2,...,Xk) +, (1) where the form of the true response function f is unknown and is an error term that represents the sources of variability not captured by f. Assume the over different runs are independent and have mean zero and variance 2. Response surface study

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8 STA 536 – Chapter 10 Response Surface Methodology coded variables: x 1, x 2,..., x k. The linear function x = (X x i )/c i transforms original (actual) levels x ic i, x i c i, x i, x i + c i, x i +c i to coded levels, 1, 0, 1,. In coded form y = f(x 1, x 2,..., x k ) +. (2) Response surface methodology (Box and Wilson, 1951) involves experimentation, modeling, data analysis and optimization.

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9 STA 536 – Chapter 10 Response Surface Methodology Three stages of RSM: Initial screening stage (to identify a few important factors from many factors) Sequential search stage (to identify an optimum design region) A final stage of response surface study (to obtain an accurate approximation of the response surface)

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10 STA 536 – Chapter 10 Response Surface Methodology Designs involved in three stages Highly fractionated designs 2 kp, 3 kp, Plackett-Burman designs and orthogonal arrays first-order designs such as resolution III 2 kp designs and Plackett-Burman designs (with some center points). second-order designs such as central composite designs.

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11 STA 536 – Chapter 10 Response Surface Methodology Models A first-order model is A second-order model is

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12 STA 536 – Chapter 10 Response Surface Methodology Fig. 10.2 Sequential Exploration of the Response Surface Q: Which direction to search? Q: When to switch from a first-order experiment to a second-order experiment? Q: Which direction to search? Q: When to switch from a first-order experiment to a second-order experiment?

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13 STA 536 – Chapter 10 Response Surface Methodology Initial first order expriments

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14 STA 536 – Chapter 10 Response Surface Methodology Two steepest ascent searches

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15 STA 536 – Chapter 10 Response Surface Methodology First order experiments

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16 STA 536 – Chapter 10 Response Surface Methodology

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17 STA 536 – Chapter 10 Response Surface Methodology 10.3 From First-Order Experiments to Second-Order Experiments: Steepest Ascent Search and Rectangular Grid Search The first-order experiment uses a two-level orthogonal design with run size n f and n c center point runs are added. = the sample average over the factorial runs = the sample average at the center points Coded values: 1 and +1 for the low and high levels of the factorial design and 0 for the level of the center point. 10.3.1 Curvature Check

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18 STA 536 – Chapter 10 Response Surface Methodology Under the second-order model in (4), Then That is, we can use the difference to test if the overall curvature is zero. Note variance of is

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19 STA 536 – Chapter 10 Response Surface Methodology If the curvature check is not significant, the search may continue with the use of another first-order experiment and steepest ascent. Otherwise, it should be switched to a second-order experiment. Two purposes of adding center points to a first-order experiment (i) it allows the check of the overall curvature effect, (ii) it provides an unbiased estimate of the process error variance.

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20 STA 536 – Chapter 10 Response Surface Methodology 10.3.2 Steepest Ascent Search Suppose the fitted first-order model (3) is Taking the partial derivative of ˆy with respect to xi, The steepest ascent direction (for maximization) is The steepest ascent direction (for minimization) is

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21 STA 536 – Chapter 10 Response Surface Methodology Example: To maximize the yield of a chemical reaction whose factors are time and temperature. The first-order design is a 2 2 design with two center points (see runs 1-6 in Table 10.3 and Fig. 10.3). The and + levels are: 75, 85 minutes for time and 180 o, 190 o C for temperature. The center points (0, 0) correspond to 80 minutes and 185 o C. The contours of the true response surface is unknown.

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22 STA 536 – Chapter 10 Response Surface Methodology Table 10.3 Design Matrix and Yield Data for First-Order Design Run Time Temperature Yield

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23 STA 536 – Chapter 10 Response Surface Methodology Fig. 10.3 First-Order Experiment With Steepest Ascent time

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24 STA 536 – Chapter 10 Response Surface Methodology Analysis To analyze the first-order experiment, the following model is fitted: where x1 and x2 are coded time and temperature from runs 1-6 of Table 10.3. Note that x 1 2 = x 2 2 so that the coefficient for x 1 2 is 11 + 22, which is the measure of overall curvature. Table 10.4 Analysis of Initial First-Order Experiment

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25 STA 536 – Chapter 10 Response Surface Methodology Analysis of Initial First-Order Experiment There is no indication of interaction and curvature, which suggests that a steepest ascent search should be conducted. The steepest ascent direction is proportional to (7.622, 8.942), or equivalently, (1, 1.173). Increasing time in steps of 2 units or 10 minutes is chosen. The results for three steps appear as runs 7-9 in Table 10.3. Times (2,4,6) corresponds to 90, 100 and 110 minutes and temperatures (1.173)(2, 4, 6) correspond to 173.27, 161.54 and 149.81 o C.

