Exact Analysis of Discrete Data

✍ Scribed by Karim F. Hirji

• Publisher: Chapman and Hall/CRC
• Year: 2005
• Tongue: English
• Leaves: 539
• Category: Library

✦ Synopsis

Researchers in fields ranging from biology and medicine to the social sciences, law, and economics regularly encounter variables that are discrete or categorical in nature. While there is no dearth of books on the analysis and interpretation of such data, these generally focus on large sample methods. When sample sizes are not large or the data are otherwise sparse, exact methods--methods not based on asymptotic theory--are more accurate and therefore preferable.This book introduces the statistical theory, analysis methods, and computation techniques for exact analysis of discrete data. After reviewing the relevant discrete distributions, the author develops the exact methods from the ground up in a conceptually integrated manner. The topics covered range from univariate discrete data analysis, a single and several 2 x 2 tables, a single and several 2 x K tables, incidence density and inverse sampling designs, unmatched and matched case -control studies, paired binary and trinomial response models, and Markov chain data. While most chapters focus on statistical theory and applications, three chapters deal exclusively with computational issues. Detailed worked examples appear throughout the book, and each chapter includes an extensive problem set. Written at an elementary to intermediate level, Exact Analysis of Discrete Data is accessible to anyone having taken a basic course in statistics or biostatistics, bringing to them valuable material previously buried in specialized journals.

