Response Surface Methodology

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.

• Preface xiii 1 Introduction 11.1 Response Surface Methodology, 11.1.1 Approximating Response Functions, 21.1.2 The Sequential Nature of RSM, 71.1.3 Objectives and Typical Applications of RSM, 91.1.4 RSM and the Philosophy of Quality Improvement, 111.2 Product Design and Formulation (Mixture Problems), 111.3 Robust Design and Process Robustness Studies, 121.4 Useful References on RSM, 122 Building Empirical Models 132.1 Linear Regression Models, 132.2 Estimation of the Parameters in Linear Regression Models, 142.3 Properties of the Least Squares Estimators and Estimation of 𝜎2, 222.4 Hypothesis Testing in Multiple Regression, 242.4.1 Test for Significance of Regression, 242.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients, 272.5 Confidence Intervals in Multiple Regression, 312.5.1 Confidence Intervals on the Individual Regression Coefficients β, 322.5.2 A Joint Confidence Region on the Regression Coefficients β, 322.5.3 Confidence Interval on the Mean Response, 332.6 Prediction of New Response Observations, 352.7 Model Adequacy Checking, 362.7.1 Residual Analysis, 362.7.2 Scaling Residuals, 382.7.3 Influence Diagnostics, 422.7.4 Testing for Lack of Fit, 432.8 Fitting a Second-Order Model, 472.9 Qualitative Regressor Variables, 552.10 Transformation of the Response Variable, 61Exercises, 663 Two-Level Factorial Designs 813.1 Introduction, 813.2 The 22 Design, 823.3 The 23 Design, 943.4 The General 2k Design, 1033.5 A Single Replicate of the 2k Design, 1083.6 2k Designs are Optimal Designs, 1253.7 The Addition of Center Points to the 2k Design, 1303.8 Blocking in the 2k Factorial Design, 1353.8.1 Blocking in the Replicated Design, 1353.8.2 Confounding in the 2k Design, 1373.9 Split-Plot Designs, 141Exercises, 1464 Two-Level Fractional Factorial Designs 1614.1 Introduction, 1614.2 The One-Half Fraction of the 2k Design, 1624.3 The One-Quarter Fraction of the 2k Design, 1744.4 The General 2k−p Fractional Factorial Design, 1844.5 Resolution III Designs, 1884.6 Resolution IV and V Designs, 1974.7 Alias Structures in Fractional Factorial and Other Designs, 1984.8 Nonregular Fractional Factorial Designs, 2004.8.1 Nonregular Fractional Factorial Designs for 6, 7, and 8 Factors in 16 Runs, 2034.8.2 Nonregular Fractional Factorial Designs for 9 Through 14 Factors in 16 Runs, 2094.8.3 Analysis of Nonregular Fractional Factorial Designs, 2134.9 Fractional Factorial Split-Plot Designs, 2164.10 Summary, 219Exercises, 2205 Process Improvement with Steepest Ascent 2335.1 Determining the Path of Steepest Ascent, 2345.1.1 Development of the Procedure, 2345.1.2 Practical Application of the Method of Steepest Ascent, 2375.2 Consideration of Interaction and Curvature, 2415.2.1 What About a Second Phase?, 2445.2.2 What Happens Following Steepest Ascent?, 2445.3 Effect of Scale (Choosing Range of Factors), 2455.4 Confidence Region for Direction of Steepest Ascent, 2475.5 Steepest Ascent Subject to a Linear Constraint, 2505.6 Steepest Ascent in a Split-Plot Experiment, 254Exercises, 2626 The Analysis of Second-Order Response Surfaces 2736.1 Second-Order Response Surface, 2736.2 Second-Order Approximating Function, 2746.2.1 The Nature of the Second-Order Function and Second-Order Surface, 2746.2.2 Illustration of Second-Order Response Surfaces, 2766.3 A Formal Analytical Approach to the Second-Order Model, 2776.3.1 Location of the Stationary Point, 2786.3.2 Nature of the Stationary Point (Canonical Analysis), 