The present book initiates the concept of robust response surface designs, along with the relevant regression and positive data analysis techniques. Response surface methodology(RSM), well-known in literature,is widely used in every field of science and technology such as biology, natural(physical/chemical),environmental,medical, agricultural, quality engineering,etc. RSM is the most popular experimental data generating, modeling, and optimization technique in every field of science. It is a particular case of robust response surface methodology(RRSM).RSM has many limitations, and RRSM aims to overcome many of such limitations. Thus, RRSM will be much better than RSM. It is intended for anyone who knows basic concepts of experimental designs and regression analysis.This is the first unique book on RRSM. Every chapter is unique regarding its contents, presentation,and organization.Problems on robust response surface designs such as rotatability, sloperotatability, weak rotatability, optimality, and along with the method of esti mation of model parameters, positive data analysis techniques are considered in this book. Some real examples on lifetime responses, resistivity, replicated measures, medical,demog- raphy, hydrogeology data,etc.,are analyzed. Some examples(considered in this book)on design of experiments do not satisfy the classical assumptions of response surface methodology.This book is intended as an introductory book on robust response surface designs.The related topics of RRSM,such as regression analysis for correlated errors and positive data analysis, have been included in this book. It is aimed primarily at theoretical and practical statisticians. The range of topics and applications gives this book broad appeal both to theoreticians and practicing professionals in a variety of fields of science. This book will be for a second course in design of experiments along with the regression and data analysis. It is intended to allow a broad group of quality engineers,scientists(in any field), medical practitioners, demographers,economists and statisticians to learn about RRSM, regression and positive data analysis, including both the theory and how to apply it.The main emphasis of this book is on examining the concept of rotatability, weak rotatability, D-optimality, sloperotatability, weak slope-rotatability,D-optimalslope rotatability, regression analysis with correlated errors,and positive data analysis.All the contents of this book are taken from the research articles of the author and his collaborators.Thus,the book is unique in every sense. The contents included in this book are described below chapter wise.The problems included in this book are described in Chapter 1. Short literature reviews related to these problems is presented herein. A detailed summary of each chapter is given. Concepts of robust response surface designs are also clearly explained in this chapter.Chapter 2 starts with correlated first-order regression designs. The concept of robust first-order rotatable and D-optimal designs are described.The rotatability and D-optimality conditions of first-order regression designs are described for correlated errors.These condi tions are further simplified for different error variancecovariance structures(i.e.,intraclass, inter-class,compound symmetry, tridiagonal,and autocorrelated), which are commonly encountered in practical situations. Some construction methods of robust rotatable and D. optimal designs are described for each error structure.Chapter 3 describes second-order regression designs for correlated errors.Robust secondorder rotatability conditions are examined for a general error variancecovariance structure. Secondorder rotatability conditions are simplified for intraclass,inter-class,compound symmetry, tridiagonal, and autocorrelated error structures. Robust second-order rotatable designs (RSORDs) are constructed for each of the above error structures.Chapter 4 includes location-scale lifetime models which are derived based on constant variance assumption from a lifetime distribution,such as log-normal or exponential. Assuming a first-or secondorder location-scale model with correlated errors, first or secondorder rotatability and D-optimality conditions are derived for a general error varianccovariance structure.It is observed that the designs are invariant of these two non-normal distributions, but they depend on the error variancecovariance structure. Similar results as in Chapters 2 and 3 are presented for both the non-normal distributions,and for some error structure as mentioned in Chapter 2.

