Description
Book SynopsisDetails methods for computing valid limits of detection.
Table of ContentsPreface xv
Acknowledgment xix
About the Companion Website xx
1 Background 1
1.1 Introduction 1
1.2 A Short List of Detection Limit References 2
1.3 An Extremely Brief History of Limits of Detection 2
1.4 An Obstruction 3
1.5 An Even Bigger Obstruction 3
1.6 What Went Wrong? 4
1.7 Chapter Highlights 5
References 5
2 Chemical Measurement Systems and their Errors 9
2.1 Introduction 9
2.2 Chemical Measurement Systems 9
2.3 The Ideal CMS 10
2.4 CMS Output Distributions 12
2.5 Response Function Possibilities 12
2.6 Nonideal CMSs 15
2.7 Systematic Error Types 15
2.7.1 What Is Fundamental Systematic Error? 16
2.7.2 Why Is an Ideal Measurement System Physically Impossible? 16
2.8 Real CMSs, Part 1 17
2.8.1 A Simple Example 18
2.9 Random Error 19
2.10 Real CMSs, Part 2 21
2.11 Measurements and PDFs 22
2.11.1 Several Examples of Compound Measurements 22
2.12 Statistics to the Rescue 23
2.13 Chapter Highlights 24
References 24
3 The Response, Net Response, and Content Domains 25
3.1 Introduction 25
3.2 What is the Blank’s Response Domain Location? 27
3.3 False Positives and False Negatives 28
3.4 Net Response Domain 29
3.5 Blank Subtraction 29
3.6 Why Bother with Net Responses? 31
3.7 Content Domain and Two Fallacies 31
3.8 Can an Absolute Standard Truly Exist? 33
3.9 Chapter Highlights 34
References 34
4 Traditional Limits of Detection 37
4.1 Introduction 37
4.2 The Decision Level 37
4.3 False Positives Again 38
4.4 Do False Negatives Really Matter? 40
4.5 False Negatives Again 40
4.6 Decision Level Determination Without a Calibration Curve 41
4.7 Net Response Domain Again 41
4.8 An Oversimplified Derivation of the Traditional Detection Limit, XDC 42
4.9 Oversimplifications Cause Problems 43
4.10 Chapter Highlights 43
References 43
5 Modern Limits of Detection 45
5.1 Introduction 45
5.2 Currie Detection Limits 46
5.3 Why were p and q Each Arbitrarily Defined as 0.05? 48
5.4 Detection Limit Determination Without Calibration Curves 49
5.5 A Nonparametric Detection Limit Bracketing Experiment 49
5.6 Is There a Parametric Improvement? 51
5.7 Critical Nexus 52
5.8 Chapter Highlights 53
References 53
6 Receiver Operating Characteristics 55
6.1 Introduction 55
6.2 ROC Basics 55
6.3 Constructing ROCs 57
6.4 ROCs for Figs 5.3 and 5.4 59
6.5 A Few Experimental ROC Results 60
6.6 Since ROCs may Work Well, Why Bother with Anything Else? 64
6.7 Chapter Highlights 65
References 65
7 Statistics of an Ideal Model CMS 67
7.1 Introduction 67
7.2 The Ideal CMS 67
7.3 Currie Decision Levels in all Three Domains 70
7.4 Currie Detection Limits in all Three Domains 71
7.5 Graphical Illustrations of eqns 7.3–7.8 72
7.6 An Example: are Negative Content Domain Values Legitimate? 74
7.7 Tabular Summary of the Equations 76
7.8 Monte Carlo Computer Simulations 77
7.9 Simulation Corroboration of the Equations in Table 7.2 78
7.10 Central Confidence Intervals for Predicted x Values 80
7.11 Chapter Highlights 81
References 81
8 If Only the True Intercept is Unknown 83
8.1 Introduction 83
8.2 Assumptions 83
8.3 Noise Effect of Estimating the True Intercept 83
8.4 A Simple Simulation in the Response and NET Response Domains 84
8.5 Response Domain Effects of Replacing the True Intercept by an Estimate 86
