Response Surface Methodology
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Raymond H. Myers, PhD, is Professor Emeritus in the Department of Statistics at Virginia Polytechnic Institute and State University. He has more than 40 years of academic experience in the areas of experimental design and analysis, response surface analysis, and designs for nonlinear models. A Fellow of the American Statistical Association (ASA) and the American Society for Quality (ASQ), Dr. Myers has authored numerous journal articles and books, including Generalized Linear Models: with Applications in Engineering and the Sciences, Second Edition, also published by Wiley.
Douglas C. Montgomery, PhD, is Regents' Professor of Industrial Engineering and Arizona State University Foundation Professor of Engineering. Dr. Montgomery has more than 30 years of academic and consulting experience and his research interest includes the design and analysis of experiments. He is a Fellow of the ASA and the Institute of Industrial Engineers, and an Honorary Member of the ASQ. He has authored numerous journal articles and books, including Design and Analysis of Experiments, Eighth Edition; Generalized Linear Models: with Applications in Engineering and the Sciences, Second Edition; Introduction to Introduction to Linear Regression Analysis, Fifth Edition; and Introduction to Time Series Analysis and Forecasting, Second Edition, all published by Wiley.
Christine M. Anderson-Cook, PhD, is a Research Scientist and Project Leader in the Statistical Sciences Group at the Los Alamos National Laboratory, New Mexico. Dr. Anderson-Cook has over 20 years of academic and consulting experience, and has written numerous journal articles on the topics of design of experiments, response surface methodology and reliability. She is a Fellow of the ASA and the ASQ.
Content
1 Introduction 1
1.1 Response Surface Methodology, 1
1.1.1 Approximating Response Functions, 2
1.1.2 The Sequential Nature of RSM, 7
1.1.3 Objectives and Typical Applications of RSM, 9
1.1.4 RSM and the Philosophy of Quality Improvement, 11
1.2 Product Design and Formulation (Mixture Problems), 11
1.3 Robust Design and Process Robustness Studies, 12
1.4 Useful References on RSM, 12
2 Building Empirical Models 13
2.1 Linear Regression Models, 13
2.2 Estimation of the Parameters in Linear Regression Models, 14
2.3 Properties of the Least Squares Estimators and Estimation of ¿¿¿¿2, 22
2.4 Hypothesis Testing in Multiple Regression, 24
2.4.1 Test for Significance of Regression, 24
2.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients, 27
2.5 Confidence Intervals in Multiple Regression, 31
2.5.1 Confidence Intervals on the Individual Regression Coefficients ß, 32
2.5.2 A Joint Confidence Region on the Regression Coefficients ß, 32
2.5.3 Confidence Interval on the Mean Response, 33
2.6 Prediction of New Response Observations, 35
2.7 Model Adequacy Checking, 36
2.7.1 Residual Analysis, 36
2.7.2 Scaling Residuals, 38
2.7.3 Influence Diagnostics, 42
2.7.4 Testing for Lack of Fit, 43
2.8 Fitting a Second-Order Model, 47
2.9 Qualitative Regressor Variables, 55
2.10 Transformation of the Response Variable, 61
Exercises, 66
3 Two-Level Factorial Designs 81
3.1 Introduction, 81
3.2 The 22 Design, 82
3.3 The 23 Design, 94
3.4 The General 2k Design, 103
3.5 A Single Replicate of the 2k Design, 108
3.6 2k Designs are Optimal Designs, 125
3.7 The Addition of Center Points to the 2k Design, 130
3.8 Blocking in the 2k Factorial Design, 135
3.8.1 Blocking in the Replicated Design, 135
3.8.2 Confounding in the 2k Design, 137
3.9 Split-Plot Designs, 141
Exercises, 146
4 Two-Level Fractional Factorial Designs 161
4.1 Introduction, 161
4.2 The One-Half Fraction of the 2k Design, 162
4.3 The One-Quarter Fraction of the 2k Design, 174
4.4 The General 2k-p Fractional Factorial Design, 184
4.5 Resolution III Designs, 188
