Discrete Multivariate Analysis: Theory and Practice

DISCRETE MULTIVARIATE ANALYSIS: THEORY AND PRACTICE......Page 1
Springerlink......Page 0
Half-title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Contents......Page 8
1.2 Why a Book?......Page 12
1.4 Sketch of the Chapters......Page 13
1.6 How to Proceed from Here......Page 18
2.1.1 Structure......Page 20
2.1.2 Models......Page 21
2.2 Two Dimensions—The Fourfold Table......Page 22
2.2.1 Possible constraints for one arrangement......Page 23
2.2.2 Effect of rearranging the data......Page 26
2.2.3 The log-linear model......Page 27
2.2.4 Differences between log-linear and other models......Page 33
2.3.1 The log-linear model......Page 35
2.3.2 Cross-product ratios and linear contrasts......Page 37
2.3.3 Effect of combining categories......Page 38
2.3.4 Different sampling schemes......Page 40
2.3.5 The logit model......Page 41
2.4 Models for Three-Dimensional Arrays......Page 42
2.4.1 The 2 × 2 × 2 table......Page 43
2.4.2 The I × J × K model......Page 46
2.5 Models for Four or More Dimensions......Page 53
2.5.1 Parameters in the 2[sup(s)] table......Page 54
2.5.2 Uses and interpretation of models......Page 56
2.5.3 Collapsing arrays......Page 58
2.6 Exercises......Page 59
2.7 Appendix: The Geometry of a 2 × 2 Table......Page 60
2.7.1 The tetrahedron of reference......Page 61
2.7.2 Surface of independence......Page 62
2.7.3 The surface of constant association......Page 63
2.7.4 Line of constant margins......Page 64
3.1 Introduction......Page 68
3.1.2 Completeness......Page 69
3.1.3 Notation......Page 72
3.2.1 Usual schemes......Page 73
3.2.2 Other sampling schemes......Page 74
3.3.1 Three dimensions and Poisson or simple multinomial sampling......Page 75
3.3.2 The hierarchy principle......Page 78
3.3.3 Generalized method of deriving sufficient statistics......Page 79
3.3.4 Birch\'s results......Page 80
3.3.5 Product multinomial sampling......Page 81
3.3.6 Exercises......Page 83
3.4.1 Two types of models......Page 84
3.4.2 Direct estimates in three and four dimensions......Page 85
3.4.3 Rules for detecting existence of direct estimates......Page 87
3.4.4 Theorems yielding classifying rules......Page 89
3.5 Iterative Proportional Fitting of Log-Linear Models......Page 94
3.5.1 Iterative proportional fitting in three dimensions......Page 95
3.5.2 The general iterative proportional fitting algorithm......Page 102
3.5.3 Exercises......Page 107
3.6 Classical Uses of Iterative Proportional Fitting......Page 108
3.7.1 Relabeling......Page 113
3.7.2 Partitioning......Page 116
3.7.3 Fitting by stages......Page 118
3.7.4 Fitting incomplete models to seemingly complete data......Page 122
3.8.1 Three dimensions......Page 125
3.8.2 General form for models with all configurations of same dimension......Page 130
3.8.3 Models with closed-form cell estimates......Page 132
3.8.4 Exercises......Page 133
4.1 Introduction......Page 134
4.2.1 Pearson X² compared with likelihood statistic G²......Page 135
4.2.2 Difference between direct models......Page 140
4.2.4 Exercises......Page 141
4.3.1 Two methods of adjustment......Page 142
4.3.3 Adjusting for more than one variable......Page 144
4.4.1 Standardized cell residuals......Page 147
4.4.2 Standardized parameter estimates......Page 152
4.4.3 Assessing magnitude of an interaction......Page 157
4.5 Choosing a Model......Page 166
4.5.1 Initial computations......Page 167
4.5.2 Stepwise procedures......Page 176
4.5.4 Exercises......Page 179
4.6.1 Notation......Page 180
4.62 Two basic rules for direct models......Page 181
4.6.3 Partitioning for other direct models......Page 182
4.6.4 Partitioning for indirect models......Page 183
4.6.5 Partitioning illustrated on Ries–Smith data......Page 184
4.6.6 Exercises......Page 186
5.1.2 Background......Page 188
5.2.1 Quasi independence......Page 189
5.2.2 Connectivity and separability......Page 193
5.2.3 Sampling models......Page 196
5.2.4 Degrees of freedom for quasi independence......Page 198
5.2.5 Iterative solution for MLEs under quasi independence......Page 199
5.2.6 Direct estimation procedures for quasi independence......Page 203
5.2.7 Equivalence of closed-form and iterative estimates......Page 212