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26 STA 536 – Chapter 10 Response Surface Methodology The next experiment A first-order experiment of runs 1-6 in Table 10.5 and Fig. 10.4, centered at 100 minutes and 161.54 o C (coded (0, 0)) Coded 1 and +1 values correspond to ±5 o C and ± 5 minutes run 5 is from the second step of the steepest ascent search (run 8 in Table 10.3)

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27 STA 536 – Chapter 10 Response Surface Methodology Table 10.5 Design Matrix and Yield Data for Second-Order Design Run Time Temperature Yield

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28 STA 536 – Chapter 10 Response Surface Methodology Fig. 10.4 Second-Order Experiment

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29 STA 536 – Chapter 10 Response Surface Methodology Fitting the regression model with runs 1–6 leads to Table 10.6 Analysis of Follow-up First-Order Experiment

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30 STA 536 – Chapter 10 Response Surface Methodology Second-Order Experiment There are significant interaction and curvature effects. This suggests augmenting the first-order design so that runs at the axial points of a central composite design are performed. See runs 7-10 in Table 10.5 and the end points of the cross in Figure 10.4. The axial points correspond to 92.95 and 107.05 minutes for time and 154.49 and 168.59 o C for temperatures, i.e., ±sqrt(2) in coded units.

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31 STA 536 – Chapter 10 Response Surface Methodology Fitting a second-order model with runs 1–10 leads to Table 10.7 Analysis of Second-Order Experiment

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32 STA 536 – Chapter 10 Response Surface Methodology The estimated response surface model is suggests that moving in a north-east direction would increase the yield, namely by increasing the time and temperature.

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33 STA 536 – Chapter 10 Response Surface Methodology Fig. 10.5 Estimated Response Surface

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34 STA 536 – Chapter 10 Response Surface Methodology 10.3.3 Rectangular Grid Search Fig. 10.6 Search for the Peak of Unknown Response Curve (dashed curve): initial design over [1, 1], second design over [0,1] Initial design: A = 1,C = 0,B = 1 second design: C = 0,D = 0.5,B = 1.

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35 STA 536 – Chapter 10 Response Surface Methodology Table 10.8 Factors and Levels, Ranitidine Screening Experiments

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36 STA 536 – Chapter 10 Response Surface Methodology Table 10.9 Design Matrix, Ranitidine Screening Experiment

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37 STA 536 – Chapter 10 Response Surface Methodology employed the rectangular grid search strategy in two first-order (i.e., screening) experiments The first screening experiment was based on a 16-run 2 IV 62 design(with I = 1235 = 2346 = 1456) plus four runs at the center point (see Tables 10.8 and 10.9). a very high overall curvature, =1607011= 16059, from the first experiment. suggest that one or more factors would attain the minimum within the range. In the second experiment, the factor range is narrowed to the left or right half (see the right half of Table 10.8). methanol and buffer concentration (NaH 2 PO 4 ) are not significant at the 10% level in both screening experiments. Temperature is significant at the 10% level for the second experiment but was dropped for further study due to practical difficulty. Table 10.10 shows the analysis results for the remaining three factors, pH, voltage and -CD. The ranitidine experiment

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38 STA 536 – Chapter 10 Response Surface Methodology Table 10.10 Selected Analysis Results, Ranitidine Screening Experiments Because the t statistics for the second experiment have a negative sign, the CEF is a decreasing function over the ranges of the three factors. This suggests that the factor range should be extended to the right by taking the rightmost level of the second screening design to be the center point of a new central composite design (see Tables 10.1)

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39 STA 536 – Chapter 10 Response Surface Methodology 10.4 Analysis of Second-Order Response Surfaces Read it! (but skip 10.4.1) 10.5 Analysis of the Ranitidine Experiment Analysis of data from Table 10.2. Run 7 is dropped in the analysis (due to a blockage occurred in the separation). The response CEF is transformed to the natural log scale (due to the large range). Fitting the second-order model (without run 7) leads to

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40 STA 536 – Chapter 10 Response Surface Methodology Table 10.11 Analysis of Ranitidine Experiment (Run 7 Dropped) only factors pH and voltage are important.

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41 STA 536 – Chapter 10 Response Surface Methodology Fig. 10.9 Estimated Response Surface (Run 7 Dropped)

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42 STA 536 – Chapter 10 Response Surface Methodology A follow-up second-order experiment in pH and voltage The range of pH (A) was narrowed. The levels are (4.19, 4.50, 5.25, 6.00, 6.31) The levels of voltage (B) are (11.5, 14.0, 20.0, 26.0, 28.5) The coded values are (1.41, 1, 0, 1, +1.41).

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43 STA 536 – Chapter 10 Response Surface Methodology Table 10.12 Final Second-Order Ranitidine Experiment Fitting the second-order model

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44 STA 536 – Chapter 10 Response Surface Methodology Table 10.13 Analysis of Final Second- Order Ranitidine Experiment

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45 STA 536 – Chapter 10 Response Surface Methodology Fig. 10.10 Estimated Response Surface, Final Second- Order Ranitidine Experiment

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46 STA 536 – Chapter 10 Response Surface Methodology The final response surface model (based on Table 10.13) is In matrix form, where

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47 STA 536 – Chapter 10 Response Surface Methodology Differentiating ˆy in (6) with respect to x and setting it to 0, leads to the solution which is called the stationary point of the quadratic surface. The minimum value is y s = 1.1104, achieved at x s (pH of 4.87 and voltage of 23.84) (see Fig. 10.10).

