Response Surfaces, Mixtures, and Ridge Analyses, Second Edition

859 Pages·2013·25.74 MB·English

by  George E. P. Box

Response Surfaces, Mixtures,
and Ridge Analyses

Response Surfaces, Mixtures,
and Ridge Analyses
Second Edition
GEORGE E. P. BOX
Departments of Industrial Engineering and Statistics
University of Wisconsin
Madison, Wisconsin
NORMAN R. DRAPER
Department of Statistics
University of Wisconsin
Madison, Wisconsin
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LibraryofCongressCataloginginPublicationData:
Box,GeorgeE.P.
Responsesurfaces, mixtures,andridgeanalysesrGeorgeE.P.Box,NormanR.Draper.
--2nded.
p.cm.
Includes bibliographical references andindex.
ISBN978-0-470-05357-7Žcloth.
1.Experimental design.2.Responsesurfaces ŽStatistics.3.Mixturedistributions
ŽProbabilitytheory.4.RidgeregressionŽStatistics.I.Draper, NormanRichard.II.Title.
III.Title:Empiricalmodel-buildingandresponsesurfaces.
QA279.B67 2007
519.5(cid:1)(cid:2)dc22 2006043975
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
Contents
Preface to the Second Edition xiii
1. Introduction to Response Surface Methodology 1
1.1. Response Surface Methodology ŽRSM., 1
1.2. Indeterminancy of Experimentation, 3
1.3. Iterative Nature of the Experimental Learning Process, 6
1.4. Some Classes of Problems ŽWhich, How, Why., 9
1.5. Need for Experimental Design, 12
1.6. Geometric Representation of Response Relationships, 13
1.7. Three Kinds of Applications, 14
2. The Use Of Graduating Functions 17
2.1. Approximating Response Functions, 17
2.2. An Example, 24
Appendix 2A. A Theoretical Response Function, 27
3. Least Squares for Response Surface Work 29
3.1. The Method of Least Squares, 29
3.2. Linear Models, 30
3.3. Matrix Formulas for Least Squares, 34
3.4. Geometry of Least Squares, 40
3.5. Analysis of Variance for One Regressor, 43
3.6. Least Squares for Two Regressors, 46
3.7. Geometryofthe Analysis ofVariance forTwoRegressors, 48
3.8. Orthogonalizing the Second Regressor, Extra Sum of Squares
Principle, 49
3.9. Generalization to p Regressors, 53
3.10. Bias in Least-Squares Estimators Arising from an Inadequate
Model, 57
3.11. Pure Error and Lack of Fit, 60
3.12. Confidence Intervals and Confidence Regions, 63
v
vi CONTENTS
3.13. Robust Estimation, Maximum Likelihood, and Least
Squares, 68
Appendix 3A. Iteratively Reweighted Least Squares, 76
Appendix3B. Justification ofLeast Squares bytheGauss(cid:1)Markov
Theorem; Robustness, 77
Appendix 3C. Matrix Theory, 78
Appendix 3D. Nonlinear Estimation, 86
Appendix 3E. Results Involving VŽˆy., 88
Exercises, 89
4. Factorial Designs at Two Levels 92
4.1. The Value of Factorial Designs, 92
4.2. Two-Level Factorials, 94
4.3. A 26 Design Used in a Study of Dyestuffs Manufacture, 101
4.4. DiagnosticCheckingoftheFitted Models,26 Dyestuffs
Example, 108
4.5. Response Surface Analysis of the 26 Design Data, 109
Appendix 4A. Yates’Method for Obtaining the Factorial Effects
for a Two-Level Design, 111
Appendix 4B. Normal Plots on Probability Paper, 113
Appendix4C. ConfidenceRegionsforContourPlanesŽseeSection
4.5., 118
Exercises, 119
5. Blocking and Fractionating 2k Factorial Designs 129
5.1. Blocking the 26 Design, 129
5.2. Fractionating the 26 Design, 133
5.3. Resolution of a 2kyp Factorial Design, 138
5.4. Construction of 2kyp Designs of Resolution III and IV, 139
5.5. Combination of Designs from the Same Family, 144
5.6. Screening,Using2kyp DesignsŽInvolvingProjectionstoLower
Dimensions., 148
5.7. Complete Factorials Within Fractional Factorial Designs, 150