4.6 Resolution IV and V Designs, 197
4.7 Alias Structures in Fractional Factorial and Other Designs, 198
4.8 Nonregular Fractional Factorial Designs, 200
4.8.1 Nonregular Fractional Factorial Designs for 6, 7, and 8 Factors in 16 Runs, 203
4.8.2 Nonregular Fractional Factorial Designs for 9 Through 14 Factors in 16 Runs, 209
4.8.3 Analysis of Nonregular Fractional Factorial Designs, 213
4.9 Fractional Factorial Split-Plot Designs, 216
4.10 Summary, 219
Exercises, 220
5 Process Improvement with Steepest Ascent 233
5.1 Determining the Path of Steepest Ascent, 234
5.1.1 Development of the Procedure, 234
5.1.2 Practical Application of the Method of Steepest Ascent, 237
5.2 Consideration of Interaction and Curvature, 241
5.2.1 What About a Second Phase?, 244
5.2.2 What Happens Following Steepest Ascent?, 244
5.3 Effect of Scale (Choosing Range of Factors), 245
5.4 Confidence Region for Direction of Steepest Ascent, 247
5.5 Steepest Ascent Subject to a Linear Constraint, 250
5.6 Steepest Ascent in a Split-Plot Experiment, 254
Exercises, 262
6 The Analysis of Second-Order Response Surfaces 273
6.1 Second-Order Response Surface, 273
6.2 Second-Order Approximating Function, 274
6.2.1 The Nature of the Second-Order Function and Second-Order Surface, 274
6.2.2 Illustration of Second-Order Response Surfaces, 276
6.3 A Formal Analytical Approach to the Second-Order Model, 277
6.3.1 Location of the Stationary Point, 278
6.3.2 Nature of the Stationary Point (Canonical Analysis), 278
6.3.3 Ridge Systems, 282
6.3.4 Role of Contour Plots, 286
6.4 Ridge Analysis of the Response Surface, 289
6.4.1 Benefits of Ridge Analysis, 290
6.4.2 Mathematical Development of Ridge Analysis, 291
6.5 Sampling Properties of Response Surface Results, 296
6.5.1 Standard Error of Predicted Response, 296
6.5.2 Confidence Region on the Location of the Stationary Point, 299
6.5.3 Use and Computation of the Confidence Region on the Location of the Stationary Point, 300
6.5.4 Confidence Intervals on Eigenvalues in Canonical Analysis, 304
6.6 Further Comments Concerning Response Surface Analysis, 307
Exercises, 307
7 Multiple Response Optimization 325
7.1 Balancing Multiple Objectives, 325
7.2 Strategies for Multiple Response Optimization, 338
7.2.1 Overlaying Contour Plots, 339
7.2.2 Constrained Optimization, 340
7.2.3 Desirability Functions, 341
7.2.4 Pareto Front Optimization, 343
7.2.5 Other Options for Optimization, 349
7.3 A Sequential Process for Optimization-DMRCS, 350
7.4 Incorporating Uncertainty of Response Predictions into Optimization, 352
Exercises, 357
8 Design of Experiments for Fitting Response Surfaces-I 369
8.1 Desirable Properties of Response Surface Designs, 369
8.2 Operability Region, Region of Interest, and Metrics for Desirable Properties, 371
8.2.1 Metrics for Desirable Properties, 372
8.2.2 Model Inadequacy and Model Bias, 373
8.3 Design of Experiments for First-Order Models and First-Order Models with Interactions, 375
8.3.1 The First-Order Orthogonal Design, 376
8.3.2 Orthogonal Designs for Models Containing Interaction, 378
8.3.3 Other First-Order Orthogonal Designs-The Simplex Design, 381
8.3.4 Definitive Screening Designs, 385
8.3.5 Another Variance Property-Prediction Variance, 389
8.4 Designs for Fitting Second-Order Models, 393
8.4.1 The Class of Central Composite Designs, 393
8.4.2 Design Moments and Property of Rotatability, 399
8.4.3 Rotatability and the CCD, 403
8.4.4 More on Prediction Variance-Scaled, Unscaled, and Estimated, 406
8.4.5 The Face-Centered Cube in Cuboidal Regions, 408
8.4.6 Choosing between Spherical and Cuboidal Regions, 411
8.4.7 The Box-Behnken Design, 413
8.4.8 Definitive Screening Designs for Fitting Second-Order Models, 417
8.4.9 Orthogonal Blocking in Second-Order Designs, 422
Exercises, 434
9 Experimental Designs for Fitting Response Surfaces-II 451
9.1 Designs that Require a Relatively Small Run Size, 452
