Exact Analysis of Discrete Data

Exact Analysis of Discrete Data......Page 1
List of figures......Page 9
List of tables......Page 10
Abbreviations......Page 14
Foreword......Page 15
Contents......Page 4
1.2 Discrete Random Variables......Page 19
1.3 Probability Distributions......Page 23
1.4 Polynomial Based Distributions......Page 24
1.5 Binomial Distribution......Page 28
1.6 Poisson Distribution......Page 30
1.7 Negative Binomial Distribution......Page 32
1.8 Hypergeometric Distribution......Page 33
1.9 A General Representation......Page 35
1.10 The Multinomial Distribution......Page 36
1.11 The Negative Trinomial......Page 38
1.12 Sufficient Statistics......Page 39
1.13 The Polynomial Form......Page 41
1.14 Relevant Literature......Page 42
1.15 Exercises......Page 43
2.2 One Parameter Inference......Page 46
2.3 Tail Probability and Evidence......Page 48
2.5 Mid-p Evidence Function......Page 52
2.7 Matters of Significance......Page 54
2.8 Confidence Intervals......Page 58
2.9 Illustrative Examples......Page 61
2.10 Design and Analysis......Page 64
2.11.1 The mid-p......Page 67
2.11.3 One-Sided Procedures......Page 68
2.12 Exercises......Page 69
3.2 Two-Sided Inference......Page 72
3.3 Twice the Smaller Tail Method......Page 76
3.4 Examples......Page 78
3.5 The Likelihood Function......Page 79
3.6 The Score Method......Page 81
3.7 Additional Illustrations......Page 83
3.8 Likelihood Ratio and Wald Methods......Page 85
3.9 Three More Methods......Page 87
3.9.2 Distance from the Center Method......Page 88
3.9.3 The Combined Tails Method......Page 89
3.10 Comparative Computations......Page 90
3.11 The ABC of Reporting......Page 92
3.12 Additional Comments......Page 95
3.13 At the Boundary......Page 97
3.15 Relevant Literature......Page 99
3.16 Exercises......Page 100
4.2 Computing Principles......Page 104
4.3 Combinatorial Coefficients......Page 105
4.4 Polynomial Storage and Evaluation......Page 110
4.5 Computing Distributions......Page 114
4.6.1 The Main Equations......Page 120
4.6.2 Other Equations......Page 121
4.7 Iterative Methods......Page 123
4.8 Relevant Literature......Page 129
4.9 Exercises......Page 130
5.2 Design and Analysis......Page 133
5.3 Modes of Inference......Page 138
5.5 The One Margin Fixed Design......Page 139
5.6 The Overall Total Fixed Design......Page 142
5.7 The Nothing Fixed Design......Page 146
5.8 A Retrospective Design......Page 148
5.9 The Inverse Sampling Design......Page 150
5.10 Unconditional Analysis......Page 151
5.11 Conditional Analysis......Page 155
5.12 Comparing Two Rates......Page 160
5.13 Points to Ponder......Page 163
5.14 Derivation of Test Statistics......Page 165
5.14.1 Study Design......Page 167
5.14.3 The 2 × 2 Table......Page 168
5.16 Exercises......Page 169
6.2 Sources of Variability......Page 174
6.3 On Stratification......Page 175
6.4.1 An RCT with Stratified Randomization......Page 177
6.4.2 A Cross Sectional Design......Page 178
6.4.4 A Follow Up Study......Page 179
6.5.1 The Product Binomial Model......Page 180
6.5.3 The Retrospective Binomial Model......Page 183
6.5.4 The Poisson Person Time Model......Page 184
6.6 Conventional Analysis......Page 185
6.7 Conditional Analysis......Page 188
6.8 An Example......Page 189
6.9 A Second Example......Page 191
6.10 On Case-Control Sampling......Page 192
6.11 Anatomy of Interactions......Page 196
6.13 Exercises......Page 198
7.2 Exact Unconditional Analysis......Page 204
7.3 Randomized Inference......Page 209
7.4 Exact Power......Page 211
7.5 Exact Coverage......Page 219
7.6 The Fisher and Irwin Tests......Page 221
