Equivalence and Noninferiority Tests for Quality, Manufacturing and Test Engineers

v

Contents

Preface .................................................................................................................... vii

The Author ..............................................................................................................ix

Acknowledgments .................................................................................................xi

Introduction to Hypothesis Testing ................................................................. xiii

1. Proportions and Binomial Random Variables .........................................1

Test 1.1 Single Binomial Proportion (One-Sided; Probability

of“Success”) ...................................................................................1

Test 1.2 Single Binomial Proportion (Two-Sided) ...................................4

Test 1.3 Difference of Two Proportions (Two-Sided) ..............................7

2. Means .............................................................................................................. 15

Test 2.1 Single Mean (One-Sided) ........................................................... 15

Test 2.2 Comparison of Two Means—Two Independent

Samples, Fixed Δ Paradigm (One-Sided) .................................20

Test 2.3 Comparison of Two Means—Proportional Difference

Paradigm for Two Independent Samples (One-Sided) ..........27

Test 2.4 Single Mean (Two-Sided) ........................................................... 32

Test 2.5 A Special Case of Single Mean

(Two-Sided)—Regression Slope ................................................ 36

Test 2.6.1 Comparison of Two Means—Two Independent Samples

(Two-Sided) ..................................................................................42

Test 2.6.2 A Special Case of Comparison of Two

Means—Two Independent Samples (Two-Sided),

Two-Level Factorial Validation/Verication Experiment .....45

Test 2.7 Comparison of Two Means—Proportional Difference

Paradigm for Two Independent Samples (Two-Sided) .......... 53

3. Variances (Standard Deviations) and Coefcients of Variation .........57

Test 3.1 Single Variance (One-Sided) ...................................................... 57

Test 3.2 Comparison of Two Variances (One-Sided) ............................ 61

Test 3.3 Single Coefcient of Variation (One-Sided) .............................66

4. Exponential Rate Parameters .....................................................................71

Test 4.1 Single Exponential Rate Parameter (One-Sided) .................... 71

Test 4.2 Two Exponential Rate Parameters (One-Sided) ...................... 76

5. Capability Indices ........................................................................................79

Test 5.1 C

pk

...................................................................................................79

Test 5.2 C

p

....................................................................................................86

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6. Multivariate ...................................................................................................91

Test 6.1 Multivariate—Single Mean Vector (Two-Sided) .....................91

7. Reliability .......................................................................................................97

Test 7.1 Comparison of Reliability Using Parametric Models ............97

Test 7.2 Nonparametric Test for Comparison of Reliability .............. 106

Test 7.3 Accelerated Life Test with Type I Censoring ........................ 109

8. Sample-Based Criteria ............................................................................... 115

Test 8.1 Single Proportion ......................................................................115

Test 8.2 Single Mean ................................................................................ 116

Test 8.3 Relative Difference between Two Means ............................... 117

Test 8.4 Single Standard Deviation ....................................................... 121

9. Comparing Sequences of Points ..............................................................123

Test 9.1 Two Sequences, Variability Known ........................................123

Test 9.2 Two Sequences, Variability Unknown ................................... 127

10. Calculating Sample Sizes..........................................................................135

11. Computer Code—Power Curves .............................................................. 139

Test 1.1 Single Binomial Proportion (One-Sided—Probability

of“Success”) ..............................................................................139

Test 2.1 Single Mean (One-Sided) .........................................................141

Test 2.2 Comparison of Two Means—Two Independent

Samples, Fixed Δ Paradigm (One-Sided) ...............................144

Test 2.3 Comparison of Two Means—Proportional Difference

Paradigm for Two Independent Samples (One-Sided)........148

Test 3.1 Single Variance (One-Sided) .................................................... 151

Test 3.2 Comparison of Two Variances (One-Sided) ..........................154

Test 3.3 Single Coefcient of Variation (One-Sided) ........................... 157

Test 4.1 Single Exponential Rate Parameter (One-Sided) .................. 160

Test 4.2 Two Exponential Rate Parameters (One-Sided) .................... 164

Test 5.1 C

pk

................................................................................................. 168

Test 5.2 C

p

.................................................................................................. 171

Test 6.1 Multivariate—Single Mean Vector ......................................... 174

Test 9.1 Two Sequences, Variability Known ........................................ 176

Test 9.2 Two Sequences, Variability Unknown ................................... 180

References ........................................................................................................... 185

Appendix I: Review of SomeProbability and Statistics Concepts ......... 187

Appendix II ......................................................................................................... 193