9.1.1 The Hoke Designs, 452
9.1.2 Koshal Design, 454
9.1.3 Hybrid Designs, 455
9.1.4 The Small Composite Design, 458
9.1.5 Some Saturated or Near-Saturated Cuboidal Designs, 462
9.1.6 Equiradial Designs, 463
9.2 General Criteria for Constructing, Evaluating, and Comparing Designed Experiments, 465
9.2.1 Practical Design Optimality, 467
9.2.2 Use of Design Efficiencies for Comparison of Standard Second-Order Designs, 474
9.2.3 Graphical Procedure for Evaluating the Prediction Capability of an RSM Design, 477
9.3 Computer-Generated Designs in RSM, 488
9.3.1 Important Relationship Between Prediction Variance and Design Augmentation for D-Optimality, 491
9.3.2 Algorithms for Computer-Generated Designs, 494
9.3.3 Comparison of D-, G-, and I-Optimal Designs, 497
9.3.4 Illustrations Involving Computer-Generated Design, 499
9.3.5 Computer-Generated Designs Involving Qualitative Variables, 508
9.4 Multiple Objective Computer-Generated Designs for RSM, 517
9.4.1 Pareto Front Optimization for Selecting a Design, 518
9.4.2 Pareto Aggregating Point Exchange Algorithm, 519
9.4.3 Using DMRCS for Design Optimization, 520
9.5 Some Final Comments Concerning Design Optimality and Computer-Generated Design, 525
Exercises, 527
10 Advanced Topics in Response Surface Methodology 543
10.1 Effects of Model Bias on the Fitted Model and Design, 543
10.2 A Design Criterion Involving Bias and Variance, 547
10.2.1 The Case of a First-Order Fitted Model and Cuboidal Region, 550
10.2.2 Minimum Bias Designs for a Spherical Region of Interest, 556
10.2.3 Simultaneous Consideration of Bias and Variance, 558
10.2.4 How Important Is Bias?, 558
10.3 Errors in Control of Design Levels, 560
10.4 Experiments with Computer Models, 563
10.4.1 Design for Computer Experiments, 567
10.4.2 Analysis for Computer Experiments, 570
10.4.3 Combining Information from Physical and Computer Experiments, 574
10.5 Minimum Bias Estimation of Response Surface Models, 575
10.6 Neural Networks, 579
10.7 Split-Plot Designs for Second-Order Models, 581
10.8 RSM for Non-Normal Responses-Generalized Linear Models, 591
10.8.1 Model Framework: The Link Function, 592
10.8.2 The Canonical Link Function, 593
10.8.3 Estimation of Model Coefficients, 593
10.8.4 Properties of Model Coefficients, 595
10.8.5 Model Deviance, 595
10.8.6 Overdispersion, 597
10.8.7 Examples, 598
10.8.8 Diagnostic Plots and Other Aspects of the GLM, 605
Exercises, 609
11 Robust Parameter Design and Process Robustness Studies 619
11.1 Introduction, 619
11.2 What is Parameter Design?, 619
11.2.1 Examples of Noise Variables, 620
11.2.2 An Example of Robust Product Design, 621
11.3 The Taguchi Approach, 622
11.3.1 Crossed Array Designs and Signal-to-Noise Ratios, 622
11.3.2 Analysis Methods, 625
11.3.3 Further Comments, 630
11.4 The Response Surface Approach, 631
11.4.1 The Role of the Control × Noise Interaction, 631
11.4.2 A Model Containing Both Control and Noise Variables, 635
11.4.3 Generalization of Mean and Variance Modeling, 638
11.4.4 Analysis Procedures Associated with the Two Response Surfaces, 642
11.4.5 Estimation of the Process Variance, 651
11.4.6 Direct Variance Modeling, 655
11.4.7 Use of Generalized Linear Models, 657
11.5 Experimental Designs For RPD and Process Robustness Studies, 661
11.5.1 Combined Array Designs, 661
11.5.2 Second-Order Designs, 663
11.5.3 Other Aspects of Design, 665
11.6 Dispersion Effects in Highly Fractionated Designs, 672
11.6.1 The Use of Residuals, 673
11.6.2 Further Diagnostic Information from Residuals, 674
11.6.3 Further Comments Concerning Variance Modeling, 680
Exercises, 684
12 Experiments with Mixtures 693
12.1 Introduction, 693
12.2 Simplex Designs and Canonical Mixture Polynomials, 696
12.2.1 Simplex Lattice Designs, 696
12.2.2 The Simplex-Centroid Design and Its Associated Polynomial, 704
12.2.3 Augmentation of Simplex Designs with Axial Runs, 707
12.3 Response Trace Plots, 716