List of Figures xiii

List of Tables xv

Preface xix

Author xxiii

1 INTRODUCTION 1

1.1THE PROBLEM AND PERSPECTIVE 1

1.2A BRIEF REVIEW OF THE LITERATURE 3

1.3 EXISTING LITERATURE IN THE DIRECTION OF PRESENT RESEARCH MONOGRAPH 6

1.4 ROBUST REGRESSION DESIGNS 7

1.5SUMMARY OF THE RESEARCH MONOGRAPH 9

1.6CONCLUDING REMARKS 13

2 ROBUST FIRST-ORDER DESIGNS 15

2.1 INTRODUCTION AND OVERVIEW 16

2.2 FIRST-ORDER CORRELATED MODEL 16

      2.2.1 Model 16

      2.2.2 Analysis and rotatability 17

      2.2.3 Robust rotatable and optimum designs 18

2.3ROBUST FIRST-ORDER DESIGNS FOR INTRA-CLASS STRUCTURE 20

      2.3.1 Comparison between RFORD and FORD 21

2.4ROBUST FIRST-ORDER DESIGNS FOR INTER-CLASS STRUCTURE 21

      2.4.1 Optimum robust first-order designs under inter-class structure 23

      2.4.2 RFORD and D-ORFOD under inter-class structure 24

      2.5 ROBUST FIRST-ORDER DESIGNS FOR GENERALIZED INTERCLASS STRUCTURE 26

      2.6 ROBUST FIRST-ORDER DESIGNS FOR COMPOUND SYMMETRY STRUCTURE 27

      2.6.1 Optimum robust first-order designs under compound symmetry structure 28

      2.6.2 RFORD and D-ORFOD under compound symmetry structure 28

      2.7 ROBUST FIRST-ORDER DESIGNS FOR TRI-DIAGONAL STRUCTURE 30

      2.7.1 Optimum robust first-order designs under tri-diagonal structure 31

      2.7.2 RFORD and D-ORFOD under tri-diagonal structure 31

2.8ROBUST FIRST-ORDER DESIGNS FOR AUTOCORRELA STRUCTURE 33

      2.8.1 Optimum robust designs under autocorrelated structure 34

      2.8.2 RFORD and nearly D-ORFOD under autocorrelated structure 35

2.9CONCLUDING REMARKS 40

3 ROBUST SECOND-ORDER DESIGNS 45

3.1 INTRODUCTION AND OVERVIEW 46

3.2 SECOND-ORDER CORRELATED 46

      3.2.1 Model 46

      3.2.2 Analysis 46

3.3 ROBUST SECOND-ORDER ROTATABILITY 48

      3.3.1 Robust second-order rotatability conditions 48

      3.3.2 Robust second-order rotatable non-singularity condition 50

      3.3.3 Robust second-order rotatable and optimum designs 51

3.4ROBUST SECOND-ORDER DESIGNS FOR. INTRA-CLASS STRUCTURE 52

      3.4.1 Second-order rotatability conditions under intra-class structure 52

      3.4.2 Non-singularity condition under intra-class structure 53

      3.4.3 Estimated response variance under intra-class structure 53

      3.4.4 Optimum RSORD under intra-class structure 54

3.5ROBUST SECOND-ORDER DESIGNS FOR INTER-CLASS STRUCTURE 54

      3.5.1 Second-order rotatability conditions under inter-class structure 55

      3.5.2 RSORD construction methods under inter-class structure 57

3.6ROBUST SECOND-ORDER DESIGNS FOR COMPOUND SYMMETRY STRUCTURE 63

      3.6.1 Second-order rotatability conditions under compound symmetry Structure 63

      3.6.2 RSORD construction for compound symmetry structure 66

3.7ROBUST SECOND-ORDER DESIGNS FOR TRI-DIAGONAL STRUCTURE 66

      3.7.1 Second-order rotatability conditions under tri-diagonal structure 66

      3.7.2 RSORD construction for tri-diagonal structure 67

3.8ROBUST SECOND-ORDER DESIGNS FOR AUTOCORRELATED STRUCTURE 71

      3.8.1 Second-order rotatability conditions for autocorrelated structure 71

      3.8.2 RSORD construction for autocorrelated structure 73

3.9 CONCLUDING REMARKS 75

4 ROBUST REGRESSION DESIGNS FOR NON-NORMAL DISTRIBUTIONS 77

4.1INTRODUCTION AND OVERVIEW 77

4.2CORRELATED ERROR MODELS FOR NON-NORMAL DISTRIBUTIONS

      4.2.1 Correlated error models for log-normal distribution 79

      4.2.2 Correlated error models for exponential distribution 80

4.3 ROBUST FIRST-ORDER DESIGNS FOR LOG-NORMAL AND EXPONENTIAL DISTRIBUTIONS 81

4.4 ROBUST FIRST-ORDER DESIGNS FOR TWO NON-NORMAL DISTRIBUTIONS 83

      4.4.1 Compound symmetry correlation structure 83

      4.4.2 Inter-class correlation structure 85

      4.4.3 Intra-class correlation structure 85

      4.4.4 Tri-diagonal correlation structure 86

4.5ROBUST SECOND-ORDER DESIGNS FOR TWO NON-NORMAL DISTRIBUTIONS 87

4.6 CONCLUDING REMARKS 88

5 WEAKLY ROBUST ROTATABLE DESIGNS 89

5.1 INTRODUCTION AND OVERVIEW 89

5.2 DEVIATION FROM ROTATABILITY 90

5.3 WEAKLY ROBUST FIRST-ORDER ROTATABLE DESIGNS 91

      5.3.1 Robust first-order rotatability measure based on dispersion matrix 91

      5.3.2 Robust first-order rotatability measure based on moment matrix 92

      5.3.3 Comparison between robust first-order rotatable and weakly robust rotatable designs 94

5.4WEAKLY ROBUST SECOND-ORDER ROTATABLE DESIGNS 96

      5.4.1 Robust second-order rotatability measure based on moment matrix 97

      5.4.2 Robust second-order rotatability measure based on polar transformation 99

      5.4.3Comparison between robust second-order rotatable and weakly robust rotatable designs 101

