Chapter 8: Split-Plot Designs

Split-plot designs were originally developed by Fisher (1925,Statistical Methods for Research Workers) for use in agricultural where factors are differentiated with respect to the ease with which they can be changed from experimental run to experimental run. This may be due to the fact that a particular treatment is expensive or time-consuming to change or it may be due to the fact that the is to be run in a large batches and batches can be subdivided later for additional treatments. A split-plot design may be considered as a special case of the two-factor randomized design where one wants to obtain more precise information about one factor and also about the between the two factors. The second factor being of secondary importance to the experimenter.

Example 1 Consider a study of the effects of four irrigation methods (A1, A2, A3, and A4) and three fertilizers (B1, B2, and B3). Two replicates are planned to carried out for each of the 12 combinations. A completely randomized design will randomly assign a combination of irrigation and fertilization methods to one of the 24 fields. This may be easy for fertilization but it is hard for irrigation where sprinklers are arranged in a certain way. What is usually done is to use just eight plots of land, chosen so that each can be irrigated at a set level without affecting the irrigation of the others. These large plots are called the whole plots. Randomly assign irrigation levels to each whole plot so that each irrigation level is assigned to exactly two plots. Irrigation is called the whole-plot factor. Divide each whole plot into three plots. Each plot is called a subplot. Within each whole plot, randomly assign the three fertilizers to the three subplots. Fertilizer is called the subplot factor. Note that, in this split-plot design, the first assigns the four irrigation types to eight fields (whole plots), then, the second randomization is conducted to assign the three fertilizers to three subplots within each field.

1 whole plot factor

A3 A1 A2 A4 A2 A1 A4 A3 subplotfactor

B3 B2 B1 B1 B2 B3 B1 B2

B2 B1 B2 B3 B1 B2 B3 B3

B1 B3 B3 B2 B3 B1 B2 B1

Example 2 Box et al. (2005, for experimenters: Design, inovation, and Discovery, 2nd) described a prototypical split-plot experiment with one easy- to-change factor and one hard-to-change factor. The experiment was designed to study the corrosion resistance of steel bars treated with four coatings, C1, o o o C2, C3, and C4, at three furnance temperatures, 360 C, 370 C, and 380 C. Furnance temperature is the hard-to-change factor because of the time it takes to reset the furnance and reach new equilibrium temperature. Once the equilibrium temperature is reached, four steel bars with randomly assigned coatings C1, C2, C3, and C4 are randomly positioned in the furnance and heated. Here are the data.

2 Temperature Coatings Whole-plot (oC) (randomized order) 1 360 C2 C3 C1 C4 73 83 67 89 2 370 C1 C3 C4 C2 65 87 86 91 3 380 C3 C1 C2 C4 147 155 127 212 4 380 C4 C3 C2 C1 153 90 100 108 5 370 C4 C1 C3 C2 150 140 121 142 6 360 C1 C4 C2 C3 33 54 8 46

Split-Plot Experiments with CRD in Whole Plots (CRSP)

Split-plot designs have three main characteristics:

1. The levels of all the factors are not randomly determined and reset for each experimental run. Usually the experimenter holds a factor at a particular setting and then run all the combinations of the other factors.

2. The size of the experimental unit is not the same for all experimental factors. Usually the experimenter applies one factor to a larger unit or group of units involving combinations of the other factors.

3. There is a restriction on the of the treatment combina- tions to the experimental units. Usually, there is something that prevents assigning the treatments to the units completely randomly.

Example 3 Recipes for chocolate and orange cookies include exactly the same ingredients up to the point where the syrup was added to the batch. However, after the cookies were baked, the chocolate cookies had an appealing round and plump appearance, while the orange cookies spread during the baking process and became thin, flat, and unappealing. A was devised to determine if there was a way to change the process of making the orange cookies that would reduce the spreading during baking. The factors that were chosen to be varied were A: the amount of shortening in the dough batch (80% of what the recipe called for or 100%), B: the baking temperature (below, at or above the temperature called for by the recipe), and C: The temperature of the cookies sheet upon which the cookies were placed to be baked (hot out of the oven or cooled to room temperature). A response that could quantify the objective of the experiment was the diameter of the baked cookies. A completely randomized design for this 3 22 factorial experiment with- out replicates needs 12 experimental units. The× experimental units is a batch of cookie doughs. However, the cooking-making process consists of two steps:

3 mixing cookie dough batch and baking cookies. The amount of shortening was a hard-to-change factor because each time it was changed it required making a new batch of cookie dough, while the baking temperature and tray temperature were relative easier to vary. Therefore, an alternative way of designing the experiments would be to follow a two-step plan. First, plan to make four batches of cookie dough and randomly assign two batches to use 80% of the recipe recommended amount of shorten and two batches to recieve the full amount of shortening recommended by the recipe. This represents a completely randomized design in one factor and the experimental unit is a batch of cookie dough. This first step is called the whole plot design and the whole-plot factor is A: the amount of shortening. Next, bake six trays of cookies from each batch of dough and completely ran- domized the six combinations of bake temperature and tray temperature to the six trays of cookies within each batch. This is a randomized block 3 2 facto- rial within each batch of cookie dough and is called the sub-plot design× . The sub-plot block is the batch of cookies, the sub-plot experimental unit is a tray of cookies and the sub-plot factors are B: the bake temperature and C: the tray temperature. The combination of the whole-plot and sub-plot design is called a split-plot design with CRD in the whole plots or CRSP. By designing the experiment in this way only four batches of cookies are required in tead of 12 and there are replicate whole-plot and sub-plot experimental units from which of two experimental errors can be estimated. It will be showns that there is less power for testing the shortening effect than there would be with the 12 batch completely randomized designed, but there is actually more power for detecting the baking temperature, tray temperature, and interaction due to the by batch.

Here are more examples from Kowalski, S.M. and Potcner, K.J. (2003, How to recognize a split-plot experiment, Quality Progress, 36(11), 60-66). In the printing press process, blanket type is the hard-to-change factor.

4 The Printing Press

In the experiment of water resistance property of wood, pretreatment type is the hard-to-change factor.

5 Water Resistance Property of Wood

In the baking process of plastic strength, oven temperature is the hard-to-change factor.

6 Strength of Plastic in Baking Process

Chapter 10. Response Surface Methodology

Response surface methodology (RSM) is a collection of statistical and mathe- matical techniques useful for developing, improving, and optimizing processes. It is also important applications in the design, development, and formulation of new products, as well as in the improvement of existing product designs. Response surface methodology was developed by Box and Wilson (1951, On the experimental attainment of optimum conditions, JRSS-B, 13, 1-45) to aid the improvement of manufacturing processes in the chemical industry. The purpose was to optimize chemical reactions to obtain, for example, high yield and purity at low cost. This was accomplished through the use of sequential experimentation involving factors such as temperature, pressure, duration of reaction, and proportion of reactants. The same methodology can be used to model or optimize any response that is affected by the levels of one or more quantitative factors. Response surface methods generally refer to a complete package of statistical design and analysis tools that are used for the following steps. 1. Design and collection of data to fit an equation to approximate the rela- tionship between factors and responses

