Scan Statistics (eBook)

Contents 7
Preface 15
Contributors 17
List of Tables 20
List of Figures 23
Joseph Naus: Father of the Scan Statistic 27
1.1 Naus (1963): Ph.D. Thesis 28
1.2 The Early Papers Touching All Aspects of the Problem: 1965– 1968 31
1.3 Joseph Naus’s Students in 1967–1978, Exploitation of Ballot Problem Results, Broadening of Problem 35
1.4 Two Key Publications, 1979–1982 40
1.5 Later Work, Briefly Noted 42
References 46
Precedence-Type Tests for the Comparison of Treatments with a Control 52
2.1 Introduction 52
2.2 Review of Precedence-Type Tests 54
2.3 Test Statistics for Comparing k 1 Treatments with Control 58
2.4 Exact Power Under Lehmann Alternative 66
2.5 Discussion 67
2.6 Illustrative Example 69
Appendix A: Probability Mass Function of (M2, . . . , Mk ) Under the Null Hypothesis 70
Appendix B: Probability Mass Function of (M2, . . . , Mk ) Under the Lehmann Alternative 73
References 78
Extreme Value Results for Scan Statistics 80
3.1 Introduction 80
3.2 Definitions and Notation 82
3.3 The Binary Scan Statistic 85
3.4 Scan Statistic Exceedances 96
References 109
Boundary Crossing Probability Computations in the Analysis of Scan Statistics 111
4.1 Introduction 111
4.2 Theoretical Developments 112
4.3 Applications in Spatial Scan Statistics 116
4.4 Recent Applications in Genomics 121
4.5 Concluding Remarks 127
References 128
Approximations for Two-Dimensional Variable Window Scan Statistics 133
5.1 Introduction 133
5.2 Two-Dimensional Discrete Scan Statistics 134
5.3 Variable Window Discrete-Type Scan Statistics 141
5.4 Numerical Results 145
5.5 Summary 149
References 150
Applications of Spatial Scan Statistics: A Review 153
6.1 Introduction 153
6.2 Brief Methodological Overview 154
6.3 Applications in Medical Imaging 156
6.4 Applications in Cancer Epidemiology 156
6.5 Applications in Infectious Disease Epidemiology 158
6.6 Applications in Parasitology 160
6.7 Other Medical Applications 161
6.8 Applications in Veterinary Medicine 162
6.9 Applications in Forestry 162
6.10 Applications in Geology 163
6.11 Applications in Astronomy 163
6.12 Applications in Psychology 164
6.13 Applications to Accidents 164
6.14 Applications in Criminology and Warfare 164
6.15 Applications in Demography 165
6.16 Applications in the Humanities 165
6.17 Scan Statistic Software 165
6.18 Discussion 166
References 166
Extensions of the Scan Statistic for the Detection and Inference of Spatial Clusters 177
7.1 Introduction 177
7.2 Irregularly Shaped Spatial Clusters 178
7.3 Data-Driven Spatial Cluster Detection Models 187
7.4 Applications 191
References 191
1-Dependent Stationary Sequences and Applications to Scan Statistics 202
8.1 Introduction 202
8.2 Application of the Approximations (8.6) and ( 8.7) to One- Dimensional Scan Statistics 207
8.3 Application of the Method to Two-Dimensional Scan Statistics 211
References 214
Scan Statistics in Genome-Wide Scan for Complex Trait Loci 217
9.1 Introduction 217
9.2 Methods 218
9.3 Applications 219
9.4 Discussion 221
References 222
On Probabilities for Complex Switching Rules in Sampling Inspection 225
10.1 Introduction 225
10.2 Notation and Finite Markov Chain Imbedding 227
10.3 Main Results 228
10.4 Numerical Examples of Switching Rules 232
10.5 Summary and Discussion 238
References 240
Bayesian Network Scan Statistics for Multivariate Pattern Detection 242
11.1 Introduction 242
11.2 The Multivariate Bayesian Scan Statistic 249
11.3 The Agent-Based Bayesian Scan Statistic 256
11.4 The Anomalous Group Detection Method 261
References 267
ULS Scan Statistic for Hotspot Detection with Continuous Gamma Response 271
12.1 Introduction 272
12.2 Basic Ideas 273
12.3 ULS Scan Statistic 274
12.4 Computational Aspects 276
12.5 Testing Significance of the Scan Statistic 278
12.6 Gamma Response Model 278
12.7 Details of Software Implementation 280
12.8 Construction of the ULS Scan Tree 283
12.9 A Case Study 285
12.10 Conclusions 287
References 288
False Discovery Control for Scan Clustering 291
13.1 Introduction 291
13.2 The Basics of Multiple Testing 292
13.3 The Method 294
13.4 Clusters Shaving for Bias Correction 298
13.5 Power Increase Through Multiple Bandwidths 300
13.6 Examples 301
References 306
Martingale Methods for Patterns and Scan Statistics 308
14.1 Introduction 308
14.2 Patterns in an Independent Sequence 309
14.3 Compound Patterns and Gambling Teams 313
14.4 Patterns in Markov Dependent Trials 318
14.5 Applications to Scans 327
14.6 Concluding Remarks 335
References 335
How Can Pattern Statistics Be Useful for DNA Motif Discovery? 337
15.1 Introduction 337
15.2 Words with Exceptional Frequency 338
15.3 Words with Exceptional Distribution 357
15.4 More Sophisticated Patterns 360
15.5 Ongoing Research and Open Problems 364
References 365
Occurrence of Patterns and Motifs in Random Strings 369
16.1 Introduction 369
16.2 Patterns: Discrete-Time Models 371
16.3 Patterns: General Discrete-Time and Continuous- Time Models 374
16.4 Compound Patterns 377
References 382
Detection of Disease Clustering 386
17.1 Introduction 386
17.2 Temporal Clustering 387
17.3 Spatial Clustering 394
17.4 Discussion 403
References 405
Index 409