5.2.8 Exercises......Page 213
5.3.1 Composite models......Page 217
5.32 Breaking apart an I × I table......Page 218
5.3.3 Exercise......Page 220
5.4.1 Quasi-log-linear models......Page 221
5.4.2 Separability in many dimensions......Page 223
5.4.3 MLEs and degrees of freedom......Page 227
5.4.4 Examples of incomplete multiway tables......Page 231
5.5.1 The model from Section 5.3......Page 236
5.5.2 Other multiplicative models......Page 237
5.5.3 Exercise......Page 239
6.1 Introduction......Page 240
6.1.1 Practical examples......Page 241
6.2.1 Notation......Page 242
6.2.2 Basic approach......Page 243
6.2.3 Incomplete-table approach......Page 244
6.2.4 Exercises......Page 246
6.3.2 Unconditional estimation......Page 247
6.4.1 Notation......Page 248
6.4.2 Models relating the three samples......Page 249
6.4.3 Choosing a model and estimating N......Page 250
6.4.4 Estimates of N and their variances......Page 251
6.4.6 Some examples......Page 253
6.4.7 Exercises......Page 256
6.5.1 Basic framework......Page 257
6.5.2 Formulas for estimating N......Page 258
6.5.5 Asymptotic variances for N̂[sub(C)]......Page 259
6.5.6 Bounds for asymptotic variances......Page 262
6.5.7 A five-sample example......Page 263
6.6 Discussion......Page 265
7.1.2 Beyond fitting a Markov model......Page 268
7.1.3 Markov models for aggregate data......Page 271
7.2.2 Approach to stationarity......Page 272
7.2.3 Estimation of transition probabilities......Page 273
7.2.4 Are the transition probabilities stationary?......Page 276
7.3.1 Estimating transition probabilities......Page 278
7.3.3 Assessing the order of a chain......Page 280
7.4 Markov Models with a Single Sequence of Transitions......Page 281
7.5.1 Markov models for cross-classified states......Page 284
7.5.2 Sequences of unrepeated events......Page 286
7.5.3 Different chains for different strata......Page 288
8.1 Introduction......Page 292
8.2.2 Symmetry......Page 293
8.2.3 Quasi symmetry......Page 297
8.2.4 Marginal homogeneity......Page 304
8.2.5 Zero diagonals in square tables......Page 307
8.2.6 Exercises......Page 309
8.3.2 Conditional symmetry and marginal homogeneity......Page 310
8.3.3 Complete symmetry......Page 311
8.3.4 Quasi symmetry......Page 314
8.3.5 Marginal homogeneity......Page 317
8.4 Summary......Page 320
9.1 Introduction......Page 322
9.2.2 Why simplicity?......Page 323
9.2.3 Precision of estimates in I × J tables......Page 324
9.3 Searching for Sampling Models......Page 326
9.4 Fitting and Testing Using the Same Data......Page 328
9.5 Too Good a Fit......Page 335
9.6.2 Chi square under the alternative......Page 340
9.6.3 Comparing chi squares......Page 341
9.7 Data Anomalies and Suppressing Parameters......Page 343
9.8.2 Some basic theory......Page 348
9.8.3 Some applications......Page 350
10.1 Introduction......Page 354
10.2 The Information-Theoretic Approach......Page 355
10.2.1 Model generation......Page 356
10.2.2 Estimating expected values......Page 357
10.3.1 Minimizing Pearson\'s chi square......Page 359
10.3.3 Minimizing Neyman\'s chi square......Page 360
10.3.4 Neyman\'s chi square with linearized constraints......Page 362
10.3.5 Equivalence of Neyman chi square and weighted least squares approaches......Page 363
10.3.6 Weighted least squares estimation......Page 364
10.4.1 The discrete-variable linear logistic model......Page 368
10.4.3 The mixed-variable linear logistic model......Page 369
10.4.6 Scoring categorical variables versus categorizing continuous variables......Page 371
10.5 Testing Via Partitioning of Chi Square......Page 372
10.6.1 2 × 2 tables......Page 375
10.6.2 The combination of 2 × 2 tables......Page 376
10.7 Analyses Based on Transformed Proportions......Page 377
10.7.3 Arc sin transformation for proportions......Page 378
10.7.5 Transformations for multiresponse data......Page 381
10.8 Necessary Developments......Page 382
11.1.2 Measures of association in general......Page 384
11.2.1 Why deal separately with 2 × 2 tables?......Page 387
11.2.2 Measures based on the cross-product ratio......Page 388
11.2.3 Measures which are functions of the correlation coefficient......Page 391
11.2.5 Geometry of measures for 2 × 2 tables......Page 394