7 2. to fit a model to describe the data

3. Examination of the fitted relationship through graphical and numerical techniques.

The most extensive applications of RSM are in the industrial world, par- ticularly in situations where several input variables potentially influence some performance measure or quality characteristic of the product or process. The general scenario is as follows. The response is a quantitative continuous vari- able (e.g., yield, purity, cost), and the response is a smooth but unknown function of the levels of p factors (e.g., temperature, pressure), and the levels are real-valued and accurately controllable. The mean response, when plotted as a function of the treatment combinations, is a surface in p + 1 dimensions, called the response surface. x <- seq(-10, 10, length= 30) y <- x f <- function(x,y) {-sqrt(x^2+y^2)} z <- outer(x, y, f) persp(x, y, z, theta = 30, phi = 30, expand = 0.5, xlab="A",ylab="B",zlab="") title("Response Surface")

8 Response Surface

B

A

In response surface methods, the relationship between the response and the factors is assumed to be a nonlinear equation given by y = f(x) + ǫ. For two factors we could write the relationship as y = f(x1,x2)+ǫ, where f is a nonliner function and ǫ is the random error. When f is unknown the nonlinear relationship f can be approximated near any point (x10,x20) using the Taylor series approximation of two variables, i.e.,

f(x1,x2) f(x10,x20) ≈ ∂f(x1,x2) +(x1 x10) x1=x10,x2=x20 − ∂x1 |

9 ∂f(x1,x2) +(x1 x20) x1=x10,x2=x20 − ∂x2 | 2 2 (x1 x10) ∂ f(x1,x2) + − 2 x1=x10,x2=x20 2 ∂x1 | 2 2 (x1 x20) ∂ f(x1,x2) + − 2 x1=x10,x2=x20 2 ∂x2 | 2 (x1 x10)(x1 x20) ∂ f(x1,x2) + − − x1=x10,x2=x20 , 2 ∂x1∂x2 | which leads to a general quadratic equation of the form

2 2 y = β0 + β1x1 + β2x2 + β11x1 + β12x1x2 + β22x2 + ǫ.

With k factors the general quadratic equation can be written in the form

k k k−1 k 2 y = β0 + βixi + βiixi + βij xixj + ǫ i=1 i=1 i=1 j=i+1 X X X X and unless the function f is known, this equation forms the basis of response surface mehtods. Response surface designs were created to provide data to approximate this equation and mathematical tools were created to explore the fitted surface represented by this equation. The objective of obtaining a response surface is twofold.

To locate a feasible treatment combination of independent variables for • which the mean response is maximized or minimized or equal to a specific target value.

To estimate the response surface in the vicinity of this good location or • region, in order to better understand the local effects of the factors on the mean response.

A Ranitidine Separation Experiments

Morris et al. (1997, Journal of Chromatography A 766, 245-254) reported on an experiment that studied important factors in the separation of ranitidine and related products by capillary electrophoresis. Ranitidine hydrochloride is the active ingredient in Zantac, a popular treatment for ulcers. Eletrophoresis is the process used to separate ranitidine and four related compounds. From screening experiments, the investigators identified three factors as important: pH of the buffer solution, the voltage used in electrophoresis, and the concentration of α- CD, a component of the buffer solution. The factor levels are given below. The coded design matrix and response chromatographic exponential function (CEF) is a quality measure in term of separation achieved and time of final separation; the goal is to minimize CEF.

10 Factor Levels A. pH 2, 3.42, 5.5, 7.58, 9 B. voltage (kV) 9.9, 14, 20, 26, 30.1 C. α-CD (mM) 0, 2, 5, 8, 10

Design Matrix and Response Data Factor Run A B C CEF ln(CEF) 1 -1 -1 -1 17.293 2.850 2 1 -1 -1 45.488 3.817 3 -1 1 -1 10.311 2.333 4 1 1 -1 11757.084 9.372 5 -1 -1 1 16.942 2.830 6 1 -1 1 25.400 3.235 7 -1 1 1 31697.199 10.364 8 1 1 1 12039.201 9.396 9 0 0 -1.67 7.474 2.011 10 0 0 1.67 6.312 1.842 11 0 -1.68 0 11.145 2.411 12 0 1.68 0 6.664 1.897 13 -1.68 0 0 16548.749 9.714 14 1.68 0 0 26351.811 10.179 15 0 0 0 9.854 2.288 16 0 0 0 9.606 2.262 17 0 0 0 8.863 2.182 18 0 0 0 8.783 2.173 19 0 0 0 8.013 2.081 20 0 0 0 8.059 2.087

The design differs from 2k−p design in two respects and It belongs to the class of central composite designs. 1. The first eight points form a 23 design. They are called cube points or corner points.

2. The next six points form three pairs of points along the three coordinate axes and are therefore called axial points or star points.

3. The last six points are at the center, called center points.

This design 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.

11 Sequential Nature of Response Surface Methodology Suppose that a scientific or engineering investigation is concerned with a process or system that involves a response y that depends on the input factors (also called input variables or process variables) X1, X2, , Xk. Their relationship can be modeled by · · · y = f(X1, X2, , X ) + ǫ, · · · k where the form of the true response function f is unknown and ǫ is an error term that represents the sources of variability not captured by f. We will

12 assume that the ǫ over different runs are independent and have mean zero and 2 σ . The Xi’s expressed in the original scale should be converted to coded variables x1, x2, , xk, which are dimensionless and have mean 0 and the same .· · · For example, the five levels of each factor in the Ranitidine experiment could be expressed as xi αci,xi ci, xi, xi +ci,xi +αci, with α > 1. By the five levels α, 1, 0, 1, α.− Another− transformation is to centerize and then standardize the− original− variable as in the regression analysis, e.g., Xij X¯.j xij = − , sj where Xij is the value of the jth variable for the ith run and X¯.j . and sj are the mean and standard deviation of the Xij’s over i. From now on, we shall assume that the input variables are in coded form and expressed as

y = f(x1,x2, ,x ) + ǫ. · · · k Often the purpose of the investigation is to minimize or maximize the response or to achieve a desired value of the response. Because f is unknown and y is observed with random error, we need to run the experiments to obtained data about f. Success of the investigation depends critically on how well f can be approximated. RSM is a strategy to achieve this goal that involves experimen- tation, modeling, data analysis, and optimization. Here is the sequential nature of RSM and these steps are illustrated by the contour plot.

1. Screening Experiment: When many variables are considered, some are likely to be inert. Use a 2k−p design or an OA. If the experimental region is far from the optimum, use the first-order model

k

y = β0 + βixi + ǫ, i=1 X to fit the data.

2. A search must be conducted over the region for the xi’s to determine if a first-order experiment should be continued or in the presence of curvature, be replaced by a more elaborate second-order experiment. Two search methods will be considered: Steepest ascent search and rectangular grid search. Based on the fitted model, find the steepest ascent direction and perform a search along this direction (called steepest ascent search). Steps 1 and 2 may be repeated until reaching the optimum region (e.g. peak of the surface).

