Response Surfaces, Mixtures, and Ridge Analyses

874 Pages·2007·17.4 MB·English

by  George E. P. Box

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

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
Copyright(cid:2)2007JohnWiley&Sons,Inc. Allrightsreserved.
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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:2)(cid:3)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
Ž .
1.1. Response Surface Methodology RSM , 1
1.2. Indeterminancy of Experimentation, 3
1.3. Iterative Nature of the Experimental Learning Process, 6
Ž .
1.4. Some Classes of Problems Which, 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:4)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
Ž
Appendix4C. ConfidenceRegionsforContourPlanes seeSection
.
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
Ž
5.9. Screening, Using Plackett and Burman Designs Involving
.
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