Negative binomial regression

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to the second edition......Page 13
New subjects discussed in the second edition......Page 15
To those who made this a better text......Page 18
1.1 What is a negative binomial model?......Page 21
1.2 A brief history of the negative binomial......Page 25
1.3 Overview of the book......Page 31
2.1 Risk and 2×2 tables......Page 35
2.2 Risk and 2×k tables......Page 38
FIRST CLASS......Page 39
2.3 Risk ratio confidence intervals......Page 40
RISK RATIO: UPPER 95% CI......Page 42
2.4 Risk difference......Page 44
2.5 The relationship of risk to odds ratios......Page 45
2.6 Marginal probabilities: joint and conditional......Page 47
Summary......Page 49
3.1 Varieties of count response model......Page 50
3.2 Estimation......Page 58
Summary......Page 61
4.1 Derivation of the IRLS algorithm......Page 63
EXPONENTIAL FAMILY PDF......Page 65
4.1.1 Solving for or U – the gradient......Page 68
Solving for…......Page 69
4.1.3 The IRLS fitting algorithm......Page 71
4.2 Newton–Raphson algorithms......Page 73
4.2.1 Derivation of the Newton–Raphson......Page 74
4.2.3 Parameterizing from  mu to xbeta......Page 77
4.2.4 Maximum likelihood estimators......Page 79
Summary......Page 80
5.1 Residuals for count response models......Page 81
5.2.1 Traditional fit tests......Page 84
5.2.1.1 R2 and pseudo-R2 Goodness-of-fit tests......Page 85
FULL MODEL......Page 86
5.2.1.4 Likelihood-ratio test......Page 87
5.2.2.1 Akaike Information Criterion......Page 88
5.2.2.2 Bayesian Information Criterion......Page 91
5.3 Validation models......Page 95
Summary......Page 96
6.1 Derivation of the Poisson model......Page 97
BINOMIAL PDF......Page 98
6.1.2 Derivation of the Poisson model......Page 99
EXPONENTIAL FAMILY PDF......Page 100
VARIANCE......Page 101
POISSON LOG-LIKELIHOOD FUNCTION (xbeta)......Page 102
6.2.1 Construction of synthetic models......Page 105
OUTPUT......Page 112
STATA......Page 113
Changes to the response......Page 114
INTERCEPT......Page 115
Changes to the predictor......Page 116
Changing response in a multivariable model......Page 117
Changing a predictor in a multivariate model......Page 119
PREDICTORS......Page 120
6.3.2 Incidence rate ratio parameterization......Page 129
CONTINUOUS IRR......Page 131
6.4 Predicted counts......Page 136
6.5 Effects plots......Page 142
6.6.1 Marginal effects for Poisson and negative binomial effects models......Page 145
Marginal effects at the mean......Page 146
Average marginal effects......Page 148
Elasticities......Page 149
6.6.2 Discrete change for Poisson and negative binomial models......Page 151
Average discrete change: partial effects......Page 152
6.7.1 Exposure in time and area......Page 154
6.7.2 Synthetic Poisson with offset......Page 156
6.7.3 Example......Page 158
Summary......Page 160
7.1 What is overdispersion?......Page 161
7.2.1 Creation of a simulated base Poisson model......Page 162
7.2.3 Outliers in data......Page 165
7.2.4 Creation of interaction......Page 169
7.2.5 Testing the predictor scale......Page 170
7.2.6 Testing the link......Page 172
MODELS DEALING WITH POISSON OVERDISPERSION......Page 177
7.3.1 Scaling of standard errors / quasi-Poisson......Page 178
SCALED SE......Page 181
QUASI-POISSON CONFIDENCE INTERVALS......Page 182
7.3.2 Quasi-likelihood variance multipliers......Page 183
QUASI-LIKELIHOOD NEGATIVE BINOMIAL......Page 184
7.3.3 Robust variance estimators......Page 188
POISSON: CLUSTERING BY PROVIDER......Page 190
7.3.4 Bootstrapped and jackknifed standard errors......Page 191
7.4 Tests of overdispersion......Page 194
Z-TESTS......Page 195
LAGRANGE MULTIPLIER TEST......Page 196
7.4.2 Boundary likelihood ratio test......Page 197
7.4.3 tests for Poisson and negative binomial models......Page 199
7.5 Negative binomial overdispersion......Page 200
POISSON......Page 201
NEGATIVE BINOMIAL (NB2)......Page 202