13 14 3. When the experimental region is near or within the optimum region, the second phase of the response surface study begins. Its main goal is to obtain an accurate approximation to the response surface in a small region around the optimum and to identify optimum process conditions. Near the optimum of the response surface, the curvature effects are the dominating terms. To capture the curvature effects, use a second-order design (like the ). Fit a second-order model

k k k 2 y = β0 + βixi + βij xixj + βiixi + ǫ, i=1 i

15 Steepest Ascent Search and Rectangular Grid Search

In a response surface study, the move from a first-order experiment to a second- order experiment often involves an iterative search of the design region and sequential experimentation. Recall that a first-order experiment becomes in- effective when its experimental region is close to the optimum of the response surface and near the optimum, the curvature effects of the response surface

16 dominate the linear effects and a second-order experiment should be conducted instead. How does the experimenter know when to switch from a first-order experiment to a second-order experiment? A simple method is to check for an overall curvature effect by adding center runs to a first-order experiment. Curvature Check Suppose that the first-order experiment is based on a two-level orthogonal design with run-size nf and nc center point runs are added to the first-order experiment. Denote the average over the factorial runs byy ¯f and the sample average at center points byy ¯c. Code the low and high levels of the factorial design by 1 and +1 and the level of the center point by 0. Note that when center points− are added, we assume that the factors are quantitative. Under the second-order model, it is easy to show that

E(¯yc) = β0 and k

E(¯yf ) = β0 + βii. i=1 X Therefore, k E(¯y y¯ ) = β , f − c ii i=1 X that is, we can use the differencey ¯ y¯ to test if the overall curvature f − c

β11 + + β · · · kk is zero. Intuitively, if the differencey ¯f y¯c is small, then center points lie on or near the plan passing through the factorial− points and there is no quadratic curvature. On the other hand, ify ¯f y¯c is large, then quadratic curvature is present. − A t test to test the significance of the overall curvature at level α is

y¯f y¯c − | 1 − 1 | 1 >tnc 1,α/2, s( + ) 2 nf nc

2 where s is the sample variance based on the nc center runs. 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. Example 4 A chemical engineer is studying the yield of a process. There are two variables of interest: reaction time (in minutes) and temperature (in oC). Because she is uncertain about the assumption of linearity over the region of exploration, the engineer decides to conduct a 22 design with a single replicate of each factorial run augmented with five center points. The data are

17 Temp Time Yield 150 30 39.3 150 40 40.9 160 30 40.0 160 40 41.5 155 35 40.3 155 35 40.5 155 35 40.7 155 35 40.2 155 35 40.6

> yf<-c(39.3,40.9,40,41.5) > yc<-c(40.3,40.5,40.7,40.2,40.6) > t<-(mean(yf)-mean(yc))/(sd(yc)*sqrt(1/length(yf)+1/length(yc))) > t [1] -0.2516098 > #p-value > pt(t,5-1,lower.tail=FALSE) [1] 0.5931296

We conclude that there was no indication of quadratic effects. That is, a first- order model Y = β0 + β1x1 + β2x2 + β12x1x2 + ǫ is appropriate (although we probably don’t need the interaction term).

Remark 1 In using the overall curvature to measure the curvature effects, we assume that the βii have the same sign. If they have different signs, it is possible k that some of the large βii cancel each other, thus yielding a very small i=1 βii. Then the test will not be able to detect the present of these curvature effects. P If it is important to detect the individual curvature effects βii, a second-order experiments should be conducted. It is also clear that adding center points to a first-order experiment serves two purposes: 1. It allows the check of the overall curvature effect. 2. It provides an unbiased estimate of the process error variance.

Steepest Ascent Search

The experimental design, model-buliding procedure, and sequential experi- mentattion that are used in searching for a region of improved response consti- tute the method of steepest ascent. The method of steepest ascent contains the following steps: 1. Fit a first-order model (a plane or a hyperplane) using an orthogonal design. Two-level desings will be quite appropriate although center runs are often recommended. 2. Compute a path of steepest ascent if maximizing the response is required. If minimum response is required, one should compute the path of steepest descent. The path of steepest ascent is computed so that one may expect the maximum increase in response. The steepest descent produces a path that results in a maximum decrease in response. 3. Conduct experimental runs along the path. That is, do either single runs or replicated runs, and observe the respnse value. The results will nor- mally show improving of the response. At some region along the path the

18 improvement will decline and eventually disappear. This stems from the deterioration of the simple first-order model once one strays too far from the initial experimental region. Often the first experimental run should be taken near the design parameter (at coordinates corresponding to a value of 1.0 in an important variable) to serve as a confirming experiment. 4. At some point where an approximation of the maximum (or minimum) response is located on the path, a base for a second experiment is chosen. The design should again be a first-order design. It is quite likely that center runs for testing curvature and degrees of freedom for interaction- type lack of fit are important at this point. 5. A second experiment is conducted and another first order model is fitted to the data. A test for lack of fit is made. If the lack of fit is not significant, a second path based on the new model is computed. This is often called a mif-course correction. Single or replicated experiments along this second path are conducted. It is quite likely that the improvement will not be as strong as that enjoyed in the first path. After improvement is diminished, one typically has a base for conducting a more elaborate experiment and a more sophisticated process.

The linear effects βi in the first-order model can be estimated by least squares estimates in regression analysis or by main effect in two-level designs. The fitted response is given by k

yˆ = βˆ0 + βˆixi. i=1 X The direction of steepest ascent is

λ(βˆ1, , βˆ ), · · · k for λ > 0. For example, if the fitted regression produces an equationy ˆ = 20+3x−1.5x2, the path of the steepest ascent will result in x1 moving in a positive direction and x2 in a negative direction. In additon, x1 will move twice as fast as x2; that is, x1 moves two units for every signle unit movement in x2. By the path of the steepest ascent we mean that which produces a maximum k 2 2 estimated response with the constraint that xi = r . In other words, of all i=1 points that are a fixed distance from the centerP of the design, we seek that point x1, x2, , xk for whichy ˆ is maximized. It is easy to see that xi = λbi. If the· · · linear approximation to the unknown response surface is adequate, the response will increase along the steepest ascent direction at least locally near the point of approximation. If the response is to be minimized, then the steepest descent direction λ(βˆ1, , βˆ ) − · · · k should be considered instead.

19 The search for a higher response continues by drawing a line from the center points of the design in the steepest ascent direction. Several runs should be taken along the steepest ascent path. The location on the path where a maximum response is observed will be taken as the center point for the next experiment. The design is again a first-order design plus nc center runs. If the curvature check is significant, the design should be augmented by star points to make it a second-order design. Example 5 A common processing step in semiconductor manufactire is plasma etching of silicon wafers. The etch rate is the typical response of interest. The process variable of interest are the anode-cathode gap (x1 in cm) and the power applied to the cathode (x2 in watt). The etch rates observed here are too low and the experimenter would like to move to a region where the etch rate is around 1000 A˚/min. Here are the data.