8.6 Response Domain Currie Decision Level and Detection Limit 88
8.7 NET Response Domain Currie Decision Level and Detection Limit 88
8.8 Content Domain Currie Decision Level and Detection Limit 89
8.9 Graphical Illustrations of the Decision Level and Detection Limit Equations 89
8.10 Tabular Summary of the Equations 90
8.11 Simulation Corroboration of the Equations in Table 8.1 91
8.12 Chapter Highlights 93
9 If Only the True Slope is Unknown 95
9.1 Introduction 95
9.2 Possible “Divide by Zero” Hazard 96
9.3 The t Test for tslope 96
9.4 Response Domain Currie Decision Level and Detection Limit 97
9.5 NET Response Domain Currie Decision Level and Detection Limit 97
9.6 Content Domain Currie Decision Level and Detection Limit 97
9.7 Graphical Illustrations of the Decision Level and Detection Limit Equations 98
9.8 Tabular Summary of the Equations 99
9.9 Simulation Corroboration of the Equations in Table 9.1 99
9.10 Chapter Highlights 101
References 101
10 If the True Intercept and True Slope are Both Unknown 103
10.1 Introduction 103
10.2 Important Definitions, Distributions, and Relationships 104
10.3 The Noncentral t Distribution Briefly Appears 105
10.4 What Purpose Would be Served by Knowing 𝛿? 106
10.5 Is There a Viable Way of Estimating 𝛿? 106
10.6 Response Domain Currie Decision Level and Detection Limit 107
10.7 NET Response Domain Currie Decision Level and Detection Limit 107
10.8 Content Domain Currie Decision Level and Detection Limit 108
10.9 Graphical Illustrations of the Decision Level and Detection Limit Equations 108
10.10 Tabular Summary of the Equations 109
10.11 Simulation Corroboration of the Equations in Table 10.3 109
10.12 Chapter Highlights 109
References 111
11 If Only the Population Standard Deviation is Unknown 113
11.1 Introduction 113
11.2 Assuming 𝜎0 is Unknown, How may it be Estimated? 114
11.3 What Happens if 𝜎0 is Estimated by s0? 114
11.4 A Useful Substitution Principle 116
11.5 Response Domain Currie Decision Level and Detection Limit 116
11.6 NET Response Domain Currie Decision Level and Detection Limit 117
11.7 Content Domain Currie Decision Level and Detection Limit 117
11.8 Major Important Differences From Chapter 7 117
11.9 Testing for False Positives and False Negatives 120
11.10 Correction of a Slightly Misleading Figure 121
11.11 An Informative Screencast 121
11.12 Central Confidence Intervals for 𝜎 and s 122
11.13 Central Confidence Intervals for YC and YD 122
11.14 Central Confidence Intervals for XC and XD 123
11.15 Tabular Summary of the Equations 123
11.16 Simulation Corroboration of the Equations in Table 11.1 123
11.17 Chapter Highlights 125
References 125
12 If Only the True Slope is Known 127
12.1 Introduction 127
12.2 Response Domain Currie Decision Level and Detection Limit 127
12.3 NET Response Domain Currie Decision Level and Detection Limit 128
12.4 Content Domain Currie Decision Level and Detection Limit 128
12.5 Graphical Illustrations of the Decision Level and Detection Limit Equations 128
12.6 Tabular Summary of the Equations 128
12.7 Simulation Corroboration of the Equations in Table 12.1 129
12.8 Chapter Highlights 129
13 If Only the True Intercept is Known 131
13.1 Introduction 131
13.2 Response Domain Currie Decision Level and Detection Limit 132