Summary......Page 203
8.1 Varieties of negative binomial......Page 205
8.2 Derivation of the negative binomial......Page 207
8.2.1 Poisson–gamma mixture model......Page 208
NB – HESSIAN – alpha......Page 212
8.2.2 Derivation of the GLM negative binomial......Page 213
LINK, CUMULANT, SCALE......Page 215
NEGATIVE BINOMIAL VARIANCE......Page 216
MEAN, VARIANCE, AND DERIVATIVE......Page 217
NB-C LOG-LIKELIHOOD – xbeta......Page 218
8.3 Negative binomial distributions......Page 219
TOP TO BOTTOM ON VERTICAL AXIS; MEAN  = 0.5, 1, 2, 5, 10......Page 220
8.4 Negative binomial algorithms......Page 227
8.4.1 NB-C: canonical negative binomial......Page 228
8.4.2 NB2: expected information matrix......Page 230
8.4.3 NB2: observed information matrix......Page 235
DEFINING zo......Page 236
8.4.4 NB2: R maximum likelihood function......Page 238
Summary......Page 239
9.1 Poisson versus negative binomial......Page 241
NEGATIVE BINOMIAL (NB2)......Page 242
9.2 Synthetic negative binomial......Page 245
NEGATIVE BINOMIAL – NB2......Page 252
AVERAGE MARGINAL EFFECT......Page 256
MARGINAL EFFECT AT MEAN: AGE......Page 257
9.4 Binomial versus count models......Page 259
Example 1: Modeling number of marital affairs......Page 268
Example 2: Heart procedures......Page 279
Example 3: Titanic survival data......Page 283
Example 4: Health reform data......Page 289
Summary......Page 302
10 Alternative variance parameterizations......Page 304
10.1.1 Derivation of the geometric......Page 305
10.1.2 Synthetic geometric models......Page 306
10.1.3 Using the geometric model......Page 310
10.1.4 The canonical geometric model......Page 314
CANONICAL LOG-LIKELIHOOD......Page 315
CANONICAL GEOMETRIC MODEL......Page 316
10.2.1 NB1 as QL-Poisson......Page 318
10.2.2 Derivation of NB1......Page 321
10.2.3 Modeling with NB1......Page 324
PARAMETERS......Page 325
10.2.4 NB1: R maximum likelihood function......Page 326
10.3.1 NB-C overview and formulae......Page 328
10.3.2 Synthetic NB-C models......Page 331
STATA......Page 332
10.3.3 NB-C models......Page 335
NB-C MODEL......Page 337
NB2 MODEL......Page 338
10.4 NB-H: Heterogeneous negative binomial regression......Page 339
HETEROGENEOUS NB2: ALPHA == 0.5......Page 340
10.5 The NB-P model: generalized negative binomial......Page 343
10.6 Generalized Waring regression......Page 348
GENERALIZED WARING PDF......Page 349
GENERALIZED WARING MODEL......Page 351
GENERALIZED WARING MODEL......Page 352
10.7 Bivariate negative binomial......Page 353
BIVARIATE NEGATIVE BINOMIAL......Page 354
OUTPUT......Page 356
10.8 Generalized Poisson regression......Page 357
10.9 Poisson inverse Gaussian regression (PIG)......Page 361
10.10 Other count models......Page 363
Summary......Page 364
11.1 Zero-truncated count models......Page 366
ZERO-TRUNCATED NEGATIVE BINOMIAL......Page 372
NB-C......Page 373
11.2 Hurdle models......Page 374
11.2.1 Theory and formulae for hurdle models......Page 376
11.2.2 Synthetic hurdle models......Page 377
11.2.3 Applications......Page 379
NEGATIVE BINOMIAL – COMPLEMENTARY LOGLOG HURDLE......Page 380
ZERO-TRUNCATED NEGATIVE BINOMIAL MODEL......Page 382
NEGATIVE BINOMIAL–LOGIT......Page 383
NEGATIVE BINOMIAL – POISSON HURDLE MODEL......Page 386
LOGIT......Page 389
11.3.1 Overview of ZIP/ZINB models......Page 390
11.3.2 ZINB algorithms......Page 391
Zero-inflated negative binomial–probit......Page 392
11.3.3 Applications......Page 394
11.3.4 Zero-altered negative binomial......Page 396
11.3.5 Tests of comparative fit......Page 397
11.3.6 ZINB marginal effects......Page 399
AVERAGE DISCRETE CHANGE – BINARY (Logit) COMPONENT......Page 401
11.4 Comparison of models......Page 402
Summary......Page 405
12.1 Censored and truncated models – econometric parameterization......Page 407
12.1.1 Truncation......Page 408
LEFT-TRUNCATED POISSON: CUT = 3......Page 411
12.1.2 Censored models......Page 415
CENSORED NEGATIVE BINOMIAL......Page 416