      5.4.4 Illustrations 102

      5.4.5Comparison between two robust second-order rotatability measures Qk(d) and Pk(d) 105

5.5 CONCLUDING REMARKS 106

6 ROBUST SECOND-ORDER SLOPE ROTATABILITY 109

6.1INTRODUCTION AND OVERVIEW 109

6.2 SECOND-ORDER SLOPE-ROTATABILITY WITH UNCORRELATEI ERRORS 111

      6.2.1 Second-order slope-rotatability conditions for uncorrelated errors 111

      6.2.2 Modified second-order slope-rotatability conditions for uncorrelated errors 111

6.3ROBUST SECOND-ORDER SLOPE ROTATABILITY CONDITIONS 112

6.4MODIFIED SECOND-ORDER SLOPE ROTATABLE DESIGN WITH CORRELATED ERRORS 115

6.5ROBUST SECOND-ORDER SLOPE ROTATABLE AND MODIFI SLOPE ROTATABLE DESIGNS UNDER INTRA-CLASS STRUCTURE 117

6.6 ILLUSTRATIONS 118

6.7CONCLUDING REMARKS 124

7 OPTIMAL ROBUST SECOND-ORDER SLOPE ROTATABLE DESIGNS 125

7.1 INTRODUCTION AND SUMMARY 125

7.2 ESTIMATION OF DERIVATIVES 127

7.3ROBUST SECOND-ORDER SLOPE-ROTATABILITY OVER ALL DIRECTIONS 128

7.4ROBUST SECOND-ORDER SYMMETRIC BALANCED DESIGN 131

7.5ROBUST SLOPE-ROTATABILITY WITH EQUAL MAXIMUM DIRECTIONAL VARIANCE 132

7.6D-OPTIMAL ROBUST SECOND-ORDER SLOPE-ROTATI DESIGNS 136

7.7 ROBUST SLOPE ROTATABLE DESIGNS OVER ALL DIRECTIONS,WITH EQUAL MAXIMUM DIRECTIONAL VARIANCE AND D-OPTIMAL SLOPE 137

7.8 CONCLUDING REMARKS 141

8 ROBUST SECOND-ORDER SLOPE-ROTATABILITY MEASURES 143

8.1 INTRODUCTION AND OVERVIEW 143

8.2 ROBUST SECOND-ORDER SLOPE-ROTATABILITY MEASURES 144

      8.2.1 Robust second-order slope-rotatability measure along axial directions 144

      8.2.2 Robust second-order slope-rotatability measure over all directions 148

      8.2.3Robust second-order slope-rotatability measure with equal maximum directional variance 149