Gap Power x1 x2 Y 1.20 275 -1 -1 775 1.60 275 +1 -1 670 1.20 325 -1 +1 890 1.60 325 +1 +1 730 1.40 300 0 0 745 1.40 300 0 0 760 1.40 300 0 0 780 1.40 300 0 0 720 Step 1: Check the curvature yf<-c(775, 670, 890,730) #center response yc<-c(745,760,780,720) #t statistic t<-(mean(yf)-mean(yc))/(sd(yc)*sqrt(1/length(yf)+1/length(yc))) #p-value pt(t,length(yc)-1,lower.tail=FALSE) [1] 0.2315778 Step 2: Fit a model y<-c(yf,yc) x1<-c(-1,1,-1,1,0,0,0,0) x2<-c(-1,-1,1,1,0,0,0,0) plasma.mod<-lm(y~x1+x2) summary(plasma.mod)

Coefficients Estimate Std. Error t value Pr(>|t|) Intercept 758.750 8.839 85.84 4.07e-09 *** x1 -66.250 12.500 -5.30 0.00319 ** x2 43.750 12.500 3.50 0.01728 *

Residual : 25 on 5 degrees of freedom Multiple R-squared: 0.8897,Adjusted R-squared: 0.8456 F-: 20.17 on 2 and 5 DF, p-value: 0.004039

Therefore, the first -order model is yˆ = 758.75 66.25x1 + 43.75x2. − Step 3: Construct the path of steepest ascent. Since the sign of x1 is negative and the sign of x2 is positive, we shall decrease the gap and increase the power in order ot increase the etch rate. Furthermore, for every unit of change in gap, we shall change the power by 43.75/66.25 = 0.66 unit. Consequently, if we choose the gap step size in coded units to be ∆x1 = 1.0, then the power − step size in coded units is ∆x2 = 0.66. Here are the results of applying steepest ascent to this process.

Natural variables Coded variables y Point Gap Power x1 x2 Base 1.40 300 0 0 ∆ -0.20 16.5 -1 0.66 Base+∆ 1.20 316.5 -1 0.66 845 Base+2∆ 1.00 333.0 -2 1.32 950 Base+3∆ 0.80 349.5 -3 1.98 1040

20 In practice, it is always a good idea to run the first point on the path near to the original experimental region as a confirmation test to ensure that conditions experienced during the original experiment have not changed. Clearly we have arrived at a region close to the desited etch rate of 1000 A˚/min at base plus 3∆. In general, when there are more than two factors, the following algorithm de- termines the coordinates of a point on the path of steepest ascent.

1. Choose a step size in one of the process variables, say ∆xi. Usually, we select the variable we know the most about or we select the variable that has the largest (or nearly largest) absolute regression coefficient b . | i| 2. The step size in other variables is

bj ∆xj = , bi/(∆xi) for j = 1, 2, ,k, i = j. · · · 6 3. Convert the ∆xj from the coded variables to the natural variables. Steepest ascent is a first-order gradient-based optimization technique. It works very well when starting a long way from the optimum. When used near an extreme point on the true response surface, steepest ascent will usually result in a very short movement away from the starting point. This could be an indication to consider expanding the model by adding high-order terms. when second-order terms describing interaction and pure quadratic curvature begin to dominate, then continuing the ascent exercise and experimentation will be self-defeating. However, the question arises, ”What do we mean by ’dominate’?” It is possible that second-order terms can be statistically significant, yet the first-order approximation allows a reasonably successful experimental strategy. One must keep in mind that ”statistical significance” only implies that the effects are real in comparisopn with experimental error. The second-order effects may be small in magnitude compared to their first-order counterparts. As a result, there will certainly be situations where one should compute the path and take experimental trials even though certain second-order effects are significant. Example 6 In a chemical process to maximize the reaction yield, four factors, A (amount of reactant A, 10, 15 grams), B (reaction time, 1 and 2 minutes), C (amount of reactant C, 25 and 35 grams), and D (temperature, 75 and 85 oC). A 24−1 fractional factorial design was used and the following data were produced.

Run A B C D Yield 1 -1 -1 -1 -1 62.0 2 -1 -1 1 1 57.0 3 -1 1 -1 1 62.2 4 -1 1 1 -1 64.7 5 1 -1 -1 1 61.8 6 1 -1 1 -1 64.5 7 1 1 -1 -1 69.0 8 1 1 1 1 66.3

21 A<-rep(c(-1,1),c(4,4)) B<-rep(rep(c(-1,1),c(2,2)),2) C<-rep(c(-1,1),4) D<-A*B*C y<-c(62.0,57.0,62.2,64.7,61.8,64.5,69.0,66.3)

> summary(lm(y~A+B+C+D))

Coefficients Estimate Std. Error t value Pr(>|t|) (Intercept) 63.4375 0.2486 255.136 1.33e-07 *** A 1.9625 0.2486 7.893 0.00424 ** B 2.1125 0.2486 8.496 0.00342 ** C -0.3125 0.2486 -1.257 0.29777 D -1.6125 0.2486 -6.485 0.00744 **

Residual standard error: 0.7033 on 3 degrees of freedom Multiple R-squared: 0.9834,Adjusted R-squared: 0.9614 F-statistic: 44.53 on 4 and 3 DF, p-value: 0.005276

If we choose one gram of reactant A as the basis for computation of points along the path, this corresponds to 1/2.5 = 0.4 design unit. As a result, the corresponding movements in other design variables are 2.1125/1.9625)(0.4) = 0.4306 design unit for B, 0.3125/1.9625)(0.4) = 0.0637 design unit for C , and 1.6125/1.9625)(0.4) =− 0.3287 design unit for− D. Here are the coordinates on the− path of steepest ascent− in natural variables.

Point A B C D Yield Base 12.5 1.5 30 80 ∆ 1.0 0.215 -0.319 -1.643 (0.4306)(0.5) (-0.0637)(5) (-0.3287)(5) Base+∆ 13.5 1.715 29.681 78.357 Base+2∆ 14.5 1.930 29.362 76.714 Base+3∆ 15.5 2.145 29.043 75.071 Base+4∆ 16.5 2.360 28.724 73.428 74.0 Base+6∆ 18.5 2.790 28.086 70.142 77.0 Base+8∆ 20.5 3.220 27.448 73.856 81.0 Base+9∆ 21.5 3.435 27.129 65.213 78.7 Example 7 Consider the following chemical reaction whose factors are time and temperature to maximize the yield. The first-order design is a 22 design with two center points. The levels of factors are given below. Factor Levels Time (in minutes) 75, 80, 85 Temperature (oC) 180, 185, 190 The design matrix and yield data are Run Time Temperature Yield 1 -1 -1 65.60 2 -1 1 45.59 3 1 -1 78.72 4 1 1 62.96 5 0 0 64.78 6 0 0 64.33 Step 1: To check the curvature, it is easy to see that

y¯f y¯c 63.295 64.555 | 1 − 1 | 1 = | − 1 1 |1 = 4.572, s( + ) 2 0.3182 ( + ) 2 nf nc × 4 2 or we can fit the following model 2 y = β0 + β1x1 + β2x2 + β12x1x2 + (β11 + β22)x1.

22 Time<-c(-1,-1,1,1,0,0) Temperature<-c(-1,1,-1,1,0,0) Yield<-c(65.6,45.59,78.72,62.96,64.78,64.33) summary(lm(Yield~Time*Temperature+I(Time^2)))

Coefficients Estimate Std. Error t value Pr(>|t|) Intercept 64.5550 0.2250 286.911 0.00222 Time 7.6225 0.1591 47.910 0.01329 Temperature -8.9425 0.1591 -56.207 0.01133 Time:Temperature 1.0625 0.1591 6.678 0.09462 Time^2 -1.3375 0.2756 -4.854 0.12935

Since there is no indication of interaction and curvature, which suggests that a steepest ascent search should be conducted. According to the model, the steepest ascent direction is proportional to (7.6225, 8.9425), or equivalently, (1, 1.173), that is , one unit increase in time will accompany− an 1.173 unit decrease− in tem- perature. Step 2: Increasing time of 2 units or 10 minutes is used because a step of 1 unit would give a point near the south-east corner of the first-order design. The results for three steps are as follows.

