Response Surfaces, Mixtures, and Ridge Analyses

Response Surfaces, Mixtures, and Ridge Analyses......Page 4
Contents......Page 8
Preface to the Second Edition......Page 16
1.1. Response Surface Methodology (RSM)......Page 18
1.2. Indeterminancy of Experimentation......Page 20
1.3. Iterative Nature of the Experimental Learning Process......Page 23
1.4. Some Classes of Problems (Which, How, Why)......Page 26
1.5. Need for Experimental Design......Page 29
1.6. Geometric Representation of Response Relationships......Page 30
1.7. Three Kinds of Applications......Page 31
2.1. Approximating Response Functions......Page 34
2.2. An Example......Page 41
Appendix 2A. A Theoretical Response Function......Page 44
3.1. The Method of Least Squares......Page 46
3.2. Linear Models......Page 47
3.3. Matrix Formulas for Least Squares......Page 51
3.4. Geometry of Least Squares......Page 57
3.5. Analysis of Variance for One Regressor......Page 60
3.6. Least Squares for Two Regressors......Page 63
3.7. Geometry of the Analysis of Variance for Two Regressors......Page 65
3.8. Orthogonalizing the Second Regressor, Extra Sum of Squares Principle......Page 66
3.9. Generalization to p Regressors......Page 70
3.10. Bias in Least-Squares Estimators Arising from an Inadequate Model......Page 74
3.11. Pure Error and Lack of Fit......Page 77
3.12. Confidence Intervals and Confidence Regions......Page 80
3.13. Robust Estimation, Maximum Likelihood, and Least Squares......Page 85
Appendix 3A. Iteratively Reweighted Least Squares......Page 93
Appendix 3B. Justification of Least Squares by the Gauss–Markov Theorem; Robustness......Page 94
Appendix 3C. Matrix Theory......Page 95
Appendix 3D. Nonlinear Estimation......Page 103
Appendix 3E. Results Involving V(y)......Page 105
Exercises......Page 106
4.1. The Value of Factorial Designs......Page 109
4.2. Two-Level Factorials......Page 111
4.3. A 2(6) Design Used in a Study of Dyestuffs Manufacture......Page 118
4.4. Diagnostic Checking of the Fitted Models, 2(6) Dyestuffs Example......Page 125
4.5. Response Surface Analysis of the 2(6) Design Data......Page 126
Appendix 4A. Yates’ Method for Obtaining the Factorial Effects for a Two-Level Design......Page 128
Appendix 4B. Normal Plots on Probability Paper......Page 130
Appendix 4C. Confidence Regions for Contour Planes (see Section 4.5)......Page 135
Exercises......Page 136
5.1. Blocking the 2(6) Design......Page 146
5.2. Fractionating the 2(6) Design......Page 150
5.3. Resolution of a 2(k–p) Factorial Design......Page 155
5.4. Construction of 2(k–p) Designs of Resolution III and IV......Page 156
5.5. Combination of Designs from the Same Family......Page 161
5.6. Screening, Using 2(k–p) Designs (Involving Projections to Lower Dimensions)......Page 165
5.7. Complete Factorials Within Fractional Factorial Designs......Page 167
5.8. Plackett and Burman Designs for n = 12 to 60 (but not 52)......Page 169
5.9. Screening, Using Plackett and Burman Designs (Involving Projections to Lower Dimensions)......Page 172
5.10. Efficient Estimation of Main Effects and Two-Factor Interactions Using Relatively Small Two-Level Designs......Page 175
5.11. Designs of Resolution V and of Higher Resolution......Page 177
5.12. Application of Fractional Factorial Designs to Response Surface Methodology......Page 183
5.13. Plotting Effects from Fractional Factorials on Probability Paper......Page 184
Exercises......Page 185
6. The Use of Steepest Ascent to Achieve Process Improvement......Page 204
6.1. What Steepest Ascent Does......Page 205
6.2. An Example. Improvement of a Process Used in Drug Manufacture......Page 206
6.3. General Comments on Steepest Ascent......Page 209
6.4. A Confidence Region for the Direction of Steepest Ascent......Page 211
6.5. Steepest Ascent Subject to a Constraint......Page 214