5.8. Plackett and Burman Designs for ns12 to 60 Žbut not 52.,
152
5.9. Screening, Using Plackett and Burman Designs ŽInvolving
Projections to Lower Dimensions., 155
5.10. Efficient EstimationofMainEffects andTwo-Factor
Interactions Using Relatively Small Two-Level Designs, 158
5.11. Designs of Resolution V and of Higher Resolution, 160
5.12. ApplicationofFractional Factorial DesignstoResponse
Surface Methodology, 166
CONTENTS vii
5.13. PlottingEffects fromFractional Factorials onProbability
Paper, 167
Exercises, 168
6. The Use of SteepestAscent to Achieve Process Improvement 187
6.1. What Steepest Ascent Does, 188
6.2. AnExample. ImprovementofaProcessUsedinDrug
Manufacture, 189
6.3. General Comments on Steepest Ascent, 192
6.4. A Confidence Region for the Direction of Steepest Ascent,
194
6.5. Steepest Ascent Subject to a Constraint, 197
Appendix 6A. Effects of Changes in Scales on Steepest Ascent,
201
Appendix 6B. Placement ofFirst-Order Designs Within Spherical
and Cuboidal Regions of Interest, 204
Exercises, 207
7. Fitting Second-Order Models 211
7.1. Fitting and Checking Second-Degree Polynomial Graduating
Functions, 211
7.2. A Comprehensive Check of the Need for Transformation in
the Response Variable, 220
7.3. Interim Summary, 222
7.4. Further Examination of Adequacy of Fit; Higher-Order
Terms, 223
7.5. Robust Methods in Fitting Graduating Functions, 228
Appendix 7A. Fitting of First- and Second-Degree Models, 230
Appendix 7B. Checking for Nonadditivity with Tukey’sTest, 233
Appendix 7C. Third-Order Biases in Estimates for Second-Order
Model, Using the 33 Design, 234
Appendix7D. AnalysisofVariancetoDetectPossibleThird-Order
Terms, 236
Appendix7E. Orthogonalization ofEffects forThree-Level
Designs, 238
Appendix7F. OrthogonalEffects AssociatedwithIndividual
Degrees of Freedom and Related Normal
Plots, 241
Appendix 7G. Yates’Algorithm for 3k Factorial Designs, 241
Appendix 7H. Desirable Features and Placement ofSecond Order
DesignsforSpherical andCuboidalRegionsof
Interest, 244
Exercises, 248
viii CONTENTS
8. Adequacy of Estimation and the Use of Transformation 271
8.1. What Can Transformations Do?, 271
8.2. A Measure of Adequacy of Function Estimation, 277
8.3. ChoosingtheResponseMetrictoValidate Distribution
Assumptions, 281
8.4. Estimating response transformation from the data, 288
8.5. Transformation of the predictor variables, 292
Appendix 8A. Proof of a Result Used in Section 8.2, 298
Exercises, 299
9. Exploration of Maxima and Ridge Systems with Second-Order
Response Surfaces 303
9.1. The Need for Second-Order Surfaces, 303
9.2. Example: Representation ofPolymerElasticity as aQuadratic
Function of Three Operating Variables, 304
9.3. Examination of the Fitted Surface, 311
9.4. InvestigationofAdequacyofFit:IsolationofResidualDegrees
of Freedom for the Composite Design, 313
Exercises, 318
10. Occurrence and Elucidation of Ridge Systems, I 321
10.1. Introduction, 321
10.2. Factor Dependence, 324
10.3. Elucidation of Stationary Regions, Maxima, and Minima by
Canonical Analysis, 329
Appendix 10A. A Simple Explanation of Canonical Analysis, 337
11. Occurrence and Elucidation of Ridge Systems, II 341
11.1. Ridge Systems Approximated by Limiting Cases, 341
11.2. Canonical Analysis in the A Form to Characterize Ridge
Phenomena, 344
11.3. Example 11.3: A Consecutive Chemical System Yielding a
Near-Stationary Planar Ridge Maximum, 348
11.4. Example11.4: ASmallReactorStudyYieldingaRisingRidge
Surface, 353
11.5. Example 11.5: A Stationary Ridge in Five Variables, 359