Run Time (in minutes) Temperature (oC) Yield 7 2 (90) 2 ×−1.173 (173.27) 89.73 8 4 (100) 4 ×−1.173 (161.54) 93.04 9 6 (110) 6 ×−1.173 (149.81) 75.06 Because 100 minutes and 161.54oC correspond to the maximum yield along the path, they would be a good center point for the next experiment. Step 3: A first-order experiment is run as step 1 in the new center point. The data are Run Time Temperature Yield 1 -1 -1 91.21 2 -1 1 94.17 3 1 -1 87.46 4 1 1 94.38 5 0 0 93.04 6 0 0 93.06 Fit the model

2 y = β0 + β1x1 + β2x2 + β12x1x2 + (β11 + β22)x1 to the new data.

Time<-c(-1,-1,1,1,0,0) Temperature<-c(-1,1,-1,1,0,0) Yield<-c(91.21,94.17,87.46,94.38,93.04,93.06) summary(lm(Yield~Time*Temperature+I(Time^2)))

Coefficients Estimate Std. Error t value Pr(>|t|) (Intercept) 93.050000 0.010000 9305.0 6.84e-05 Time -0.885000 0.007071 -125.2 0.00509 Temperature 2.470000 0.007071 349.3 0.00182 Time:Temperature 0.990000 0.007071 140.0 0.00455 Time^2 -1.245000 0.012247 -101.7 0.00626

Since the interaction and curvature effects are significant, this suggests augment- ing the first order design so that runs at the axial points of a central composite design are performed. The axial points correspond to 92.95 and 107.05 minutes for time and 154.49 and 168.59oC for temperature, i.e, √2 in coded units. ±

23 Run Time Temperature Yield 1 -1 -1 91.21 2 -1 1 94.17 3 1 -1 87.46 4 1 1 94.38 5 0 0 93.04 6 0 0 93.06 7 -1.41 0 93.56 8 1.41 0 91.17 9 0 -1.41 88.74 10 0 1.41 95.08 The second order model is

2 2 yˆ = 93.05 0.87x1 + 2.36x2 + 0.99x1x2 0.43x 0.65x . − − 1 − 2 Time<-c(-1,-1,1,1,0,0,-sqrt(2),sqrt(2),0,0) Temperature<-c(-1,1,-1,1,0,0,0,0,-sqrt(2),sqrt(2)) Yield<-c(91.21,94.17,87.46,94.38,93.04,93.06, 93.56,91.17,88.74,95.08) summary(lm(Yield~Time*Temperature+I(Time^2) +I(Temperature^2)))

Coefficients Estimate Std. Error t value Pr(>|t|) (Intercept) 93.0500 0.2028 458.904 1.35e-10 *** Time -0.8650 0.1014 -8.532 0.00104 ** Temperature 2.3558 0.1014 23.236 2.03e-05 *** I(Time^2) -0.4256 0.1341 -3.174 0.03374 * I(Temperature^2) -0.6531 0.1341 -4.870 0.00822 ** Time:Temperature 0.9900 0.1434 6.905 0.00231 **

Residual standard error: 0.2868 on 4 degrees of freedom Multiple R-Squared: 0.9942, Adjusted R-squared: 0.9869 F-statistic: 137.1 on 5 and 4 DF, p-value: 0.0001464

Rectangular Grid Search

The search strategy is illustrated by the following graph. Star with a first-order experiment with widely spaced factor levels, use the sign and magnitude of the main effect for each factor to narrow the search to smaller and then per- form a first- or second-order experiment over smaller region of factor levels. When the design is closer to the peak of the unknown response curve, the mag- nitude of the main effect becomes smaller. Because this strategy searches over different rectangular subregions of the factoregion, it is called the rectangular grid search method. x<-seq(-1,1,0.1) plot(x,-(x-1)^2,type="l",lty=2,axes=F, xlab="Coded Levels",ylab="response",xlim=c(-1.2,1)) axis(1,at=c(-1,0,0.5,1)) abline(v=-1.1) text(c(-1.05,0,0.5,1),c(-3.9,-0.9,-0.15,0.1), labels=c("A","C","D","B")) segments(-1,-4,1,0) segments(0,-1,1,0) title("Rectangular Grid Search")

24 Rectangular Grid Search

B D

C response

A

−1.0 0.0 0.5 1.0

Coded Levels

For the ranitidine experiment, two first-order designs was employed to six fac- 6−2 tors. The first screening experiment was based on a 16-run 2IV design with I = 1235 = 2346 = 1456 plus four runs at the center point. A very high over- all curvature,y ¯f y¯c = 16070 11 = 16059 was reported. This large positive value suggests that− one or more− factors would attain the minimum within range. Because the response CEF is to be minimized, the factor range is narrowed to the left or right half, depending on whether the leftmost or rightmost level has the lower CEF. For example, the factor methanol and its leftmost level 4 has a lower CEF. Therefore, in the second screening experiment the center level 12 become the new rightmost level, the leftmost level remains the same, and the new center level is 8. First Experiment Second Experiment Factor −1 0 +1 −1 0 +1 A. Methoanol (%) 4 12 20 4 8 12 B. α-CD (mM) 1 5.5 10 1 3.25 5.5 C. pH 2 5.5 9 2 3.75 5.5 D. voltage (kV) 10 20 30 10 15 20 E. temperature (oC) 24 42 60 24 33 42 F. NaH2PO4 (mM) 20 35 50 20 27.5 35

As reported by Morris et al. (1997), methanol and buffer concentration (NaH2P O4) are not significant at the 10% level in both screening experiments. The temper- ature is significant at the 10% level for the second experiment but was dropped for further study because the existing instrument was unable to adjust the tem- perature effectively over a sufficient length of capillary tube. For the remaining three factors, pH, voltage, and α-CD, the t statistics and p values are

First Experiment Second Experiment Factor t p Value t p Value pH 0.650 0.56 -3.277 0.002 voltage (kV) -5.456 0.01 -0.398 0.71 α-CD (mM) 2.647 0.08 -0.622 0.56

25 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 second experiment to be the center point of a new central composite design.

Analysis of Second-Order Response Surfaces

Using the least squares method to estimate the parameters in the model, the fitted second-order model is

k k k ˆ ˆ ˆ ˆ 2 y = β0 + βixi + βijxixj + βiixi , i=1 i

T T yˆ = βˆ0 + x b + x Bx,

T T where x = (x1, ,xk), b = (βˆ1, , βˆk), and B is the k k symmetric matrix · · · · · · × ˆ 1 ˆ 1 ˆ β11 2 β12 2 β1k 1 ˆ ˆ · · · 1 ˆ 2 β12 β22 2 β2k B =  . . · · · .  . . . .    1 ˆ 1 ˆ · · · ˆ   2 β1k 2 β2k βkk   · · ·  Differentiatingy ˆ with respect to x and setting it to zero, ∂yˆ = b + 2Bx = 0, ∂x leads to the solution 1 −1 xs = B b, −2 which is called the stationary points of the quadratic surface (or system). The nature of the quadratic surface around the stationary point is better understood in a new coordinate system. Let P be the k k matrix whose columns are the standardized eigenvectors of B. Then ×

