Complex multivariate testing problems are frequently encountered in many scientific disciplines, such as engineering, medicine and the social sciences. As a result, modern statistics needs permutation testing for complex data with low sample size and many variables, especially in observational studies.The Authors give a general overview on permutation tests with a focus on recent theoretical advances within univariate and multivariate complex permutation testing problems, this book brings the reader completely up to date with today’s current thinking.Key Features:Examines the most up-to-date methodologies of univariate and multivariate permutation testing.Includes extensive software codes in MATLAB, R and SAS, featuring worked examples, and uses real case studies from both experimental and observational studies.Includes a standalone free software NPC Test Release 10 with a graphical interface which allows practitioners from every scientific field to easily implement almost all complex testing procedures included in the book.Presents and discusses solutions to the most important and frequently encountered real problems in multivariate analyses.A supplementary website containing all of the data sets examined in the book along with ready to use software codes.Together with a wide set of application cases, the Authors present a thorough theory of permutation testing both with formal description and proofs, and analysing real case studies. Practitioners and researchers, working in different scientific fields such as engineering, biostatistics, psychology or medicine will benefit from this book.
Author(s): Fortunato Pesarin, Luigi Salmaso
Edition: 1
Year: 2010
Language: English
Pages: 448
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;Прикладная математическая статистика;
Cover......Page 1
Contents......Page 9
Preface......Page 17
Notation and Abbreviations......Page 21
1.1 On Permutation Analysis......Page 25
1.2.1 Nonparametric Family of Distributions......Page 28
1.2.2 The Permutation Testing Principle......Page 29
1.4 When and Why Conditioning is Appropriate......Page 31
1.5 Randomization and Permutation......Page 33
1.6 Computational Aspects......Page 34
1.7 Basic Notation......Page 35
1.8.1 Modelling Responses......Page 37
1.8.3 Further Aspects......Page 39
1.8.4 The Student’s t-Paired Solution......Page 40
1.8.5 The Signed Rank Test Solution......Page 41
1.9.1 General Aspects......Page 42
1.9.2 The Permutation Sample Space......Page 43
1.9.3 The Conditional Monte Carlo Method......Page 44
1.9.4 Approximating the Permutation Distribution......Page 46
1.10 A Two-Sample Problem......Page 47
1.10.1 Modelling Responses......Page 48
1.10.3 The Permutation Solution......Page 49
1.10.5 Problems and Exercises......Page 52
1.11.1 Modelling Responses......Page 53
1.11.2 Permutation Solutions......Page 54
1.11.3 Problems and Exercises......Page 56
2.1.1 Notation and Basic Assumptions......Page 57
2.1.2 The Conditional Reference Space......Page 59
2.1.3 Conditioning on a Set of Sufficient Statistics......Page 63
2.2.1 General Aspects......Page 65
2.2.2 Randomized Permutation Tests......Page 66
2.2.4 The p-Value......Page 67
2.2.5 A CMC Algorithm for Estimating the p-Value......Page 68
2.3 Some Useful Test Statistics......Page 69
2.4 Equivalence of Permutation Statistics......Page 71
2.4.1 Some Examples......Page 73
2.4.2 Problems and Exercises......Page 74
2.5 Arguments for Selecting Permutation Tests......Page 75
2.6 Examples of One-Sample Problems......Page 77
2.6.1 A Problem with Repeated Observations......Page 83
2.6.2 Problems and Exercises......Page 87
2.7 Examples of Multi-Sample Problems......Page 88
2.8.1 General Aspects......Page 98
2.8.2 A Solution Based on Score Transformations......Page 100
2.8.3 Typical Goodness-of-Fit Solutions......Page 101
2.8.4 Extension to Non-Dominance Alternatives and C Groups......Page 103
2.9 Problems and Exercises......Page 104