11.6. The Economic Importance of the Dimensionality of Maxima
and Minima, 363
11.7. A Method for Obtaining a Desirable Combination of Several
Responses, 364
Appendix 11A. Calculations for the Analysis of Variance, Table
11.5, 366
Exercises, 366
CONTENTS ix
12. RidgeAnalysisforExamining Second-OrderFittedModels,
Unrestricted Case 391
12.1. Introduction, 391
12.2. Standard Ridge Analysis About the Origin xs0, 392
12.3. Worked Example of Ridge Analysis About the Origin
xs0, 394
12.4. Ridge Analysis Around a Selected Focus f, 398
12.5. Worked Example of Ridge Analysis About a Selected
Focus f, 399
12.6. Behavior of Ridge Paths in General, 400
12.7. A Ridge Exploration Solution Using Minitab, 400
Exercises, 404
13. Design Aspects of Variance, Bias, and Lack of Fit 413
13.1. The Use of Approximating Functions, 413
13.2. The Competing Effects of Bias and Variance, 415
13.3. Integrated Mean Squared Error, 418
13.4. Region of Interest and Region of Operability in k
Dimensions, 419
13.5. AnExtension ofthe Idea ofaRegionofInterest: The Weight
Function, 422
13.6. Designs That Minimize Squared Bias, 424
13.7. Ensuring Good Detectability of Lack of Fit, 429
13.8. Checks Based on the Possibility of Obtaining a Simple Model
by Transformation of the Predictors, 436
Appendix 13A. Derivation of Eq. Ž13.5.8., 446
Appendix13B. NecessaryandSufficient DesignConditionsto
Minimize Bias, 445
Appendix 13C. Minimum Bias Estimations, 448
Appendix 13D. A Shrinkage Estimation Procedure, 449
Appendix 13E. Conditions for Efficacy of Transformation on the
Predictor Variables, 450
Appendix 13F. The Third-Order Columns of the X Matrix for a
Composite Design, 455
Exercises, 457
14. Variance-Optimal Designs 461
14.1. Introduction, 461
14.2. Orthogonal Designs, 462
14.3. The Variance Function, 464
14.4. The Alphabetic Optimality Approach, 471
14.5. Alphabetic Optimality in the Response Surface Context, 476
x CONTENTS
15. Practical Choice of a Response Surface Design 483
15.1. Introduction, 483
15.2. First-Order Designs, 486
15.3. Second-Order Composite Designs, 488
15.4. Second-Order Designs Requiring Only Three Levels, 495
15.5. Screening Designs with Three Levels and Their Projections
into Fewer Factors, 499
15.6. Designs Requiring Only a Small Number of Runs, 500
Exercises, 504
16. Response Surfaces for Mixture Ingredients 509
16.1. Mixture Spaces, 509
16.2. First-Order Mixture Models, 513
16.3. Second-Order Mixture Models, 516
16.4. Third-Order Canonical Mixture Models, 520
16.5. Design ofExperiments forCanonical PolynomialModels, 521
16.6. Simplex Centroid Designs and the Related Polynomials, 525
Appendix 16A. Examples of Second-Order Response Contours for
qs3 Mixture Ingredients, 526
Appendix 16B. Examples of Special Cubic Response Contours for
qs3 Mixture Ingredients, 529
Appendix 16C. Examples of a General Cubic Response Surface
and a Quartic Response Surface for qs3 Mixture
Ingredients, 529
Exercises, 531
17. Mixture Experiments in Restricted Regions 535
17.1. Introduction, 535
17.2. Lower Bounds and Pseudocomponents, 537
17.3. Upper Bounds and Pseudocomponents, 542
17.4. Upper and Lower Bounds, 544
Exercises, 550
18. Other Mixture Methods and Topics 553
18.1. Using Ratios of Ingredients as Predictor Variables, 553
18.2. The Kissell and Marshall Example, 555
18.3. Mixture Designs and Models When the Total Amount is
Important, 561
18.4. DesignsforComponent-AmountModelsObtained by
Projection of Standard Simplex Lattice and Simplex-Centroid
Mixture Designs, 570
18.5. DesignsinOrthogonalBlocksforComponent-Amount
Models, 578