T P BP = Λ, where Λ = diag(λ1, ,λk) is a diagonal matrix and λi are the eigenvalues of B associated with the· · · ith column of P. The of the eigenvectors ensures that PTP = PPT = I. By translating the predicted model to the new center, namely stationary point, and rotating to new axes associated with the eigenvectors, i.e., T Z = x xs, v = P z, − we have T T yˆ = βˆ0 + (z + xs) b + (z + xs) B(z + xs)

26 T T T T T = (βˆ0 + xs b + xs Bxs) + z Bz + z b + 2z Bxs T =y ˆs + z Bz, T wherey ˆs is the fitted response at xs. By rotating z to v = (v1, , vk) and noting that z = PPTz = Pv, · · ·

k T T T 2 yˆ =y ˆs + v P BPv =y ˆs + v Λv =y ˆs + λivi . i=1 X This method of translation and rotation to simplify a quadratic model is called canonical analysis. The behavior ofy ˆ around the stationary pointy ˆs can be nicely described in terms of the eigenvalues λi’s of B and the canonical variables vi’s. From the signs of λi’s, we can classify the second-order response surface into two broad types:

1. When the λi’s are of the same signs, the contours around the stationary point are elliptical. It is called an elliptic system. When the signs are negative, the stationary point is a point of maximum response. When the signs are positive, the stationary point is a point of minimum response.

2. When the λi’s are of mixed signs, the contours around the stationary point are hyperbolic. It is called a hyperbolic system. The stationary point is a saddle point.

27 Elliptic System Hyperbolic System

0 70 64 56 50 46 20 60 70 82 −20 5 45 30 65 85 72 68 62 58 15 50 80 66 40 55 75 78 76 74 80 95 78 74 80 72 76 82 66 70 56 62 68

Temperature 90 Temperature 50 44 50 58 64 85 30 36 80 75 45 35 25 10 42 48 52 60 54 60 12 20 28 34 40 70 65 55 40 30 20 5 −5 −20 140 160 180 200 140 160 180 200

70 80 90 100 110 120 130 70 80 90 100 110 120 130

Time Time

Stationary Ridge System Rising Ridge System

20 0 −10 −20 0 90 −40 30 50 50 30 −10 10 80 70 60 40 40 20 100

90 80 90 80 70

Temperature 20 60 Temperature 60 30 −20 50 30 −30 0 50 40 −10 −70 70 −40 −10 10 40 20 10 0 −20 −40 −60 −80 140 160 180 200 140 160 180 200

70 80 90 100 110 120 130 70 80 90 100 110 120 130

Time Time

par(mfrow=c(2,2)) Time<-seq(-3,1.5,len=25) Temperature<-seq(-3,1.5,len=25) f<-function(x,y){94+4*x-8*y-4*x^2-12*y^2+12*x*y} z<-outer(Time, Temperature,f) contour(Time, Temperature,z,nlevels = 30,xlab="Time", ylab="Temperature",axes=F) axis(1,at=seq(-3,1.5,len=7),labels=seq(70,130,10)) axis(2,at=seq(-3,1.5,len=7),labels=seq(140,200,10)) title("Elliptic System")

Time<-seq(-1.5,1.5,len=25) Temperature<-seq(-1.5,1.5,len=25) f<-function(x,y){80+4*x+8*y-2*x^2-12*y^2-12*x*y} z<-outer(Time, Temperature,f) contour(Time, Temperature,z,nlevels = 30,xlab="Time", ylab="Temperature",axes=F) axis(1,at=seq(-1.5,1.5,len=7),labels=seq(70,130,10)) axis(2,at=seq(-1.5,1.5,len=7),labels=seq(140,200,10)) title("Hyperbolic System")

#A very small eigenvalue implies considerable elongation #of the response surface in that canonical direction

28 #resulting in a ridge system #If both eigenvalues are negative but the first is close to zero Time<-seq(-2.5,2.5,len=25) Temperature<-seq(-2.5,2.5,len=25) f<-function(x,y){95+4*x+8*y-3*x^2-12*y^2-12*x*y} z<-outer(Time, Temperature,f) contour(Time, Temperature,z,nlevels = 20,xlab="Time", ylab="Temperature",axes=F) axis(1,at=seq(-2.5,2.5,len=7),labels=seq(70,130,10)) axis(2,at=seq(-2.5,2.5,len=7),labels=seq(140,200,10)) title("Stationary Ridge System")

#If both eigenvalues are negative but the second is close to zero Time<-seq(-2.5,2.5,len=25) Temperature<-seq(-2.5,2.5,len=25) f<-function(x,y){80+4*x+12*y-3*x^2-12*y^2+12*x*y} z<-outer(Time, Temperature,f) contour(Time, Temperature,z,nlevels = 20,xlab="Time", ylab="Temperature",axes=F) axis(1,at=seq(-2.5,2.5,len=7),labels=seq(70,130,10)) axis(2,at=seq(-2.5,2.5,len=7),labels=seq(140,200,10)) title("Rising Ridge System")

Analysis of the Ranitidine Experiment

Morris et al (1997) fitted the second-order model and found pH and voltage are important. Because it was difficult to locate the minimum from the plot, a follow-up second-order experiment is performed. Since pH was the most signifi- cant factor, the range of pH was narrowed and the five levels of pH and voltage are (4.19, 4.50, 5.25, 6.00, 6.31) and (11.5, 14, 20, 26, 28.5), respectively. The design matrix and data are Run pH Voltage ln CEF 1 0 -1.41 6.943 2 1 -1 6.248 3 -1.41 0 2.100 4 0 0 2.034 5 0 0 2.009 6 0 0 2.022 7 1 1 3.252 8 1.41 0 9.445 9 0 1.41 1.781 10 0 0 1.925 11 -1 -1 2.390 12 -1 1 2.066 13 0 0 2.113 pH<-c(0,1,-1.41,0,0,0,1,1.41,0,0,-1,-1,0) voltage<-c(-1.41,-1,0,0,0,0,1,0,1.41,0,-1,1,0) lnCEF<-c(6.943,6.248,2.100,2.034,2.009,2.022, 3.252,9.445,1.781,1.925,2.390,2.066,2.113) summary(lm(lnCEF~pH+voltage+I(pH^2)+I(voltage^2)+pH:voltage))

Coefficients Estimate Std. Error t value Pr(>|t|) Intercept 2.0244 0.5524 3.665 0.00802 pH 1.9308 0.4374 4.414 0.00310 voltage -1.3288 0.4374 -3.038 0.01890 pH^2 1.4838 0.4703 3.155 0.01605 voltage^2 0.7743 0.4703 1.646 0.14371 pH:voltage -0.6680 0.6176 -1.082 0.31529

29 Residual standard error: 1.235 on 7 degrees of freedom Multiple R-Squared: 0.8554, Adjusted R-squared: 0.7521 F-statistic: 8.282 on 5 and 7 DF, p-value: 0.007461

For illustration purpose, consider the full model of the ranitidine experiment.