11.3.1 Measures based on chi square......Page 396
11.3.2 Proportional reduction in error measures......Page 398
11.3.4 A proportion of explained variance measure......Page 400
11.3.5 Measures insensitive to marginal totals......Page 403
11.4.1 The problem......Page 404
11.4.2 A measure of agreement......Page 406
11.4.3 Measuring conditional agreement......Page 408
11.4.4 Studying disagreement......Page 409
11.4.5 Agreement among more than two observers......Page 411
12.1.1 Central formulas......Page 412
12.1.2 Historical background......Page 414
12.2.1 Bayes estimators......Page 415
12.2.2 The risk criterion......Page 417
12.2.4 Geometric motivation for the use of Bayes and pseudo-Bayes estimators......Page 419
12.3.1 Special asymptotics for sparse rnultinomials......Page 421
12.3.2 Standard asymptotics for multinomials......Page 424
12.4.1 Binomial ease......Page 427
12.4.2 Trinomial case......Page 428
12.5 Data-Dependent λ\'s......Page 430
12.5.1 Decomposing λ......Page 431
12.5.2 Small-sample results......Page 432
12.5.3 λ and K̂ at work......Page 436
12.6 Another Example: Two Social Mobility Tables......Page 437
12.7.1 Further results when λ is fixed a priori......Page 440
12.7.2 Further work when λ is data dependent......Page 442
12.7.3 Some advice about pseudo-Bayes estimators......Page 444
13.2 The Binomial Distribution......Page 446
13.2.1 Moments......Page 447
13.2.2 Point estimation of parameters......Page 448
13.3 The Poisson Distribution......Page 449
13.3.1 Moments......Page 450
13.3.4 A conditioning property......Page 451
13.4.1 Moments......Page 452
13.4.2 Marginal and conditional distributions......Page 455
13.4.3 Estimation of parameters......Page 456
13.4.4 Relationship between Poisson and multinomial estimators......Page 457
13.5 The Hypergeometric Distribution......Page 459
13.5.2 Estimation of parameters......Page 460
13.6 The Multivariate Hypergeometric Distribution......Page 461
13.6.2 Marginal and conditional distributions......Page 462
13.7 The Negative Binomial Distribution......Page 463
13.7.2 Estimation of parameters......Page 464
13.8 The Negative Multinomial Distribution......Page 465
13.8.2 Marginal and conditional distributions......Page 466
13.8.3 Estimation of parameters......Page 467
14.1 Introduction......Page 468
14.2 The O, o Notation......Page 469
14.2.1 Conventions in the use of the O, o notation......Page 470
14.2.2 An example of the use of the O, o notation......Page 471
14.3.1 Convergence in distribution......Page 474
14.3.2 Convergence in probability to a constant......Page 476
14.3.3 Methods of demonstrating convergence in distribution......Page 478
14.4.2 Definition of X[sub(n)] = O[sub(p)](b[sub(n)])......Page 486
14.4.3 Methods for determining the stochastic order of a sequence......Page 487
14.4.4 Convergence in distribution and o[sub(p)](1)......Page 489
14.4.5 Chernoff–Pratt theory of stochastic order......Page 490
14.5 Convergence of Moments......Page 495
14.5.2 The order of moments and o[sub(p)]......Page 496
14.6 The δ Method for Calculating Asymptotic Distributions......Page 497
14.6.1 The one-dimensional version of the δ method......Page 498
14.6.2 Variance-stabilizing transformations......Page 502
14.6.3 Multivariate versions of the δ method......Page 503
14.6.4 Some uses for asymptotic variances and covariances......Page 509
14.6.5 Exercises......Page 511
14.7.1 Notation......Page 513
14.7.2 Estimation in multinomial models......Page 515
14.7.3 Multinomial goodness-of-fit tests......Page 518
14.7.4 Exercise......Page 519
14.8.1 The regularity conditions......Page 520
14.8.2 Expansion of the maximum likelihood estimator......Page 522
14.9.1 Asymptotic equivalence or three goodness-of-fit statistics......Page 524
14.9.2 The limiting chi square distribution of X²......Page 527
14.9.4 The asymptotic variance of distance measures......Page 529
14.9.5 The limiting distribution of X² in some complex cases......Page 531
14.9.6 Nested models......Page 535
14.9.7 Approximate percentage points for the χ² distribution......Page 538
14.9.8 Exercises......Page 540
References......Page 542
Index to Data Sets......Page 554
Author Index [missing]......Page 558
Subject Index......Page 562