3.1.1 One-Sided Alternatives......Page 107
3.1.2 Two-Sided Alternatives......Page 114
3.2.1 Definition and Algorithm for the Conditional Power......Page 117
3.2.3 Definition and Algorithm for the Unconditional Power: Fixed Effects......Page 121
3.2.5 Comments on Power Functions......Page 122
3.4 Permutation Confidence Interval for d......Page 123
3.4.1 Problems and Exercises......Page 127
3.5 Extending Inference from Conditional to Unconditional......Page 128
3.6 Optimal Properties......Page 130
3.6.1 Problems and Exercises......Page 131
3.7.1 Introduction......Page 132
3.7.2 Two Basic Theorems......Page 133
3.8.2 Permutation Central Limit Theorems......Page 135
3.9 Problems and Exercises......Page 137
4.1.1 General Aspects......Page 141
4.1.2 Bibliographic Notes......Page 142
4.1.3 Main Assumptions and Notation......Page 144
4.1.4 Some Comments......Page 145
4.2.1 Assumptions on Partial Tests......Page 146
4.2.2 Desirable Properties of Combining Functions......Page 147
4.2.3 A Two-Phase Algorithm for Nonparametric Combination......Page 149
4.2.4 Some Useful Combining Functions......Page 152
4.2.5 Why Combination is Nonparametric......Page 158
4.2.7 Problems and Exercises......Page 159
4.3.2 Unbiasedness......Page 161
4.3.4 Power of Combined Tests......Page 163
4.3.5 Conditional Multivariate Confidence Region for δ......Page 165
4.3.6 Problems and Exercises......Page 166
4.4.2 Asymptotic Properties......Page 167
4.5.1 Introduction......Page 170
4.5.2 Finite-Sample Consistency......Page 171
4.5.3 Some Applications of Finite-Sample Consistency......Page 176
4.6 Some Examples of Nonparametric Combination......Page 180
4.6.1 Problems and Exercises......Page 196
4.7.1 General Comments......Page 197
4.7.2 Final Remarks......Page 198
5.1 Defining Raw and Adjusted p-Values......Page 201
5.2.1 Multiple Comparison and Multiple Testing......Page 202
5.2.2 Some Definitions of the Global Type I Error......Page 203
5.3 Multiple Testing......Page 204
5.4 The Closed Testing Approach......Page 205
5.4.1 Closed Testing for Multiple Testing......Page 206
5.4.2 Closed Testing Using the MinP Bonferroni–Holm Procedure......Page 207
5.5.1 Analysis Using MATLAB......Page 210
5.5.2 Analysis Using R......Page 211
5.6.1 Analysis Using MATLAB......Page 213
5.6.2 Analysis Using R......Page 215
5.7 Weighted Methods for Controlling FWE and FDR......Page 217
5.8 Adjusting Stepwise p-Values......Page 218
5.8.2 Algorithm Description......Page 219
5.8.3 Optimal Subset Procedures......Page 220
6.1 Introduction......Page 221
6.2 The Multivariate McNemar Test......Page 222
6.2.1 An Extension of the Multivariate McNemar Test......Page 224
6.3 Multivariate Goodness-of-Fit Testing for Ordered Variables......Page 225
6.4 MANOVA with Nominal Categorical Data......Page 227
6.5.1 Formal Description......Page 228
6.5.2 Further Breaking Down the Hypotheses......Page 229
6.5.3 Permutation Test......Page 230
6.6.1 General Aspects......Page 231
6.6.2 The Multifocus Solution......Page 232
6.6.3 An Application......Page 234
6.7.1 Introduction......Page 235
6.7.2 Allelic Association Analysis in Genetics......Page 236
6.7.3 Parametric Solutions......Page 237
6.7.4 Permutation Approach......Page 238
6.8.1 General Aspects......Page 239
6.8.2 Score Transformations and Univariate Tests......Page 240
6.8.3 Multivariate Extension......Page 241
6.9.1 Introduction......Page 242
6.9.2 Tests for Comparing Heterogeneities......Page 243
6.9.3 A Case Study in Population Genetics......Page 244
6.10.1 Description of the Problem......Page 245
6.10.2 Global Satisfaction Index......Page 246
6.10.3 Multivariate Performance Comparisons......Page 248
7.1 Introduction......Page 249
7.3.1 A General Additive Model......Page 250
7.4.1 Solutions Using the NPC Approach......Page 252
7.4.3 Analysis of the Cross-Over (AB-BA) Design......Page 254
7.4.4 Analysis of a Cross-Over Design with Paired Data......Page 255