12.4 Reparameterizing Canonical Mixture Models to Contain A Constant Term (¿¿¿¿0), 716
Exercises, 720
13 Other Mixture Design and Analysis Techniques 731
13.1 Constraints on the Component Proportions, 731
13.1.1 Lower-Bound Constraints on the Component Proportions, 732
13.1.2 Upper-Bound Constraints on the Component Proportions, 743
13.1.3 Active Upper- and Lower-Bound Constraints, 747
13.1.4 Multicomponent Constraints, 758
13.2 Mixture Experiments Using Ratios of Components, 759
13.3 Process Variables in Mixture Experiments, 763
13.3.1 Mixture-Process Model and Design Basics, 763
13.3.2 Split-Plot Designs for Mixture-Process Experiments, 767
13.3.3 Robust Parameter Designs for Mixture-Process Experiments, 778
13.4 Screening Mixture Components, 783
Exercises, 785
Appendix 1 Moment Matrix of a Rotatable Design 797
Appendix 2 Rotatability of a Second-Order Equiradial Design 803
References 807
Index 821
1
INTRODUCTION
1.1 RESPONSE SURFACE METHODOLOGY
Response surface methodology (RSM) is a collection of statistical and mathematical techniques useful for developing, improving, and optimizing processes. It also has important applications in the design, development, and formulation of new products, as well as in the improvement of existing product designs.
The most extensive applications of RSM are in the industrial world, particularly in situations where several input variables potentially influence performance measures or quality characteristics of the product or process. These performance measures or quality characteristics are called the response. They are typically measured on a continuous scale, although attribute responses, ranks, and sensory responses are not unusual. Most real-world applications of RSM will involve more than one response. The input variables are sometimes called independent variables, and they are subject to the control of the engineer or scientist, at least for purposes of a test or an experiment.
Figure 1.1 shows graphically the relationship between the response variable yield (y) in a chemical process and the two process variables (or independent variables) reaction time (?1) and reaction temperature (?2). Note that for each value of ?1 and ?2 there is a corresponding value of yield y and that we may view these values of the response yield as a surface lying above the time-temperature plane, as in Fig. 1.1a. It is this graphical perspective of the problem environment that has led to the term response surface methodology. It is also convenient to view the response surface in the two-dimensional time-temperature plane, as in Fig. 1.1b. In this presentation we are looking down at the time-temperature plane and connecting all points that have the same yield to produce contour lines of constant response. This type of display is called a contour plot.
Figure 1.1 (a) A theoretical response surface showing the relationship between yield of a chemical process and the process variables reaction time (?1) and reaction temperature (?2). (b) A contour plot of the theoretical response surface.
Clearly, if we could easily construct the graphical displays in Fig. 1.1, optimization of this process would be very straightforward. By inspection of the plot, we note that yield is maximized in the vicinity of time ?1 = 4 hr and temperature ?2 = 525°C. Unfortunately, in most practical situations, the true response function in Fig. 1.1 is unknown. The field of response surface methodology consists of the experimental strategies for exploring the space of the process or independent variables (here the variables ?1 and ?2), empirical statistical modeling to develop an appropriate approximating relationship between the yield and the process variables, and optimization methods for finding the levels or values of the process variables ?1 and ?2 that produce desirable values of the responses (in this case that maximize yield).
