A Primer on Experiments with Mixtures

JOHN A. CORNELL , PhD, is Professor Emeritus of Statistics at the University of Florida. A recognized authority on the topic of experimental design, he has more than forty years of experience in both academia and industrial consulting and was awarded the Shewhart Medal by the American Society of Quality (ASQ) in 2001. A Fellow of both the ASQ and American Statistical Association, Dr. Cornell is the author of Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data , Third Edition, also published by Wiley.

Preface ix 1. Introduction 1 1.1 The Original Mixture Problem, 2 1.2 A Pesticide Example Involving Two Chemicals, 2 1.3 General Remarks About Response Surface Methods, 9 1.4 An Historical Perspective, 13 References and Recommended Reading, 17 Questions, 17 Appendix 1A: Testing for Nonlinear Blending of the Two Chemicals Vendex and Kelthane While Measuring the Average Percent Mortality (APM) of Mites, 20 2. The Original Mixture Problem: Designs and Models for Exploring the Entire Simplex Factor Space 23 2.1 The Simplex-Lattice Designs, 23 2.2 The Canonical Polynomials, 26 2.3 The Polynomial Coefficients As Functions of the Responses at the Points of the Lattices, 31 2.4 Estimating The Parameters in the { q,m } Polynomials, 34 2.5 Properties of the Estimate of the Response, y ( x ), 37 2.6 A Three-Component Yarn Example Using A {3 , 2} Simplex-Lattice Design, 38 2.7 The Analysis of Variance Table, 42 2.8 Analysis of Variance Calculations of the Yarn Elongation Data, 45 2.9 The Plotting of Individual Residuals, 48 2.10 Testing the Degree of the Fitted Model: A Quadratic Model or Planar Model?, 49 2.11 Testing Model Lack of Fit Using Extra Points and Replicated Observations, 55 2.12 The Simplex-Centroid Design and Associated Polynomial Model, 58 2.13 An Application of a Four-Component Simplex-Centroid Design: Blending Chemical Pesticides for Control of Mites, 60 2.14 Axial Designs, 62 2.15 Comments on a Comparison Made Between An Augmented Simplex-Centroid Design and a Full Cubic Lattice for Three Components Where Each Design Contains Ten Points, 66 2.16 Reparameterizing Scheffe's Mixture Models to Contain A Constant ( beta 0) Term: A Numerical Example, 69 2.17 Questions to Consider at the Planning Stages of a Mixture Experiment, 77 2.18 Summary, 78 References and Recommended Reading, 78 Questions, 80 Appendix 2A: Least-Squares Estimation Formula for the Polynomial Coefficients and Their Variances: Matrix Notation, 85 Appendix 2B: Cubic and Quartic Polynomials and Formulas for the Estimates of the Coefficients, 90 Appendix 2C: The Partitioning of the Sources in the Analysis of Variance Table When Fitting the Scheff'e Mixture Models, 91 3. Multiple Constraints on the Component Proportions 95 3.1 Lower-Bound Restrictions on Some or All of the Component Proportions, 95 3.2 Introducing L -Pseudocomponents, 97 3.3 A Numerical Example of Fitting An L -Pseudocomponent Model, 99 3.4 Upper-Bound Restrictions on Some or All Component Proportions, 101 3.5 An Example of the Placing of an Upper Bound on a Single Component: The Formulation of a Tropical Beverage, 103 3.6 Introducing U -Pseudocomponents, 107 3.7 The Placing of Both Upper and Lower Bounds on the Component Proportions, 112 3.8 Formulas For Enumerating the Number of Extreme Vertices, Edges, and Two-Dimensional Faces of the Constrained Region, 119 3.9 McLean and Anderson's Algorithm For Calculating the Coordinates of the Extreme Vertices of a Constrained Region, 123 3.10 Multicomponent Constraints, 128 3.11 Some Examples of Designs for Constrained Mixture Regions: CONVRT and CONAEV Programs, 131 3.12 Multiple Lattices for Major and Minor Component Classifications, 138 Summary, 154 References and Recommended Reading, 155 Questions, 157 4. The Analysis of Mixture Data 159 4.1 Techniques Used in the Analysis of Mixture Data, 160 4.2 Test Statistics for Testing the Usefulness of the Terms in the Scheff'e Polynomials, 163 4.3 Model Reduction, 170 4.4 An Example of Reducing the System from Three to Two Components, 173 4.5 Screening Components, 175 4.6 Other Techniques Used to Measure Component Effects, 179 4.7 Leverage and the Hat Matrix, 190 4.8 A Three-Component Propellant Example, 192 4.9 Summary, 195 References and Recommended Reading, 196 Questions, 197 5. Other Mixture Model Forms 201 5.1 The Inclusion of Inverse Terms in the Scheff'e Polynomials, 201 5.2 Fitting Gasoline Octane Numbers Using Inverse Terms in the Model, 204 5.3 An Alternative Model Form for Modeling the Additive Blending Effect of One Component In a Multicomponent System, 205 5.4 A Biological Example on the Linear Effect of a Powder Pesticide In Combination With Two Liquid Pesticides Used for Suppressing Mite Population Numbers, 212 5.5 The Use of Ratios of Components, 215 5.6 Cox's Mixture Polynomials: Measuring Component Effects, 219 5.7 An Example Illustrating the Fits of Cox's Model and Scheffe's Polynomial, 224 5.8 Fitting A Slack-Variable Model, 229 5.9 A Numerical Example Illustrating The Fits of Different Reduced Slack-Variable Models: Tint Strength of a House Paint, 233 5.10 Summary, 239 References and Recommended Reading, 240 Questions, 242 6. The Inclusion of Process Variables in Mixture Experiments 247 6.1 Designs Consisting of Simplex-Lattices and Factorial Arrangements, 249 6.2 Measuring the Effects of Cooking Temperature and Cooking Time on the Texture of Patties Made from Two Types of Fish, 251 6.3 Mixture-Amount Experiments, 256 6.4 Determining the Optimal Fertilizer Blend and Rate for Young Citrus Trees, 262 6.5 A Numerical Example of the Fit of a Combined Model to Data Collected on Fractions of the Fish Patty Experimental Design, 269 6.6 Questions Raised and Recommendations Made When Fitting a Combined Model Containing Mixture Components and Other Variables, 272 6.7 Summary, 277 References and Recommended Reading, 278 Questions, 280 Appendix 6A: Calculating the Estimated Combined Mixture Component-Process Variable Model of Eq. (6.10) Without the Computer, 282 7. A Review of Least Squares and the Analysis of Variance 285 7.1 A Review of Least Squares, 285 7.2 The Analysis of Variance, 288 7.3 A Numerical Example: Modeling the Texture of Fish Patties, 289 7.4 The Adjusted Multiple Correlation Coefficient, 293 7.5 The Press Statistic and Studentized Residuals, 293 7.6 Testing Hypotheses About the Form of the Model: Tests of Significance, 295 References and Recommended Reading, 298 Bibliography 299 Answers to Selected Questions 317 Appendix 337 Index 347