Robust Response Surfaces, Regression, and Positive Data Analyses

333 Pages·2014·9.91 MB·English

by  Rabindra Nath Das

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Robust Response Surfaces,
Regression, and
Positive Data Analyses

Robust Response Surfaces,
Regression, and
Positive Data Analyses
Rabindra Nath Das
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2014 by Taylor & Francis Group, LLC
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Version Date: 20140502
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Dedicated to
my beloved mother
and to the memory of my father

Contents
List of Figures xiii
List of Tables xv
Preface xix
Author xxiii
1 INTRODUCTION 1
1.1 THE PROBLEM AND PERSPECTIVE . . . . . . . . . . . . . . . . . . . 1
1.2 A BRIEF REVIEW OF THE LITERATURE . . . . . . . . . . . . . . . . 3
1.3 EXISTING LITERATURE IN THE DIRECTION OF PRESENT
RESEARCH MONOGRAPH . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 ROBUST REGRESSION DESIGNS . . . . . . . . . . . . . . . . . . . . . 7
1.5 SUMMARY OF THE RESEARCH MONOGRAPH . . . . . . . . . . . . 9
1.6 CONCLUDING 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
2.3 ROBUST FIRST-ORDER DESIGNS FOR INTRA-CLASS STRUCTURE 20
2.3.1 Comparison between RFORD and FORD . . . . . . . . . . . . . . 21
2.4 ROBUST 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 INTER-
CLASS 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. . . 29
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
vii
viii Contents
2.8 ROBUST FIRST-ORDER DESIGNS FOR AUTOCORRELATED
STRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.8.1 Optimum robust designs under autocorrelated structure . . . . . . 34
2.8.2 RFORD and nearly D-ORFOD under autocorrelated structure . . 35
2.9 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 ROBUST SECOND-ORDER DESIGNS 45
3.1 INTRODUCTION AND OVERVIEW . . . . . . . . . . . . . . . . . . . . 46
3.2 SECOND-ORDER CORRELATED MODEL . . . . . . . . . . . . . . . . 46
3.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
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
3.4 ROBUST 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
3.5 ROBUST 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
3.6 ROBUST SECOND-ORDERDESIGNS FOR COMPOUNDSYMMETRY
STRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6.1 Second-order rotatability conditions under compound symmetry
structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6.2 RSORD construction for compound symmetry structure . . . . . . 66
3.7 ROBUST 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-diagonalstructure . . . . . . . . . . . 67
3.8 ROBUST SECOND-ORDER DESIGNS FOR AUTOCORRELATED
STRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.8.1 Second-order rotatability conditions for autocorrelatedstructure . . 71
3.8.2 RSORD construction for autocorrelatedstructure . . . . . . . . . . 73
3.9 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 ROBUST REGRESSION DESIGNS FOR NON-NORMAL
DISTRIBUTIONS 77
4.1 INTRODUCTION AND OVERVIEW . . . . . . . . . . . . . . . . . . . . 77
4.2 CORRELATED ERROR MODELS FOR NON-NORMAL
DISTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.1 Correlated error models for log-normaldistribution . . . . . . . . . 79
4.2.2 Correlated error models for exponential distribution. . . . . . . . . 80
4.3 ROBUST FIRST-ORDER DESIGNS FOR LOG-NORMAL AND
EXPONENTIAL DISTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . 81