7.7 Some Features......Page 225
7.8 Desirable Features......Page 229
7.9 On Unconditional Analysis......Page 232
7.10 Why the Mid-p?......Page 233
7.11.1 Exact Unconditional Analysis......Page 234
7.11.3 Exact Power and Coverage......Page 235
7.12 Exercises......Page 236
8.2 Three Models......Page 241
8.3 Exact Distributions......Page 243
8.4 The COR Model......Page 248
8.5 Conditional Independence......Page 251
8.6 Trend In Odds Ratios......Page 255
8.8 Relevant Literature......Page 260
8.9 Exercises......Page 261
9.2 Models for Combining Risk......Page 266
9.3 Testing for Homogeneity......Page 269
9.4 Test Statistics......Page 271
9.5 A Worked Example......Page 273
9.6 Checking the TOR Model......Page 274
9.7 An Incidence Density Study......Page 276
9.8 Other Study Designs......Page 279
9.9 Exact Power......Page 280
9.10 Additional Issues......Page 282
9.11 Derivation......Page 284
9.13 Exercises......Page 287
10.1 Introduction......Page 291
10.2 An Ordered Table......Page 292
10.3 An Unordered Table......Page 297
10.4 Test Statistics......Page 299
10.5 An Illustration......Page 302
10.6 Checking Linearity......Page 306
10.7 Other Sampling Designs......Page 307
10.8 Incidence Density Data......Page 311
10.9 An Inverse Sampling Design......Page 315
10.10.1 Aspects of Trend Analysis......Page 317
10.10.2 Exact Power......Page 319
10.10.3 Pair-wise Comparisons......Page 320
10.10.4 Recommendations......Page 323
10.11.2 Several 2 ×K Tables......Page 324
10.11.3 Logistic Regression......Page 325
10.12 Derivation......Page 326
10.13 Relevant Literature......Page 327
10.14 Exercises......Page 328
11.2 Exhaustive Enumeration......Page 334
11.3 Monte-Carlo Simulation......Page 337
11.4 Recursive Multiplication......Page 340
11.5 Exponent Checks......Page 341
11.6 Applications......Page 345
11.7 The Fast Fourier Transform......Page 350
11.9 Exercises......Page 355
12.2 Bivariate Polynomials......Page 359
12.3 A Conditional Polynomial......Page 362
12.4 Backward Induction......Page 365
12.5 Conditional Values......Page 367
12.6.1 Application I......Page 370
12.6.4 Application IV......Page 371
12.7 Trivariate Polynomials......Page 372
12.8 An Extension......Page 375
12.9 Network Algorithms......Page 376
12.10 Power Computation......Page 378
12.12 Relevant Literature......Page 382
12.13 Exercises......Page 383
13.2.1 Combinations......Page 387
13.2.2 Compositions......Page 388
13.2.3 Partitions......Page 389
13.3 A Single Multinomial......Page 390
13.4 Trinary Response Models......Page 398
13.5 Conditional Polynomials......Page 404
13.6 Several......Page 410
13.7.1 Unordered Tables......Page 412
13.7.2 Doubly Ordered Tables......Page 415
13.7.3 Computation......Page 416
13.9 Exercises......Page 418
14.2 Matched Designs......Page 424
14.3.1 Models Without Covariates......Page 433
14.3.2 Models With Covariates......Page 434
14.3.3 One Binary Common Covariate......Page 436
14.3.4 Computation......Page 439
14.4 Markov Chain Models......Page 441
14.5 Relevant Literature......Page 451
14.6 Exercises......Page 452
15.2 Inexact Terminology......Page 457
15.3.1 The Bayesian Framework......Page 459
15.3.2 The Frequentist Framework......Page 461
15.4 Design and Analysis......Page 463
15.5.1 Paper I......Page 470
15.5.2 Paper II......Page 472
15.5.3 Paper III......Page 473
15.5.4 Comments......Page 474
15.6 Practical Inexactness......Page 476
15.7 Formal Exactness......Page 479
15.8 In Praise of Exactness......Page 485
15.9 Relevant Literature......Page 489
15.10 Exercises......Page 490
References......Page 493