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48 STA 536 – Chapter 10 Response Surface Methodology 10.6 Analysis Strategies for Multiple Responses II: Contour Plots and the Use of Desirability Functions Skipped.

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49 STA 536 – Chapter 10 Response Surface Methodology 10.7 Central Composite Designs A central composite design (Box and Wilson, 1951) of k input factors x = (x1,..., xk) consists of the following three parts: (i) n f cube points (or corner points) with x i = 1 or 1 for i = 1,..., k. They form the factorial portion of the design. (ii) n c center points with xi= 0 for i = 1,..., k. (iii) 2k star points (or axial points) of the form (0, · · ·, xi, · · ·, 0) with x i = or for i = 1,..., k. The central composite design can be used in a single experiment or in a sequential experiment.

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50 STA 536 – Chapter 10 Response Surface Methodology construct central composite design

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51 STA 536 – Chapter 10 Response Surface Methodology construct central composite design

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52 STA 536 – Chapter 10 Response Surface Methodology Choice of the Factorial Portion The second-order model The total number of distinct design points in a central composite design is N = n f + 2k + 1. require: N = n f + 2k + 1 (k + 1)(k + 2)/2.

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53 STA 536 – Chapter 10 Response Surface Methodology Theorem 10.1. In any central composite design whose factorial portion is a 2 kp design that does not use any main effect as a defining relation, the following parameters in the second-order model are estimable: 0,i, ii, i = 1,..., k, and one ij selected from each set of aliased effects for i < j. It is not possible to estimate more than one ij from each set of aliased effects. defining words of length two are allowed. words of length four are worse than words of length three. Reasons: the linear and quadratic effects can be estimated by exploiting the information in the star points. a central composite design has three or five levels for each factor. however, the star points do not contain information on the interactions ij.

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54 STA 536 – Chapter 10 Response Surface Methodology Any resolution III design whose defining relation does not contain words of length four is said to have resolution III*. Any central composite design whose factorial portion has resolution III* is a second-order design.

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55 STA 536 – Chapter 10 Response Surface Methodology Table 10A.1 A collection of small central composite designs. For k=2, use the resolution II design with I = AB or 2 2 For k=3, use the 2 III 31 or the 2 3 design For k=4, use the 2 III 41 design with I = ABD or the 2 4 design For k=5, use the 2 V 51 design For k=6, use the 2 III* 62 design with I = ABE = CDF = ABCDEF. (The minimum aberration 2 62 design has resolution IV and is not a good choice.) For k = 7, use the 2 III* 72 design I = ABCDF = DEG = ABCEFG. (The minimum aberration 2 72 design has resolution IV and is not a good choice.)

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56 STA 536 – Chapter 10 Response Surface Methodology Smaller central composite designs can be found by using the Plackett-Burman designs for the factorial portion. See Table 10A.1. A composite design consists of the cube and star points of the central composite design and its run size is n f + 2k. A composite design is a second-order design if Choice of In general, should be chosen between 1 and sqrt(k) and rarely outside this range. For =1, the star points are placed at the center of the faces of the cube. The design is therefore called the face center cube. This choice has two advantages. they are the only central composite designs that require three levels. they are effective designs if the design region is a cube. The choice of =sqrt(k) makes the star points and cube points lie on the same sphere. The design is often referred to as a spherical design. It is more suitable for spherical design region.

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57 STA 536 – Chapter 10 Response Surface Methodology Let ˆy(x) be the predicted response at x = (x1,..., xk). Definition: A design is called rotatable if Var(ˆy(x)) depends only on (x 1 2 + · · · + x k 2 ) 1/2, that is, if the accuracy of prediction of the response is the same on a sphere around the center of the design. A central composite design whose factorial portion is a 2 V kp design is rotatable if (7) Equation (7) serves as a useful guide for the choice of. For the ranitidine experiment (see Table 10.1) n f = 8 and = 1.682 =1.68(= 3.5/2.08) was chosen for factors A and B =1.67(= 5/3) was chosen for factor C

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58 STA 536 – Chapter 10 Response Surface Methodology Number of Runs at the Center Point For (which is suitable for spherical regions), at least one center point is required for the estimability of the parameters. The rule of thumb is to take 3 to 5 runs at the center point when is close to sqrt(k). When is close to 1 (which is suitable for cubical regions), 1 or 2 runs at the center point should be sufficient.

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59 STA 536 – Chapter 10 Response Surface Methodology 10.8 Box-Behnken Designs and Uniform Shell Designs Skipped. 10.9 Practical Summary Read it.

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