8.3 ILLUSTRATIONS OF ROBUST SLOPE-ROTATABILITY MEASURES 149

8.4 CONCLUDING REMARKS 153

9 REGRESSION ANALYSES WITH CORRELATED ERRORS AND APPLICATIONS 155

9.1 INTRODUCTION AND SUMMARY 156

9.2REGRESSION ANALYSES WITH COMPOUND SYMMETRY ERROR STRUCTURE 158

      9.2.1 Correlated error regression models 158

      9.2.2 Regression parameter estimation with CSES 159

      9.2.3 Hypotheses testing of regression parameters with CSES 161

      9.2.4Confidence ellipsoid of regression parameters with CSES 162

      9.2.5Index of fit with CSES 163

      9.2.6 Illustration of regression analysis with CSES 163

      9.2.7Randomized block design with CSES 166

      9.2.7.1 Background of an RBD with CSES 166

      9.2.7.2 Randomized block design model with CSES 167

      9.2.7.3 Analysis of an RBD with CSES 168

      9.2.7.4Confidence ellipsoid of treatment contrasts with CSES 170

      9.2.7.5 Multiple comparison of treatment contrasts with CSES 171

      9.2.7.6 llustration of an RBD with CSES 172

9.3REGRESSION ANALYSES WITH COMPOUND AUTOCORRELATED ERROR STRUCTURE 174

      9.3.1 Estimation of regression parameters with CAES 174

      9.3.2 Hypothesis testing of regression parameters with CAES 177

      9.3.3 Confidence ellipsoid of regression parameters with CAES 178

      9.3.4 Index of fit with CAES 179

      9.3.5 llustration of regression analysis with CAES 179

      9.3.6 Randomized block design with CAES 183

      9.3.6.1A reinforced RBD with CAES 182

      9.3.6.2An RBD model with CAES 184

      9.3.6.3A reinforced RBD analysis with CAES 185

      9.3.6.4Confidence ellipsoid of treatment contrasts with CAES 189

      9.3.6.5Multiple comparison of treatment contrasts with CAES 190

      9.3.6.6 Illustration of a reinforced RBD with CAES 191

9.4 CONCLUDING REMARKS 193

10 POSITIVE DATA ANALYSES VIA LOG-NORMAL AND GAMMA MODELS 195

10.1 INTRODUCTION AND OVERVIEW 196

10.2 DISCREPANCY IN REGRESSION ESTIMATES BETWEEN LOGNORMAL AND GAMMA MODELS FOR CONSTANT VARIANCE 198

      10.2.1 Log-normal and gamma models for constant variance 198

      10.2.2 Log-normal and gamma models for non-constant variance 199

      10.2.3 Motivating example 200

      10.2.4 Examples of different regression estimates 202

      10.2.5 Discussion about discrepancy 207

10.3DISCREPANCY OF REGRESSION PARAMETERS BETWEEN LOGNORMAL AND GAMMA MODELS FOR NON-CONSTANT VARIANCE 210

10.4DISCREPANCY IN FITTING BETWEENLOG-NORMAL AND GAMMA MODELS 217

10.5REPLICATED RESPONSES ANALYSIS IN QUALITY ENGINEERING 224

      10.5.1 Background of replicated response analyses 224

      10.5.2 Taguchi approach and dual-response approach with its extension in GLMs 225

      10.5.3 Illustrations with two real examples 227

10.6 RESISTIVITY OF UREA FORMALDEHYDE RESIN IMPROVEMENT 232

      10.6.1 Resin experiment background 233

      10.6.2 Urea formaldehyde resin experiment data 234

      10.6.2.1 Resistivity analysis of urea formaldehyde resin data 234

      10.6.2.2 Non-volatile solid analysis of urea formaldehyde resin data 235

      10.6.2.3 Viscosity analysis of urea formaldehyde resin data 236

      10.6.2.4 Acid value analysis of urea formaldehyde resin data 236

      10.6.2.5 Petroleum ether tolerance value analysis of urea formaldehyde resin data 237

10.7 DETERMINANTS OF INDIAN INFANT AND CHILD MORTALITY.

      10.7.1 Infant and child mortality background 240

      10.7.2 Infant survival time data,analysis, and interpretation 241

      10.7.3 Child survival time data, analysis, and interpretation 247

10.8AN APPLICATION OF GAMMA MODELS IN HYDROLOGY 249

      10.8.1 Background of drinking groundwater quality characteristics 249

      10.8.2 Description,analyses,and interpretation of groundwater data 251

      10.8.2.1 Analysis of chemical oxygen demand(COD) 25

      10.8.2.2 Analysis of total alkalinity (TAK) 252

      10.8.2.3 Analysis of total hardness (THD) 254

      10.8.2.4 Analysis of dissolved oxygen (DO) 355

      10.8.2.5 Analysis of electrical conductivity(EC) 256

      10.8.2.6 Analysis of chloride content (CLD) 257

10.9 AN APPLICATION OF LOG-NORMAL AND GAMMA MODELS IN MEDICAL SCIENCE 259

      10.9.1 Background of human blood biochemical parameters 260

      10.9.2 Description, analysis,and interpretations of human blood biochemical parameters 261

      10.9.2.1 Analysis of fasting serum insulin (FI) 261

      10.9.2.2 Analysis of total cholesterol (TC) 261

      10.9.2.3 Analysis of serum triglycerides (STG) 263

      10.9.2.4 Analysis of low-density lipoprotein (LDL) 263

      10.9.2.5 Analysis of high-density lipoprotein (HDL) 266

      10.9.2.6 Analysis of fasting plasma glucose level(PGL) 267

10.10 CONCLUDING REMARKS 269

11 GENERAL CONCLUSIONS AND DISCUSSIONS 271

APPENDIX 277

Bibliography 289

Index 307

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