2786.3.3 Ridge Systems, 2826.3.4 Role of Contour Plots, 2866.4 Ridge Analysis of the Response Surface, 2896.4.1 Benefits of Ridge Analysis, 2906.4.2 Mathematical Development of Ridge Analysis, 2916.5 Sampling Properties of Response Surface Results, 2966.5.1 Standard Error of Predicted Response, 2966.5.2 Confidence Region on the Location of the Stationary Point, 2996.5.3 Use and Computation of the Confidence Region on the Location of the Stationary Point, 3006.5.4 Confidence Intervals on Eigenvalues in Canonical Analysis, 3046.6 Further Comments Concerning Response Surface Analysis, 307Exercises, 3077 Multiple Response Optimization 3257.1 Balancing Multiple Objectives, 3257.2 Strategies for Multiple Response Optimization, 3387.2.1 Overlaying Contour Plots, 3397.2.2 Constrained Optimization, 3407.2.3 Desirability Functions, 3417.2.4 Pareto Front Optimization, 3437.2.5 Other Options for Optimization, 3497.3 A Sequential Process for Optimization—DMRCS, 3507.4 Incorporating Uncertainty of Response Predictions into Optimization, 352Exercises, 3578 Design of Experiments for Fitting Response Surfaces—I 3698.1 Desirable Properties of Response Surface Designs, 3698.2 Operability Region, Region of Interest, and Metrics for Desirable Properties, 3718.2.1 Metrics for Desirable Properties, 3728.2.2 Model Inadequacy and Model Bias, 3738.3 Design of Experiments for First-Order Models and First-Order Models with Interactions, 3758.3.1 The First-Order Orthogonal Design, 3768.3.2 Orthogonal Designs for Models Containing Interaction, 3788.3.3 Other First-Order Orthogonal Designs—The Simplex Design, 3818.3.4 Definitive Screening Designs, 3858.3.5 Another Variance Property—Prediction Variance, 3898.4 Designs for Fitting Second-Order Models, 3938.4.1 The Class of Central Composite Designs, 3938.4.2 Design Moments and Property of Rotatability, 3998.4.3 Rotatability and the CCD, 4038.4.4 More on Prediction Variance—Scaled, Unscaled, and Estimated, 4068.4.5 The Face-Centered Cube in Cuboidal Regions, 4088.4.6 Choosing between Spherical and Cuboidal Regions, 4118.4.7 The Box–Behnken Design, 4138.4.8 Definitive Screening Designs for Fitting Second-Order Models, 4178.4.9 Orthogonal Blocking in Second-Order Designs, 422Exercises, 4349 Experimental Designs for Fitting Response Surfaces—II 4519.1 Designs that Require a Relatively Small Run Size, 4529.1.1 The Hoke Designs, 4529.1.2 Koshal Design, 4549.1.3 Hybrid Designs, 4559.1.4 The Small Composite Design, 4589.1.5 Some Saturated or Near-Saturated Cuboidal Designs, 4629.1.6 Equiradial Designs, 4639.2 General Criteria for Constructing, Evaluating, and Comparing Designed Experiments, 4659.2.1 Practical Design Optimality, 4679.2.2 Use of Design Efficiencies for Comparison of Standard Second-Order Designs, 4749.2.3 Graphical Procedure for Evaluating the Prediction Capability of an RSM Design, 4779.3 Computer-Generated Designs in RSM, 4889.3.1 Important Relationship Between Prediction Variance and Design Augmentation for D-Optimality, 4919.3.2 Algorithms for Computer-Generated Designs, 4949.3.3 Comparison of D-, G-, and I-Optimal Designs, 4979.3.4 Illustrations Involving Computer-Generated Design, 4999.3.5 Computer-Generated Designs Involving Qualitative Variables, 5089.4 Multiple Objective Computer-Generated Designs for RSM, 5179.4.1 Pareto Front Optimization for Selecting a Design, 5189.4.2 Pareto Aggregating Point Exchange Algorithm, 5199.4.3 Using DMRCS for Design Optimization, 5209.5 Some Final Comments Concerning Design Optimality and Computer-Generated