2 yˆ = 2.0244 + 1.9308x1 1.3288x2 + 1.4838x − 1 2 +0.7743x 0.6680x1x2. 2 − Note that T b = (1.938, 1.3288), − 1.4838 0.3340 B = . 0.3340− 0.7743  −  T The stationary point xs = ( 0.5067, 0.6393), Λ = diag(1.6163, 0.6418) and eigen vectors − 0.9295 0.3687 P = . −0.3687 −0.9295  −  pH<-seq(3.5,7,length=100) voltage<-seq(7,33,length=100) estimate<-function(x1,x2) 2.0244+1.9308*((x1-5.25)/0.75) -1.3288*((x2-20)/6)+1.4838*(((x1-5.25)/0.75)^2) +0.7743*(((x2-20)/6)^2)-0.6680*((x1-5.25)/0.75)*((x2-20)/6) contour(pH,voltage,outer(pH,voltage,estimate),xlim=c(3.5,7), ylim=c(7,33),xlab="pH",ylab="Voltage",nlevels=20)

Since both λ’s are positive, ys = 1.1104 is the minimum value which is achieved at xs (pH of 4.87 and voltage 23.84).

Estimated Response Surface voltage 8 14 20 26 32

3.75 4.5 5.25 6 6.75

pH

30 Example 8 Consider the following data (Myers and Montgomery, 2002, Re- sponse Surface Methodology, p. 56) from an investigation into the effect of two o variables: reaction temperature (x1 in C) and reactant concentration (x2 in %) on the percentage conversion of a chemical process (y).

Natural variables Coded variables Temp Conc x1 x2 y 200 15 -1 -1 43 250 15 1 -1 78 200 25 -1 1 69 250 25 1 1 73 189.65 20 -1.414 0 48 260.35 20 1.414 0 78 225 12.93 0 -1.414 65 225 27.07 0 1.414 74 225 20 0 0 76 225 20 0 0 79 225 20 0 0 83 225 20 0 0 81 y<-c(43,78,69,73,48,76,65,74,76,79,83,81) x1<-c(-1,1,-1,1,-1.414,1.414,0,0,0,0,0,0) x2<-c(-1,-1,1,1,0,0,-1.414,1.414,0,0,0,0) chem.mod<-lm(y~x1+x2+I(x1^2)+I(x2^2)+x1*x2) summary(chem.mod)

#First order model lm(formula = y ~ x1 + x2)

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 70.417 2.813 25.034 1.24e-09 *** x1 9.825 3.445 2.852 0.019 * x2 4.216 3.445 1.224 0.252

Residual standard error: 9.744 on 9 degrees of freedom Multiple R-squared: 0.5169,Adjusted R-squared: 0.4096 F-statistic: 4.815 on 2 and 9 DF, p-value: 0.03785

#Second-order model

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 79.7500 1.2135 65.717 8.35e-10 *** x1 9.8255 0.8582 11.449 2.66e-05 *** x2 4.2164 0.8582 4.913 0.002675 ** I(x1^2) -8.8766 0.9596 -9.250 9.02e-05 *** I(x2^2) -5.1255 0.9596 -5.341 0.001759 ** x1:x2 -7.7500 1.2135 -6.386 0.000694 ***

Residual standard error: 2.427 on 6 degrees of freedom Multiple R-squared: 0.98,Adjusted R-squared: 0.9634 F-statistic: 58.86 on 5 and 6 DF, p-value: 5.117e-05

#contour plot temperature<-seq(185,265,length=100) concentration<-seq(12,28,length=100) estimate<-function(x1,x2) 79.75+9.8255*((x1-225)/25)+ 4.2164*((x2-20)/5)-8.8766*(((x1-225)/25)^2)- 5.1255*(((x2-20)/5)^2)-7.75*((x1-225)/25)*((x2-20)/5) contour(temperature,concentration, outer(temperature,concentration,estimate),xlim=c(185,265), ylim=c(12,28),xlab="Temperature",ylab="Concentration",nlevels=20)

Note that T b = (9.83, 4.22), 8.88 3.875 B = . −3.875 − 5.13  − −  T 1 −1 The stationary point xs = 2 B b = (0.5432, 0.0101), Λ = diag( 11.0782, 2.6740) The canonical form of the− second-order model is− − − 2 2 yˆ =y ˆs + λ1z1 + λ2z2 = 82.40 11.0782z2 2.6740z2. − 1 − 2 Since λ1 > λ2 , the surface is somewhat elongated in the z2 direction. | | | |

31 Experimental Designs for Fitting Response Surfaces

Desirable properties of response surface designs are as follows.

1. Result in a good fit of the model to the data.

2. Give sufficient information to allow a test for lack of fit.

3. Allow models of increasing order to be constructed sequentially.

4. Provide an estimate of pure experimental error.

5. Be insensitive to the presence of outliers in the data.

6. Be robust to errors in control of design levels.

7. Be cost effective.

8. Allow for experiments to be done in blocks. 9. Provide a check on the homogeneous variance assumption.

10. Provide a good distribution of var[ˆy]/σ2.

For the first-order model, orthogonal designs of two levels are appropriate. For the second-order model, central composite designs are the most popular class of second-order designs.

Central Composite Designs

Central composite designs were introduced by Box and Wilson (1951). A central composite design consists of the following three parts.

1. nf cube points (or corner points) with xi = 1, 1 for i = 1, , k. They form the factorial portion of the design. · · ·

2. n center points with x = 0 for i = 1, , k. c i · · · 3. 2k star points (or axial points) of the form (0, ,xi, , 0) with xi = α, α for i = 1, , k. · · · · · · − · · · The central composite design can be used in a single experiment or in a sequen- tial experiment depending on if the experimental region is close to the optimum region or not. How to construct a central composite design?

Choice of the factorial portion: • (k + 1)(k + 2) N = n + 2k + 1 . f ≥ 2

32 Theorem 1 In any central composite design whose factorial portion is a 2k−p design that doesn’t use any main effect as a defining relation, the following parameters in the second-order model are estimable: β0, βi, βii, i = 1, 2, , k and one β selected from each set of aliased effects for · · · ij i

Remark 2 Defining words of length two are allowed and words of length four are worse than words of length three. This is because that linear main effects are not sacrificed in the defining relation, the linear and quadratic effects can be estimated by exploiting the information in the start points. On the other hand, the star points don’t contain information on the inter- actions βij and thus cannot be used to de-alias any aliased βij’s. Defining words of length four lead to the aliasing of some βij ’s.

Any resolution III design whose defining relation doesn’t contain words of length four is said to have resolution III∗. Therefore, any central com- posite design whose factorial portion has resolution III∗ is a second-order design. Here are some

1. For k = 2, using the resolution II design with I = AB for the factorial portion leads to a second-order design. If more runs can be afforded, the 22 design could be used. 3−1 3 2. For k = 3, either the 2III or the 2 design can be chosen for the factorial portion. 4−1 4 3. For k = 4, either the 2III design with I = ABD or the 2 design 4−1 can be chosen for the factorial portion. However, 2IV is not a good choice. 5−1 4. For k = 5, either the 2V design is a good choice for the factorial portion. 6−2 5. For k = 6, either the 2III∗ design with I = ABE = CDF = ABCDEF can be chosen for the factorial portion. However, MA 6−2 design 2IV with I = ABCE = ABDF = CDEF is not a good choice. 7−2 6. For k = 7, either the 2III∗ design with I = ABCDF = DEG can 7−2 be chosen for the factorial portion. However, MA design 2IV with I = ABCF = ABDEG = CDEFG is not a good choice.

Smaller central composite designs can be found by using the Plackett- Burman design.

