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**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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**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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**The experiment used a central composite design:**

Runs 1–8 form a 23 design. Because they are on the corners of the 23 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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**Table 10.1 Factors and Levels, Ranitidine Experiment**

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**Table 10.2 Design Matrix and Response Data, Ranitidine Experiment**

cube points or corner points 23 axial points or star points center points

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Fig A Central Composite Design in Three Dimensions [cube point (dot), star point (cross), center point (circle)]

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**10.2 Sequential Nature of Response Surface Methodology**

Response surface study 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.

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coded variables: x1, x2, , xk. The linear function x = (X − xi)/ci transforms original (actual) levels xi−ci, xi − ci, xi, xi + ci, xi +ci to coded levels −, −1, 0, 1, . In coded form y = f(x1, x2, , xk) + . (2) Response surface methodology (Box and Wilson, 1951) involves experimentation, modeling, data analysis and optimization.

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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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**Designs involved in three stages**

Highly fractionated designs 2k−p, 3k−p, Plackett-Burman designs and orthogonal arrays first-order designs such as resolution III 2k−p designs and Plackett-Burman designs (with some center points). second-order designs such as central composite designs.

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Models A first-order model is A second-order model is

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**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?

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**Initial first order expriments**

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**Two steepest ascent searches**

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**First order experiments**

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10.3 From First-Order Experiments to Second-Order Experiments: Steepest Ascent Search and Rectangular Grid Search Curvature Check The first-order experiment uses a two-level orthogonal design with run size nf and nc 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.

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**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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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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**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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**Example: To maximize the yield of a chemical reaction whose factors are time and temperature.**

The first-order design is a 22 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 180o, 190oC for temperature. The center points (0, 0) correspond to 80 minutes and 185oC. The contours of the true response surface is unknown.

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**Table 10.3 Design Matrix and Yield Data for First-Order Design Run Time Temperature Yield**

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**Fig. 10.3 First-Order Experiment With Steepest Ascent time**

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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 x12= x22 so that the coefficient for x12 is 11 +22, which is the measure of overall curvature. Table 10.4 Analysis of Initial First-Order Experiment

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**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 , and oC.

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The next experiment A first-order experiment of runs 1-6 in Table 10.5 and Fig. 10.4, centered at 100 minutes and oC (coded (0, 0)) Coded −1 and +1 values correspond to ±5oC 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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**Table 10.5 Design Matrix and Yield Data for Second-Order Design Run Time Temperature Yield**

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**Fig. 10.4 Second-Order Experiment**

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**Fitting the regression model**

with runs 1–6 leads to Table 10.6 Analysis of Follow-up First-Order Experiment

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**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 and minutes for time and and oC for temperatures, i.e., ±sqrt(2) in coded units.

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**Fitting a second-order model**

with runs 1–10 leads to Table 10.7 Analysis of Second-Order Experiment

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**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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**Fig. 10.5 Estimated Response Surface**

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**10.3.3 Rectangular Grid Search**

Fig 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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**Table 10.8 Factors and Levels, Ranitidine Screening Experiments**

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**Table 10.9 Design Matrix, Ranitidine Screening Experiment**

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**The ranitidine experiment**

employed the rectangular grid search strategy in two first-order (i.e., screening) experiments The first screening experiment was based on a 16-run 2IV6−2 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, =16070−11= 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 (NaH2PO4) 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 shows the analysis results for the remaining three factors, pH, voltage and -CD.

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**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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**10.4 Analysis of Second-Order Response Surfaces**

Read it! (but skip ) 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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**Table 10.11 Analysis of Ranitidine Experiment (Run 7 Dropped)**

only factors pH and voltage are important.

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**Fig. 10.9 Estimated Response Surface (Run 7 Dropped)**

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**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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**Table 10.12 Final Second-Order Ranitidine Experiment**

Fitting the second-order model

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**Table 10.13 Analysis of Final Second-Order Ranitidine Experiment**

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**Fig. 10.10 Estimated Response Surface, Final Second-Order Ranitidine Experiment**

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**The final response surface model (based on Table 10.13) is**

In matrix form, where

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**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 ys = , achieved at xs (pH of 4.87 and voltage of 23.84) (see Fig ).

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10.6 Analysis Strategies for Multiple Responses II: Contour Plots and the Use of Desirability Functions Skipped.

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**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) nf cube points (or corner points) with xi= −1 or 1 for i = 1, , k. They form the factorial portion of the design. (ii) nc center points with xi= 0 for i = 1, , k. (iii) 2k star points (or axial points) of the form (0, · · · , xi, · · · , 0) with xi= 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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**construct central composite design**

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**construct central composite design**

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**Choice of the Factorial Portion**

The second-order model The total number of distinct design points in a central composite design is N = nf + 2k + 1. require: N = nf + 2k + 1 ≥ (k + 1)(k + 2)/2.

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Theorem In any central composite design whose factorial portion is a 2k−p 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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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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**Table 10A.1 A collection of small central composite designs.**

For k=2, use the resolution II design with I = AB or 22 For k=3, use the 2III3−1 or the 23 design For k=4, use the 2III4−1 design with I = ABD or the 24 design For k=5, use the 2V5−1 design For k=6, use the 2III*6−2 design with I = ABE = CDF = ABCDEF. (The minimum aberration 26−2 design has resolution IV and is not a good choice.) For k = 7, use the 2III*7−2 design I = ABCDF = DEG = ABCEFG. (The minimum aberration 27−2 design has resolution IV and is not a good choice.)

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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 nf + 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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**Let ˆy(x) be the predicted response at x = (x1, . . . , xk).**

Definition: A design is called rotatable if Var(ˆy(x)) depends only on (x12+ · · · + xk2)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 2Vk−p design is rotatable if (7) Equation (7) serves as a useful guide for the choice of . For the ranitidine experiment (see Table 10.1) nf = 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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**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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**10.8 Box-Behnken Designs and Uniform Shell Designs**

Skipped. 10.9 Practical Summary Read it.

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