Design, 525Exercises, 52710 Advanced Topics in Response Surface Methodology 54310.1 Effects of Model Bias on the Fitted Model and Design, 54310.2 A Design Criterion Involving Bias and Variance, 54710.2.1 The Case of a First-Order Fitted Model and Cuboidal Region, 55010.2.2 Minimum Bias Designs for a Spherical Region of Interest, 55610.2.3 Simultaneous Consideration of Bias and Variance, 55810.2.4 How Important Is Bias?, 55810.3 Errors in Control of Design Levels, 56010.4 Experiments with Computer Models, 56310.4.1 Design for Computer Experiments, 56710.4.2 Analysis for Computer Experiments, 57010.4.3 Combining Information from Physical and Computer Experiments, 57410.5 Minimum Bias Estimation of Response Surface Models, 57510.6 Neural Networks, 57910.7 Split-Plot Designs for Second-Order Models, 58110.8 RSM for Non-Normal Responses—Generalized Linear Models, 59110.8.1 Model Framework: The Link Function, 59210.8.2 The Canonical Link Function, 59310.8.3 Estimation of Model Coefficients, 59310.8.4 Properties of Model Coefficients, 59510.8.5 Model Deviance, 59510.8.6 Overdispersion, 59710.8.7 Examples, 59810.8.8 Diagnostic Plots and Other Aspects of the GLM, 605Exercises, 60911 Robust Parameter Design and Process Robustness Studies 61911.1 Introduction, 61911.2 What is Parameter Design?, 61911.2.1 Examples of Noise Variables, 62011.2.2 An Example of Robust Product Design, 62111.3 The Taguchi Approach, 62211.3.1 Crossed Array Designs and Signal-to-Noise Ratios, 62211.3.2 Analysis Methods, 62511.3.3 Further Comments, 63011.4 The Response Surface Approach, 63111.4.1 The Role of the Control × Noise Interaction, 63111.4.2 A Model Containing Both Control and Noise Variables, 63511.4.3 Generalization of Mean and Variance Modeling, 63811.4.4 Analysis Procedures Associated with the Two Response Surfaces, 64211.4.5 Estimation of the Process Variance, 65111.4.6 Direct Variance Modeling, 65511.4.7 Use of Generalized Linear Models, 65711.5 Experimental Designs For RPD and Process Robustness Studies, 66111.5.1 Combined Array Designs, 66111.5.2 Second-Order Designs, 66311.5.3 Other Aspects of Design, 66511.6 Dispersion Effects in Highly Fractionated Designs, 67211.6.1 The Use of Residuals, 67311.6.2 Further Diagnostic Information from Residuals, 67411.6.3 Further Comments Concerning Variance Modeling, 680Exercises, 68412 Experiments with Mixtures 69312.1 Introduction, 69312.2 Simplex Designs and Canonical Mixture Polynomials, 69612.2.1 Simplex Lattice Designs, 69612.2.2 The Simplex-Centroid Design and Its Associated Polynomial, 70412.2.3 Augmentation of Simplex Designs with Axial Runs, 70712.3 Response Trace Plots, 71612.4 Reparameterizing Canonical Mixture Models to Contain A Constant Term (𝛽0), 716Exercises, 72013 Other Mixture Design and Analysis Techniques 73113.1 Constraints on the Component Proportions, 73113.1.1 Lower-Bound Constraints on the Component Proportions, 73213.1.2 Upper-Bound Constraints on the Component Proportions, 74313.1.3 Active Upper- and Lower-Bound Constraints, 74713.1.4 Multicomponent Constraints, 75813.2 Mixture Experiments Using Ratios of Components, 75913.3 Process Variables in Mixture Experiments, 76313.3.1 Mixture-Process Model and Design Basics, 76313.3.2 Split-Plot Designs for Mixture-Process Experiments, 76713.3.3 Robust Parameter Designs for Mixture-Process Experiments, 77813.4 Screening Mixture Components, 783Exercises, 785Appendix 1 Moment Matrix of a Rotatable Design 797Appendix 2 Rotatability of a Second-Order Equiradial Design 803References 807Index 821