1. For k = 4, an intermediate run size between the 8-run and 16-run designs is available. 11 4 matrix. × 2. For k = 5, one 12 5 matrix and one 11 5. × ×

33 3. For k = 7, the smallest resolution III∗ design has 32 runs. one 22 7 matrix and one 23 7. × × Choice of α : If α = √k the second-order design property of the central • composite design is6 not affected by the removal of the center point from the design. That is, we can use the cube and star points of the central composite design. For α = √k, the center cannot be removed. In general, α should be chosen between 1 and √k and rarely outside this range. 1. α = 1 the face center cube design The only CCC design with three levels. Very efficient when the design region is a cube. If the choice of factor range is independent of each other, the region is naturally cuboidal. The efficiency of parameter estimates is increased by pushing the star points toward the face of the cube. 2. α = √k Spherical design. Be careful to large k. 3. A design is called rotatable if V ar(ˆy(x)) depends only on x = 2 2 || || x1 + + xk that is the accuracy of prediction of the response is the same· · · on a sphere around the center of the design. For a CCC p − design whose factorial portion is a 2k p design of resolution V, it can be shown that rotatability is guaranteed by choosing

1/4 α = nf .

1/4 For the ranitidine experiment, nf = 8 and 8 = 1.682 but α = 1.68(= 3.5/2.08) was chosen for A and α = 1.67(= 5/3) was chosen for factors B and C in order to have integer values for the voltage and α CD levels and the stated levels of pH to the second decimal place. −

Number of runs at the center: When α = √k, three to five runs at center • points; when α is close to 1, one or two runs at the center point. Between the above two, two to four runs ahould be considered. If the primary purpose of taking replicate runs at center point is to estimate the error variance, then more than four or five may be required.

Box-Behnken designs and Uniform shell designs

Box-Behnken designs: Consider the following balanced incomplete block • design with three treatments and three blocks.

Treatment Block 1 2 3 1 2 × × 3 × × × ×

34 Take the three treatments as the three input factors x1,x2,x3 in a response surface and replace the two crosses in each block by the two columns of the two-level 22 design, insert a column of zero whenever a cross doesn’t appear. Adding a few center runs leads to the construction of the Box- Behnken design for k = 3

x1 x2 x3 -1 -1 0 -1 1 0 1 -1 0 1 1 0 -1 0 -1 -1 0 1 1 0 -1 1 0 1 0 -1 -1 0 -1 1 0 1 -1 0 1 1 0 0 0 or more condense notation,

1 1 0 ±1± 0 1  ±0 1 ±1  ± ±  0 0 0      Similarly, for k = 4

1 1 0 0 ±0± 0 1 1  0 0± 0± 0   1 0 0 1   ± ±   0 1 1 0   ± ±   0 0 0 0     1 0 1 0   ± ±   0 1 0 1   ± ±   0 0 0 0      for k = 5

35 1 10 0 0 ±0± 0 1 1 0  0 1± 0± 0 1  ± ±  1 0 1 0 0   ± ±   0 0 0 1 1   ± ±   0 0 0 0     0 1 1 0 0   ± ±   1 0 0 1 0   ± ±   0 0 1 0 1   ± ±   10 0 0 1   ± ±   0 1 0 1 0   ± ±   0 0 0 0 0      for k = 6 1 1 0 1 0 0 ±0 ±1 1± 0 1 0  0± 0 ±1 1± 0 1  ± ± ±  1 0 0 1 1 0   ± ± ±   0 1 0 0 1 1   ± ± ±   1 0 1 0 0 1   ± ± ±   000000      for k = 7 0 0 0 1 1 1 0 10 0± 0± 0 ±1 1  ±0 1 0 0 1± 0 ±1  ± ± ±  1 1 0 10 0 0   ± ± ±   0 0 1 1 0 0 1   ± ± ±   1 0 1 0 1 0 0   ± ± ±   0 1 1 0 0 1 0   ± ± ±   0000000      Uniform shell designs: These are the other family of second-order designs. • For k = 2 it can be described by a regular hexagon plus the center point. The six points are uniformly spaced on the circle. The main attractive feature is the uniformity property. For k = 3 it happens to be identical to the Box-Behnken design.

0.000 0.000 1.000 0.000 -1.000 0.000 0.500 0.866 -0.500 0.866 0.500 -0.866 -0.500 -0.866 5 3

36 -1.118 -0.500 - 0.500 -0.500 -1.118 0.500 - 0.500 0.500 -1.118 -0.500 0.500 0.500 -1.118 0.500 0.500 -0.500 0.000 1.000 0.000 1.000 0.000 0.000 1.000 1.000 0.000 1.000 1.000 0.000 0.000 -1.000 1.000 0.000 0.000 0.000 1.000 -1.000 0.000 1.000 0.000 -1.000 add origin and negative 3 5 5 5

37 Analysis strategies for multiple responses II

Out-of-Spec probabilities need a variance model, this may not met in • RSM where the main interest lies in the mean predicted responsey ˆ and the experiment is rarely replicated at every design point.

If there are few responses and only two or three important input variables, • overlay and exam the contour plots for the responses over a few pairs of input variables.

If, among the responses, one is of primary importance, it can be formulated •

38 as a constrained optimization problem by optimizing this response subject to constraints imposed on the other responses. Construct a desirability function d: • 1. Nominal the best: Let t be the target and L and U be the lower and upper values that the product is considered as accepted.

α yˆ−L 1 t−L , L yˆ t, α ≤ ≤ d =  yˆ−U 2 t−U , t yˆ U,  ≤ ≤  0, yˆ < L ory>U ˆ

 2. Smaller-the-better: Let a be the smallest possible value for the re- sponse y and U be a value above which the product is considered to be unacceptable.

− α yˆ U , a yˆ U, d = a−U ≤ ≤ ( 0, y>Uˆ

3. Larger-the-better: Let U be a value above which the product is con- sidered to be nearly perfect and L be a value below which the product is considered to be unacceptable.

− α yˆ L , L yˆ U, U−L ≤ ≤ d =  0, yˆ < L

 1, y>Uˆ

Another choice is  1. Nominal the best: Let t be the target

α exp c1 yˆ t 1 , < yˆ t, d = {− | − |α } −∞ ≤ exp c2 yˆ t 2 , t yˆ < .  {− | − | } ≤ ∞ 2. Smaller-the-better: Let a be the smallest possible value for the re- sponse y. d = exp c yˆ a α , a yˆ < , {− | − | } ≤ ∞ 3. Larger-the-better: Let L be a value below which the product is con- sidered to be unacceptable.

1 exp cyˆα /exp cLα , L yˆ < , d = − {− 0}, {− } ≤yˆ < L∞  The constants c and α can be used to fine-tune the scale and shape of the desirability function. A smaller α value makes the desirability function drop more slowly from its peak. A smaller c increases the spread of the

39 desirability function by lowering the whole exponential curve between o and 1. Suppose that a fitted response model and a desirability function di are chosen for the ith response, i = 1, 2, , m. Then an overall desirability · · · function d can be defined as the of the di’s. That is

1 d = d1d2 d m . { · · · m} It can be extended to d = dw1 dw2 dwm 1 2 · · · m to reflect the possible difference in the importance of the different re- sponses, where the weight w satisfy 1 < w < 1 and w1 + + w = 1. If i i · · · m any di = 0, then d = 0. Any setting for the input factors that maximizes the d value is chosen to be one which achieves an optimal balance over the m responses.

40