1.1.1 Approximating Response Functions
In general, suppose that the scientist or engineer (whom we will refer to as the experimenter) is concerned with a product, process, or system involving a response y that depends on the controllable input variables ?1, ?2, . , ?k. These input variables are also sometimes called factors, independent variables, or process variables. The actual relationship can be written
(1.1)where the form of the true response function f is unknown and perhaps very complicated, and ? is a term that represents other sources of variability not accounted for in f. Thus ? includes effects such as measurement error on the response, other sources of variation that are inherent in the process or system (background noise, or common/special cause variation in the language of statistical process control), the effect of other (possibly unknown) variables, and so on. We will treat ? as a statistical error, often assuming it to have a normal distribution with mean zero and variance s 2. If the mean of ? is zero, then
(1.2)The variables ?1, ?2, . , ?k in Equation 1.2 are usually called the natural variables, because they are expressed in the natural units of measurement, such as degrees Celsius (°C), pounds per square inch (psi), or grams per liter for concentration. In much RSM work it is convenient to transform the natural variables to coded variables x1, x2, . , xk, which are usually defined to be dimensionless with mean zero and the same spread or standard deviation. In terms of the coded variables, the true response function (1.2) is now written as
(1.3)Because the form of the true response function f is unknown, we must approximate it. In fact, successful use of RSM is critically dependent upon the experimenter's ability to develop a suitable approximation for f. Usually, a low-order polynomial in some relatively small region of the independent variable space is appropriate. In many cases, either a first-order or a second-order model is used. For the case of two independent variables, the first-order model in terms of the coded variables is
(1.4)Figure 1.2 shows the three-dimensional response surface and the two-dimensional contour plot for a particular case of the first-order model, namely,
In three dimensions, the response surface for y is a plane lying above the x1, x2 space. The contour plot shows that the first-order model can be represented as parallel straight lines of constant response in the x1, x2 plane.
Figure 1.2 (a) Response surface for the first-order model ? = 50 + 8x1 + 3x2. (b) Contour plot for the first-order model.
The first-order model is likely to be appropriate when the experimenter is interested in approximating the true response surface over a relatively small region of the independent variable space in a location where there is little curvature in f. For example, consider a small region around the point A in Fig. 1.1b; the first-order model would likely be appropriate here.
The form of the first-order model in Equation 1.4 is sometimes called a main effects model, because it includes only the main effects of the two variables x1 and x2. If there is an interaction between these variables, it can be added to the model easily as follows:
(1.5)This is the first-order model with interaction. Figure 1.3 shows the three-dimensional response surface and the contour plot for the special case
Notice that adding the interaction term -4x1x2 introduces curvature into the response function. This leads to different rates of change of the response as x1 is changed for different fixed values of x2. Similarly, the rate of change in y across x2 varies for different fixed values of x1.
Figure 1.3 (a) Response surface for the first-order model with interaction ? = 50 + 8x1 + 3x2 - 4x1x2. (b) Contour plot for the first-order model with interaction.
Often the curvature in the true response surface is strong enough that the first-order model (even with the interaction term included) is inadequate. A second-order model will likely be required in these situations. For the case of two variables, the second-order model is
(1.6)This model would likely be useful as an approximation to the true response surface in a relatively small region around the point B in Fig. 1.1b, where there is substantial curvature in the true response function f.
Figure 1.4 presents the response surface and contour plot for the special case of the second-order model
Notice the mound-shaped response surface and elliptical contours generated by this model. Such a response surface could arise in approximating a response such as yield, where we would expect to be operating near a maximum point on the surface.
Figure 1.4 (a) Response surface for the second-order model ? = 50 + 8x1 + 3x2 - 7x12 - 3x22 - 4x1x2. (b) Contour plot for the second-order model.
The second-order model is widely used in response surface methodology for several reasons. Among these are the following:
-
The second-order model is very flexible. It can take on a wide variety of functional forms, so it will often work well as an approximation to the true response surface. Figure 1.5 shows several different response surfaces and contour plots that can be generated by a...