13.3 NET Response Domain Currie Decision Level and Detection Limit 132
13.4 Content Domain Currie Decision Level and Detection Limit 132
13.5 Tabular Summary of the Equations 133
13.6 Simulation Corroboration of the Equations in Table 13.1 133
13.7 Chapter Highlights 135
References 135
14 If all Three Parameters are Unknown 137
14.1 Introduction 137
14.2 Response Domain Currie Decision Level and Detection Limit 137
14.3 NET Response Domain Currie Decision Level and Detection Limit 138
14.4 Content Domain Currie Decision Level and Detection Limit 138
14.5 The Noncentral t Distribution Reappears for Good 138
14.6 An Informative Computer Simulation 139
14.7 Confidence Interval for xD, with a Major Proviso 142
14.8 Central Confidence Intervals for Predicted x Values 143
14.9 Tabular Summary of the Equations 143
14.10 Simulation Corroboration of the Equations in Table 14.1 143
14.11 An Example: DIN 32645 145
14.12 Chapter Highlights 146
References 147
15 Bootstrapped Detection Limits in a Real CMS 149
15.1 Introduction 150
15.2 Theoretical 151
15.2.1 Background 151
15.2.2 Blank Subtraction Possibilities 151
15.2.3 Currie Decision Levels and Detection Limits 152
15.3 Experimental 153
15.3.1 Experimental Apparatus 153
15.3.2 Experiment Protocol 153
15.3.3 Testing the Noise: Is It AGWN? 156
15.3.4 Bootstrapping Protocol in the Experiments 157
15.3.5 Estimation of the Experimental Noncentrality Parameter 160
15.3.6 Computer Simulation Protocol 160
15.4 Results and Discussion 161
15.4.1 Results for Four Standards 161
15.4.2 Results for 3–12 Standards 162
15.4.3 Toward Accurate Estimates of XD 163
15.4.4 How the XD Estimates Were Obtained 164
15.4.5 Ramifications 165
15.5 Conclusion 165
Acknowledgments 166
References 166
15.6 Postscript 167
15.7 Chapter Highlights 167
16 Four Relevant Considerations 169
16.1 Introduction 169
16.2 Theoretical Assumptions 170
16.3 Best Estimation of 𝛿 171
16.4 Possible Reduction in the Number of Expressions? 172
16.5 Lowering Detection Limits 174
16.6 Chapter Highlights 178
References 178
17 Neyman–Pearson Hypothesis Testing 181
17.1 Introduction 181
17.2 Simulation Model for Neyman–Pearson Hypothesis Testing 181
17.3 Hypotheses and Hypothesis Testing 183
17.3.1 Hypotheses Pertaining to False Positives 183
17.3.1.1 Hypothesis 1 183
17.3.1.2 Hypothesis 2 183
17.3.2 Hypotheses Pertaining to False Negatives 185
17.3.2.1 Hypothesis 3 185
17.3.2.2 Hypothesis 4 185
17.4 The Clayton, Hines, and Elkins Method (1987–2008) 189
17.5 No Valid Extension for Heteroscedastic Systems 191
17.6 Hypothesis Testing for the 𝛿critical Method 192
17.6.1 Hypothesis Pertaining to False Positives 192
17.6.1.1 Hypothesis 5 192
17.6.2 Hypothesis Pertaining to False Negatives 192
17.6.2.1 Hypothesis 6 192
17.7 Monte Carlo Tests of the Hypotheses 192
17.8 The Other Propagation of Error 193
17.9 Chapter Highlights 197
References 197
18 Heteroscedastic Noises 199
18.1 Introduction 199
18.2 The Two Simplest Heteroscedastic NPMs 199
18.2.1 Linear NPM 201
18.2.2 Experimental Corroboration of the Linear NPM 202
18.2.3 Hazards with Heteroscedastic NPMs 203
18.2.4 Example: A CMS with Linear NPM 204
18.3 Hazards with ad hoc Procedures 206
18.4 The HS (“Hockey Stick”) NPM 207
18.5 Closed-Form Solutions for Four Heteroscedastic NPMs 209
18.6 Shot Noise (Gaussian Approximation) NPM 210