Appendix 6A. Effects of Changes in Scales on Steepest Ascent......Page 218
Appendix 6B. Placement of First-Order Designs Within Spherical and Cuboidal Regions of Interest......Page 221
Exercises......Page 224
7.1. Fitting and Checking Second-Degree Polynomial Graduating Functions......Page 227
7.2. A Comprehensive Check of the Need for Transformation in the Response Variable......Page 236
7.3. Interim Summary......Page 238
7.4. Further Examination of Adequacy of Fit; Higher-Order Terms......Page 239
7.5. Robust Methods in Fitting Graduating Functions......Page 244
Appendix 7A. Fitting of First- and Second-Degree Models......Page 246
Appendix 7B. Checking for Nonadditivity with Tukey’s Test......Page 249
Appendix 7C. Third-Order Biases in Estimates for Second-Order Model, Using the 3(3) Design......Page 250
Appendix 7D. Analysis of Variance to Detect Possible Third-Order Terms......Page 252
Appendix 7E. Orthogonalization of Effects for Three-Level Designs......Page 254
Appendix 7G. Yates’ Algorithm for 3(k) Factorial Designs......Page 257
Appendix 7H. Desirable Features and Placement of Second Order Designs for Spherical and Cuboidal Regions of Interest......Page 260
Exercises......Page 264
8.1. What Can Transformations Do?......Page 287
8.2. A Measure of Adequacy of Function Estimation......Page 293
8.3. Choosing the Response Metric to Validate Distribution Assumptions......Page 297
8.4. Estimating Response Transformation from the Data......Page 304
8.5. Transformation of the Predictor Variables......Page 308
Appendix 8A. Proof of a Result Used in Section 8.2......Page 314
Exercises......Page 315
9.1. The Need for Second-Order Surfaces......Page 320
9.2. Example: Representation of Polymer Elasticity as a Quadratic Function of Three Operating Variables......Page 321
9.3. Examination of the Fitted Surface......Page 328
9.4. Investigation of Adequacy of Fit: Isolation of Residual Degrees of Freedom for the Composite Design......Page 330
Exercises......Page 335
10.1. Introduction......Page 338
10.2. Factor Dependence......Page 341
10.3. Elucidation of Stationary Regions, Maxima, and Minima by Canonical Analysis......Page 346
Appendix 10A. A Simple Explanation of Canonical Analysis......Page 354
11.1. Ridge Systems Approximated by Limiting Cases......Page 358
11.2. Canonical Analysis in the A Form to Characterize Ridge Phenomena......Page 361
11.3. Example 11.3: A Consecutive Chemical System Yielding a Near-Stationary Planar Ridge Maximum......Page 365
11.4. Example 11.4: A Small Reactor Study Yielding a Rising Ridge Surface......Page 370
11.5. Example 11.5: A Stationary Ridge in Five Variables......Page 376
11.6. The Economic Importance of the Dimensionality of Maxima and Minima......Page 380
11.7. A Method for Obtaining a Desirable Combination of Several Responses......Page 381
Exercises......Page 383
12.1. Introduction......Page 408
12.2. Standard Ridge Analysis About the Origin x = 0......Page 409
12.3. Worked Example of Ridge Analysis About the Origin x = 0......Page 411
12.4. Ridge Analysis Around a Selected Focus f......Page 415
12.5. Worked Example of Ridge Analysis About a Selected Focus f......Page 416
12.7. A Ridge Exploration Solution Using Minitab......Page 417
Exercises......Page 421
13.1. The Use of Approximating Functions......Page 430
13.2. The Competing Effects of Bias and Variance......Page 432
13.3. Integrated Mean Squared Error......Page 435
13.4. Region of Interest and Region of Operability in k Dimensions......Page 436
13.5. An Extension of the Idea of a Region of Interest: The Weight Function......Page 439
13.6. Designs That Minimize Squared Bias......Page 441
13.7. Ensuring Good Detectability of Lack of Fit......Page 446
13.8. Checks Based on the Possibility of Obtaining a Simple Model by Transformation of the Predictors......Page 453