7.6.1 Bibliographic Notes......Page 256
7.7.1 Data Missing Completely at Random......Page 257
7.8 The Permutation Approach......Page 258
7.8.1 Deletion, Imputation and Intention to Treat Strategies......Page 259
7.8.2 Breaking Down the Hypotheses......Page 260
7.9.1 Hypotheses for MNAR Models......Page 261
7.9.2 Hypotheses for MCAR Models......Page 262
7.9.3 Permutation Structure with Missing Values......Page 263
7.10.1 Partitioning the Permutation Sample Space......Page 264
7.10.2 Solution for Two-Sample MCAR Problems......Page 265
7.10.3 Extensions to Multivariate C-Sample Problems......Page 266
7.10.4 Extension to MNAR Models......Page 267
7.11 Germina Data: An Example of an MNAR Model......Page 268
7.11.2 The Permutation Solution......Page 269
7.11.4 Analysis Using R......Page 272
7.12 Multivariate Paired Observations......Page 275
7.13 Repeated Measures and Missing Data......Page 276
7.13.1 An Example......Page 277
7.14 Botulinum Data......Page 278
7.14.1 Analysis Using MATLAB......Page 280
7.14.2 Analysis Using R......Page 282
7.15.1 Analysis Using MATLAB......Page 284
7.15.2 Analysis Using R......Page 288
8.1 Multivariate Ordered Alternatives......Page 291
8.2 Testing for Umbrella Alternatives......Page 293
8.2.1 Hypotheses and Tests in Simple Stochastic Ordering......Page 294
8.2.2 Permutation Tests for Umbrella Alternatives......Page 295
8.3 Analysis of Experimental Tumour Growth Curves......Page 297
8.4.1 Introduction......Page 300
8.4.2 A Permutation Solution......Page 302
8.4.3 Analysis Using MATLAB......Page 303
8.4.4 Analysis Using R......Page 310
9.1.1 Failure Time Distributions......Page 313
9.1.2 Data Structure......Page 314
9.2 Comparison of Survival Curves......Page 315
9.3 An Overview of the Literature......Page 316
9.3.1 Permutation Tests in Survival Analysis......Page 318
9.4.1 Breaking Down the Hypotheses......Page 319
9.4.2 The Test Structure......Page 320
9.4.3 NPC Test for Treatment-Independent Censoring......Page 321
9.4.4 NPC Test for Treatment-Dependent Censoring......Page 322
9.5 An Application to a Biomedical Study......Page 324
10.1 Introduction......Page 327
10.2.1 How to Describe Shapes......Page 328
10.2.2 Multivariate Morphometrics......Page 330
10.3 Inference with Shape Data......Page 332
10.4.1 Notation......Page 333
10.4.2 Comparative Simulation Study......Page 335
10.5 NPC Analysis with Correlated Landmarks......Page 336
10.6.1 The Case Study......Page 340
10.6.3 Shape Analysis Using MATLAB......Page 343
10.6.4 Shape Analysis Using R......Page 345
11.1 Autofluorescence Case Study......Page 349
11.1.1 A Permutation Solution......Page 350
11.1.3 Analysis Using R......Page 353
11.2.1 A Permutation Solution......Page 357
11.2.2 MATLAB and R Codes......Page 359
11.3.1 Brief Overview of Permutation Tests in Two-Way ANOVA......Page 368
11.3.2 MANOVA Using MATLAB and R Codes......Page 370
12.1.1 A Case Study on Oesophageal Cancer......Page 375
12.1.3 Survival Analysis with Stratification by Propensity Score......Page 377
12.2 Integrating Propensity Score and NPC Testing......Page 378
12.2.1 Analysis Using MATLAB......Page 382
12.3.1 A Two-Sample Epidemiological Survey: Problem Description......Page 383
12.3.2 Analysing SETIG Data Using MATLAB......Page 384
12.3.3 Analysing the SETIG Data Using R......Page 386
12.3.4 Analysing the SETIG Data Using NPC Test......Page 389
12.3.5 Analysis of the SETIG Data Using SAS......Page 393
12.4 A Comparison of Three Survival Curves......Page 394
12.4.2 Survival Analysis with Stratification by Propensity Score......Page 395
12.5.1 Survival Analysis Using NPC Test......Page 399
12.5.2 Survival Analysis Using SAS......Page 401
12.6.1 Application to Lymph Node Metastases......Page 402
12.6.2 Application to Bladder Cancer......Page 404
12.6.3 NPC Results......Page 406
12.6.4 Analysis by Logistic Regression......Page 408
12.6.5 Some Comments......Page 409
References......Page 411
Index......Page 433