✦ Table of Contents

EXACT ANALYSIS OF DISCRETE DATA......Page 1
Title Page......Page 3
Copyright Page......Page 4
Dedication......Page 6
Contents......Page 8
List of figures......Page 14
List of tables......Page 16
Abbreviations......Page 20
Foreword......Page 22
1.2 Discrete Random Variables......Page 26
1.3 Probability Distributions......Page 30
1.4 Polynomial Based Distributions......Page 31
1.5 Binomial Distribution......Page 35
1.6 Poisson Distribution......Page 37
1.7 Negative Binomial Distribution......Page 39
1.8 Hypergeometric Distribution......Page 40
1.9 A General Representation......Page 42
1.10 The Multinomial Distribution......Page 43
1.11 The Negative Trinomial......Page 45
1.12 Sufficient Statistics......Page 46
1.13 The Polynomial Form......Page 48
1.14 Relevant Literature......Page 49
1.15 Exercises......Page 50
2.2 One Parameter Inference......Page 54
2.3 Tail Probability and Evidence......Page 56
2.5 Mid-p Evidence Function......Page 60
2.7 Matters of Significance......Page 62
2.8 Confidence Intervals......Page 66
2.9 Illustrative Examples......Page 69
2.10 Design and Analysis......Page 72
2.11.1 The mid-p......Page 75
2.11.3 One-Sided Procedures......Page 76
2.12 Exercises......Page 77
3.2 Two-Sided Inference......Page 80
3.3 Twice the Smaller Tail Method......Page 84
3.4 Examples......Page 86
3.5 The Likelihood Function......Page 87
3.6 The Score Method......Page 89
3.7 Additional Illustrations......Page 91
3.8 Likelihood Ratio and Wald Methods......Page 93
3.9 Three More Methods......Page 95
3.9.2 Distance from the Center Method......Page 96
3.9.3 The Combined Tails Method......Page 97
3.10 Comparative Computations......Page 98
3.11 The ABC of Reporting......Page 100
3.12 Additional Comments......Page 103
3.13 At the Boundary......Page 105
3.15 Relevant Literature......Page 107
3.16 Exercises......Page 108
4.2 Computing Principles......Page 112
4.3 Combinatorial Coefficients......Page 113
4.4 Polynomial Storage and Evaluation......Page 118
4.5 Computing Distributions......Page 122
4.6.1 The Main Equations......Page 128
4.6.2 Other Equations......Page 129
4.7 Iterative Methods......Page 131
4.8 Relevant Literature......Page 137
4.9 Exercises......Page 138
5.2 Design and Analysis......Page 142
5.3 Modes of Inference......Page 147
5.5 The One Margin Fixed Design......Page 148
5.6 The Overall Total Fixed Design......Page 151
5.7 The Nothing Fixed Design......Page 155
5.8 A Retrospective Design......Page 157
5.9 The Inverse Sampling Design......Page 159
5.10 Unconditional Analysis......Page 160
5.11 Conditional Analysis......Page 164
5.12 Comparing Two Rates......Page 169
5.13 Points to Ponder......Page 172
5.14 Derivation of Test Statistics......Page 174
5.14.1 Study Design......Page 176
5.14.3 The 2 × 2 Table......Page 177
5.16 Exercises......Page 178
6.2 Sources of Variability......Page 184
6.3 On Stratification......Page 185
6.4.1 An RCT with Stratified Randomization......Page 187
6.4.2 A Cross Sectional Design......Page 188
6.4.4 A Follow Up Study......Page 189
6.5.1 The Product Binomial Model......Page 190
6.5.3 The Retrospective Binomial Model......Page 193
6.5.4 The Poisson Person Time Model......Page 194
6.6 Conventional Analysis......Page 195
6.7 Conditional Analysis......Page 198
6.8 An Example......Page 199
6.9 A Second Example......Page 201
6.10 On Case-Control Sampling......Page 202
6.11 Anatomy of Interactions......Page 206
6.13 Exercises......Page 208
7.2 Exact Unconditional Analysis......Page 214
7.3 Randomized Inference......Page 219
7.4 Exact Power......Page 221
7.5 Exact Coverage......Page 229
7.6 The Fisher and Irwin Tests......Page 231
7.7 Some Features......Page 235
7.8 Desirable Features......Page 239
7.9 On Unconditional Analysis......Page 242
7.10 Why the Mid-p?......Page 243
7.11.1 Exact Unconditional Analysis......Page 244
7.11.3 Exact Power and Coverage......Page 245
7.12 Exercises......Page 246
8.2 Three Models......Page 252
8.3 Exact Distributions......Page 254
8.4 The COR Model......Page 259
8.5 Conditional Independence......Page 262
8.6 Trend In Odds Ratios......Page 266
8.8 Relevant Literature......Page 271
8.9 Exercises......Page 272
9.2 Models for Combining Risk......Page 278
9.3 Testing for Homogeneity......Page 281
9.4 Test Statistics......Page 283
9.5 A Worked Example......Page 285
9.6 Checking the TOR Model......Page 286
9.7 An Incidence Density Study......Page 288
9.8 Other Study Designs......Page 291
9.9 Exact Power......Page 292
9.10 Additional Issues......Page 294
9.11 Derivation......Page 296
9.13 Exercises......Page 299
10.1 Introduction......Page 304
10.2 An Ordered Table......Page 305
10.3 An Unordered Table......Page 310
10.4 Test Statistics......Page 312
10.5 An Illustration......Page 315
10.6 Checking Linearity......Page 319
10.7 Other Sampling Designs......Page 320
10.8 Incidence Density Data......Page 324
10.9 An Inverse Sampling Design......Page 328
10.10.1 Aspects of Trend Analysis......Page 330
10.10.2 Exact Power......Page 332
10.10.3 Pair-wise Comparisons......Page 333
10.10.4 Recommendations......Page 336
10.11.2 Several 2 ×K Tables......Page 337
10.11.3 Logistic Regression......Page 338
10.12 Derivation......Page 339
10.13 Relevant Literature......Page 340
10.14 Exercises......Page 341
11.2 Exhaustive Enumeration......Page 348
11.3 Monte-Carlo Simulation......Page 351
11.4 Recursive Multiplication......Page 354
11.5 Exponent Checks......Page 355
11.6 Applications......Page 359
11.7 The Fast Fourier Transform......Page 364
11.9 Exercises......Page 369
12.2 Bivariate Polynomials......Page 374
12.3 A Conditional Polynomial......Page 377
12.4 Backward Induction......Page 380
12.5 Conditional Values......Page 382
12.6.1 Application I......Page 385
12.6.4 Application IV......Page 386
12.7 Trivariate Polynomials......Page 387
12.8 An Extension......Page 390
12.9 Network Algorithms......Page 391
12.10 Power Computation......Page 393
12.12 Relevant Literature......Page 397
12.13 Exercises......Page 398
13.2.1 Combinations......Page 402
13.2.2 Compositions......Page 403
13.2.3 Partitions......Page 404
13.3 A Single Multinomial......Page 405
13.4 Trinary Response Models......Page 413
13.5 Conditional Polynomials......Page 419
13.6 Several......Page 425
13.7.1 Unordered Tables......Page 427
13.7.2 Doubly Ordered Tables......Page 430
13.7.3 Computation......Page 431
13.9 Exercises......Page 433
14.2 Matched Designs......Page 440
14.3.1 Models Without Covariates......Page 449
14.3.2 Models With Covariates......Page 450
14.3.3 One Binary Common Covariate......Page 452
14.3.4 Computation......Page 455
14.4 Markov Chain Models......Page 457
14.5 Relevant Literature......Page 467
14.6 Exercises......Page 468
15.2 Inexact Terminology......Page 474
15.3.1 The Bayesian Framework......Page 476
15.3.2 The Frequentist Framework......Page 478
15.4 Design and Analysis......Page 480
15.5.1 Paper I......Page 487
15.5.2 Paper II......Page 489
15.5.3 Paper III......Page 490
15.5.4 Comments......Page 491
15.6 Practical Inexactness......Page 493
15.7 Formal Exactness......Page 496
15.8 In Praise of Exactness......Page 502
15.9 Relevant Literature......Page 506
15.10 Exercises......Page 507
References......Page 510
Back Cover......Page 539