LEFT-CENSORED NEGATIVE BINOMIAL : CUT = 3......Page 417
RIGHT-CENSORED NEGATIVE BINOMIAL : CUT = 15......Page 418
12.2 Censored Poisson and NB2 models – survival parameterization......Page 419
CENSORED POISSON LOG-LIKELIHOOD FUNCTION......Page 420
CENSORED NEGATIVE BINOMIAL LOG-LIKELIHOOD FUNCTION......Page 421
CENSORED NB2: SURVIVAL......Page 424
CENSORED POISSON: ECONOMETRIC......Page 425
Summary......Page 426
13 Handling endogeneity and latent class models......Page 427
13.1.1 Basics of finite mixture modeling......Page 428
MARGINAL EFFECTS/DISCRETE CHANGE......Page 431
13.1.2 Synthetic finite mixture models......Page 432
13.2.1 Problems related to endogeneity......Page 436
13.2.2 Two-stage instrumental variables approach......Page 437
Stage 1: Model binary endogenous predictor......Page 438
NB2: ADJUSTED STANDARD ERRORS......Page 439
NB2: MODEL STANDARD ERRORS......Page 440
13.2.3 Generalized method of moments (GMM)......Page 441
13.2.4 NB2 with an endogenous multinomial treatment variable......Page 442
13.2.5 Endogeneity resulting from measurement error......Page 445
13.3 Sample selection and stratification......Page 448
13.3.1 Negative binomial with endogenous stratification......Page 449
13.3.2 Sample selection models......Page 453
SAMPLE CORRECTED POISSON......Page 456
SAMPLE CORRECTED NEGATIVE BINOMIAL......Page 457
13.3.3 Endogenous switching models......Page 458
13.4 Quantile count models......Page 461
Summary......Page 466
14.1 Overview of count panel models......Page 467
14.2.1 The GEE algorithm......Page 470
14.2.2 GEE correlation structures......Page 472
SCHEMATIC......Page 473
14.2.3 Negative binomial GEE models......Page 475
ML NB2 ESTIMATION – TO OBTAIN VALUE OF alpha......Page 476
GEE NB2 ESTIMATION – WITH alpha ENTERED AS A CONSTANT......Page 477
EXAMPLE......Page 478
EXAMPLE......Page 480
Unstructured correlation structure with Poisson model......Page 481
EXAMPLE......Page 482
14.2.4 GEE goodness-of-fit......Page 484
14.2.5 GEE marginal effects......Page 486
14.3 Unconditional fixed-effects negative binomial model......Page 488
14.4 Conditional fixed-effects negative binomial model......Page 494
CONDITIONAL FIXED-EFFECTS NEGATIVE BINOMIAL LOG-LIKELIHOOD......Page 495
14.5 Random-effects negative binomial......Page 498
RANDOM-EFFECTS POISSON WITH GAMMA EFFECT......Page 500
QUADRATIC NEGATIVE BINOMIAL (NB2)......Page 502
RANDOM-EFFECTS NEGATIVE BINOMIAL WITH BETA EFFECT LOG-LIKELIHOOD......Page 504
BETA-DISTRIBUTED RANDOM-EFFECTS NEGATIVE BINOMIAL......Page 506
14.6.1 Random-intercept negative binomial models......Page 508
RANDOM-EFECTS POISSON, WITH NORMALLY DISTRIBUTED INTERCEPT......Page 509
RANDOM-INTERCEPT POISSON......Page 510
RANDOM-INTERCEPT NEGATIVE BINOMIAL, WITH GAUSSIAN-DISTRIBUTED EFFECTS......Page 511
14.6.2 Non-parametric random-intercept negative binomial......Page 514
14.6.3 Random-coefficient negative binomial models......Page 516
RANDOM-COEFFICIENT POISSON......Page 517
RANDOM-COEFFICIENT NEGATIVE BINOMIAL......Page 518
Summary......Page 520
15.1 Bayesian versus frequentist methodology......Page 522
BAYES’ THEOREM......Page 523
15.2 The logic of Bayesian regression estimation......Page 526
15.3 Applications......Page 530
MAXIMUM LIKELIHOOD......Page 532
BAYESIAN NEGATIVE BINOMIAL – NON-INFORMATIVE UNIFORM......Page 533
BAYESIAN NEGATIVE BINOMIAL – INFORMATIVE PRIOR......Page 534
Appendix A: Constructing and interpreting interaction terms......Page 540
Binary × Binary interactions......Page 541
Binary × Continuous interactions......Page 543
STANDARD ERRORS......Page 544
Significance of interaction......Page 545
Categorical × Continuous interactions......Page 546
GRADUATE SCHOOL......Page 547
Continuous × Continuous interactions......Page 548
Brief look at software......Page 549
Appendix B: Data sets, commands, functions......Page 550
References and further reading......Page 552
Index......Page 561