Appendix 13B. Necessary and Sufficient Design Conditions to Minimize Bias......Page 463
Appendix 13C. Minimum Bias Estimations......Page 465
Appendix 13D. A Shrinkage Estimation Procedure......Page 466
Appendix 13E. Conditions for Efficacy of Transformation on the Predictor Variables......Page 467
Appendix 13F. The Third-Order Columns of the X Matrix for a Composite Design......Page 472
Exercises......Page 474
14.1. Introduction......Page 478
14.2. Orthogonal Designs......Page 479
14.3. The Variance Function......Page 481
14.4. The Alphabetic Optimality Approach......Page 488
14.5. Alphabetic Optimality in the Response Surface Context......Page 493
15.1. Introduction......Page 500
15.2. First-Order Designs......Page 503
15.3. Second-Order Composite Designs......Page 505
15.4. Second-Order Designs Requiring Only Three Levels......Page 512
15.5. Screening Designs with Three Levels and Their Projections into Fewer Factors......Page 516
15.6. Designs Requiring Only a Small Number of Runs......Page 517
Exercises......Page 521
16.1. Mixture Spaces......Page 526
16.2. First-Order Mixture Models......Page 530
16.3. Second-Order Mixture Models......Page 533
16.4. Third-Order Canonical Mixture Models......Page 537
16.5. Design of Experiments for Canonical Polynomial Models......Page 538
16.6. Simplex Centroid Designs and the Related Polynomials......Page 542
Appendix 16A. Examples of Second-Order Response Contours for q = 3 Mixture Ingredients......Page 543
Appendix 16C. Examples of a General Cubic Response Surface and a Quartic Response Surface for q = 3 Mixture Ingredients......Page 546
Exercises......Page 548
17.1. Introduction......Page 552
17.2. Lower Bounds and Pseudocomponents......Page 554
17.3. Upper Bounds and Pseudocomponents......Page 559
17.4. Upper and Lower Bounds......Page 561
Exercises......Page 567
18.1. Using Ratios of Ingredients as Predictor Variables......Page 570
18.2. The Kissell and Marshall Example......Page 572
18.3. Mixture Designs and Models When the Total Amount is Important......Page 578
18.4. Designs for Component-Amount Models Obtained by Projection of Standard Simplex Lattice and Simplex-Centroid Mixture Designs......Page 587
18.5. Designs in Orthogonal Blocks for Component-Amount Models......Page 595
18.6. An Example of a Projection of a Nearly D-Optimal Mixture Design into Fewer Dimensions......Page 606
Exercise......Page 610
19.1. Introduction......Page 612
19.2. A motivating experiment......Page 613
19.3. Standard Ridge Analysis Without Linear Restrictions, Brief Recapitulation of the Algebra......Page 616
19.4. Ridge Analysis with Multiple Linear Restrictions, Including Applications to Mixture Experiments......Page 617
19.5. Generalized Ridge Analysis of the Motivating Experiment......Page 621
19.6. Some Additional Considerations......Page 627
Appendix 19A. Orthogonal and Normalized Rows (Orthogonal Polynomials) for q = 3, 4, 5, and 6 Dimensions......Page 629
Appendix 19B. Adaptable Minitab Instructions for the Anik and Sukumar Example......Page 630
Exercises......Page 632
20.1. Introduction......Page 636
20.2. Canonical Reduction with no Restrictions (Recapitulation)......Page 637
20.3. Adding Linear Restrictions......Page 638
20.4. Canonical Reduction under Linear Restrictions......Page 639
20.5. Three Ingredients Mixture Example......Page 642
20.6. Five Ingredients Mixture Example......Page 644
Answers to Exercises......Page 646
Tables......Page 760
A. Tail Area of the Unit Normal Distribution......Page 761
B. Probability Points of the t Distribution......Page 762
C. Probability Points of the X(2) Distribution......Page 763
D. Percentage Points of the F Distribution......Page 765
Bibliography......Page 774
Author Index......Page 868
Subject Index......Page 872