18.7 Root Quadratic NPM 211
18.8 Example: Marlap Example 20.13, Corrected 211
18.9 Quadratic NPM 211
18.10 A Few Important Points 212
18.11 Chapter Highlights 212
References 213
19 Limits of Quantitation 215
19.1 Introduction 215
19.2 Theory 217
19.3 Computer Simulation 219
19.4 Experiment 221
19.5 Discussion and Conclusion 223
Acknowledgments 224
References 224
19.6 Postscript 225
19.7 Chapter Highlights 226
20 The Sampled Step Function 227
20.1 Introduction 227
20.2 A Noisy Step Function Temporal Response 229
20.3 Signal Processing Preliminaries 230
20.4 Processing the Sampled Step Function Response 231
20.5 The Standard t-Test for Two Sample Means When the Variance is Constant 232
20.6 Response Domain Decision Level and Detection Limit 233
20.7 Hypothesis Testing 233
20.8 Is There any Advantage to Increasing Nanalyte? 233
20.9 NET Response Domain Decision Level and Detection Limit 235
20.10 NET Response Domain SNRs 235
20.11 Content Domain Decision Level and Detection Limit 235
20.12 The RSDB–BEC Method 236
20.13 Conclusion 237
20.14 Chapter Highlights 237
References 237
21 The Sampled Rectangular Pulse 239
21.1 Introduction 239
21.2 The Sampled Rectangular Pulse Response 239
21.3 Integrating the Sampled Rectangular Pulse Response 240
21.4 Relationship Between Digital Integration and Averaging 242
21.5 What is the Signal in the Sampled Rectangular Pulse? 243
21.6 What is the Noise in the Sampled Rectangular Pulse? 243
21.7 The Noise Bandwidth 244
21.8 The SNR with Matched Filter Detection of the Rectangular Pulse 245
21.9 The Decision Level and Detection Limit 245
21.10 A Square Wave at the Detection Limit 246
21.11 Effect of Sampling Frequency 247
21.12 Effect of Area Fraction Integrated 247
21.13 An Alternative Limit of Detection Possibility 248
21.14 Pulse-to-Pulse Fluctuations 248
21.15 Conclusion 249
21.16 Chapter Highlights 250
References 250
22 The Sampled Triangular Pulse 251
22.1 Introduction 251
22.2 A Simple Triangular Pulse Shape 251
22.3 Processing the Sampled Triangular Pulse Response 253
22.4 The Decision Level and Detection Limit 254
22.5 Detection Limit for a Simulated Chromatographic Peak 254
22.6 What Should Not be Done? 256
22.7 A Bad Play, in Three Acts 256
22.8 Pulse-to-Pulse Fluctuations 258
22.9 Conclusion 258
22.10 Chapter Highlights 259
References 259
23 The Sampled Gaussian Pulse 261
23.1 Introduction 261
23.2 Processing the Sampled Gaussian Pulse Response 262
23.3 The Decision Level and Detection Limit 263
23.4 Pulse-to-Pulse Fluctuations 263
23.5 Conclusion 264
23.6 Chapter Highlights 264
References 264
24 Parting Considerations 267
24.1 Introduction 267
24.2 The Measurand Dichotomy Distraction 269
24.3 A “New Definition of LOD” Distraction 273
24.4 Potentially Important Research Prospects 274
24.4.1 Extension to Method Detection Limits 274
24.4.2 Confidence Intervals in the Content Domain 275
24.4.3 Noises Other Than AGWN 275
24.5 Summary 276
References 277
Appendix A Statistical Bare Necessities 279
Appendix B An Extremely Short Lightstone® Simulation Tutorial 299
Appendix C Blank Subtraction and the 𝜂1∕2 Factor 311
Appendix D Probability Density Functions for Detection Limits 321
Appendix E The Hubaux and Vos Method 325
Bibliography 331
Glossary of Organization and Agency Acronyms 335
Index 337
