Publication Date: March 8, 2011 | ISBN-10: 0521198151 | ISBN-13: 978-0521198158 | Edition: 2
This second edition of Hilbe's Negative Binomial Regression is a substantial enhancement to the popular first edition. The only text devoted entirely to the negative binomial model and its many variations, nearly every model discussed in the literature is addressed. The theoretical and distributional background of each model is discussed, together with examples of their construction, application, interpretation and evaluation. Complete Stata and R codes are provided throughout the text, with additional code (plus SAS), derivations and data provided on the book's website. Written for the practising researcher, the text begins with an examination of risk and rate ratios, and of the estimating algorithms used to model count data. The book then gives an in-depth analysis of Poisson regression and an evaluation of the meaning and nature of overdispersion, followed by a comprehensive analysis of the negative binomial distribution and of its parameterizations into various models for evaluating count data.



Review"As with all of Joe Hilbe's books this text is thorough and scholarly with an extensive list of references. The text is well-written and for the most part easy to understand."
Michael R. Chernick, Significance

Book DescriptionA substantial enhancement to the popular first edition, which remains the only text devoted entirely to the negative binomial model and its many variations. Written for the practising researcher, the book discusses the theoretical and distributional background of each model with numerous examples of their application.
Hardcover: 576 pages Publisher: Cambridge University Press; 2 edition (March 8, 2011) Language: English ISBN-10: 0521198151 ISBN-13: 978-0521198158

Preface to the s econd edition p age xi
1 I ntroduction 1
1.1 What is a negative binomial model? 1
1.2 A brief history of the negative binomial 5
1.3 Overview of the book 11
2 The concept of risk 15
2.1 Risk and 2 ×2 tables 15
2.2 Risk and 2 ×k tables 18
2.3 Risk ratio confidence intervals 20
2.4 Risk difference 24
2.5 The relationship of risk to odds ratios 25
2.6 Marginal probabilities: joint and conditional 27
3 Overview of count response models 30
3.1 Varieties of count response model 30
3.2 Estimation 38
3.3 Fit considerations 41
4 Methods of estimation 43
4.1 Derivation of the IRLS algorithm 43
4.1.1 Solving for ∂ L or U – the gradient 48
4.1.2 Solving for ∂
2
L 49
4.1.3 The IRLS fitting algorithm 51
4.2 Newton–Raphson algorithms 53
4.2.1 Derivation of the Newton–Raphson 54
4.2.2 GLM with OIM 57

4.2.3 Parameterizing fromµ to x

β 57
4.2.4 Maximum likelihood estimators 59
5 Assessment of count models 61
5.1 Residuals for count response models 61
5.2 Model fit tests 64
5.2.1 Traditional fit tests 64
5.2.2 Information criteria fit tests 68
5.3 Validation models 75
6 Poisson regression 77
6.1 Derivation of the Poisson model 77
6.1.1 Derivation of the Poisson from the binomial
distribution 78
6.1.2 Derivation of the Poisson model 79
6.2 Synthetic Poisson models 85
6.2.1 Construction of synthetic models 85
6.2.2 Changing response and predictor values 94
6.2.3 Changing multivariable predictor values 97
6.3 Example: Poisson model 100
6.3.1 Coefficient parameterization 100
6.3.2 Incidence rate ratio parameterization 109
6.4 Predicted counts 116
6.5 Effects plots 122
6.6 Marginal effects, elasticities, and discrete change 125
6.6.1 Marginal effects for Poisson and negative binomial
effects models 125
6.6.2 Discrete change for Poisson and negative
binomial models 131
6.7 Parameterization as a rate model 134
6.7.1 Exposure in time and area 134
6.7.2 Synthetic Poisson with offset 136
6.7.3 Example 138
7 Overdispersion 141
7.1 What is overdispersion? 141
7.2 Handling apparent overdispersion 142
7.2.1 Creation of a simulated base Poisson model 142
7.2.2 Delete a predictor 145
7.2.3 Outliers in data 145
7.2.4 Creation of interaction 149

7.2.5 Testing the predictor scale 150
7.2.6 Testing the link 152
Methods of handling r eal overdispersion 157
7.3.1 Scaling of standard errors / quasi-Poisson 158
7.3.2 Quasi-likelihood variance multipliers 163
7.3.3 Robust variance estimators 168
7.3.4 Bootstrapped and jackknifed s tandard errors 171
Tests of overdispersion 174
7.4.1 Score and Lagrange multiplier tests 175
7.4.2 Boundary likelihood ratio test 177
7.4.3 R
2
p
and R
2
pd
tests for Poisson and negative
binomial models 179
Negative binomial overdispersion 180
Negative binomial regression 185
Varieties of negative binomial 185
Derivation of the negative binomial 187
8.2.1 Poisson–gamma mixture model 188
8.2.2 Derivation of the GLM negative binomial 193
Negative binomial distributions 199
Negative binomial algorithms 207
8.4.1 NB-C: canonical negative binomial 208
8.4.2 NB2: expected information matrix 210
8.4.3 NB2: observed information matrix 215
8.4.4 NB2: R maximum likelihood function 218
Negative binomial regression: modeling 221
Poisson versus negative binomial 221
Synthetic negative binomial 225
Marginal effects and discrete change 236
Binomial versus count models 239
Examples: negative binomial regression 248
Example 1: Modeling number of marital affairs 248
Example 2: Heart procedures 259
Example 3: Titanic s urvival data 263
Example 4: Health reform data 269
Alternative variance parameterizations 284
Geometric regression: NB α = 1 285
10.1.1 Derivation of the geometric 285
10.1.2 Synthetic geometric models 286

10.1.3 Using the geometric model 290
10.1.4 The canonical geometric model 294
10.2 NB1: The linear negative binomial model 298
10.2.1 NB1 as QL-Poisson 298
10.2.2 Derivation of NB1 301
10.2.3 Modeling with NB1 304
10.2.4 NB1: R maximum likelihood function 306
10.3 NB-C: Canonical negative binomial regression 308
10.3.1 NB-C overview and formulae 308
10.3.2 Synthetic NB-C models 311
10.3.3 NB-C models 315
10.4 NB-H: Heterogeneous negative binomial regression 319
10.5 The NB-P model: generalized negative binomial 323
10.6 Generalized Waring regression 328
10.7 Bivariate negative binomial 333
10.8 Generalized Poisson regression 337
10.9 Poisson inverse Gaussian regression (PIG) 341
10.10 Other count models 343
11 Problems with zero counts 346
11.1 Zero-truncated count models 346
11.2 Hurdle models 354
11.2.1 Theory and formulae for hurdle models 356
11.2.2 Synthetic hurdle models 357
11.2.3 Applications 359
11.2.4 Marginal effects 369
11.3 Zero-inflated negative binomial models 370
11.3.1 Overview of ZIP/ZINB models 370
11.3.2 ZINB algorithms 371
11.3.3 Applications 374
11.3.4 Zero-altered negative binomial 376
11.3.5 Tests of comparative fit 377
11.3.6 ZINB marginal effects 379
11.4 Comparison of models 382
12 Censored and truncated count models 387
12.1 Censored and truncated models – econometric
parameterization 387
12.1.1 Truncation 388
12.1.2 Censored models 395

12.2 Censored Poisson and NB2 models – survival
parameterization 399
13 Handling endogeneity and latent class models 407
13.1 Finite mixture models 408
13.1.1 Basics of finite mixture modeling 408
13.1.2 Synthetic finite mixture models 412
13.2 Dealing with endogeneity and latent class models 416
13.2.1 Problems related to endogeneity 416
13.2.2 Two-stage instrumental variables approach 417
13.2.3 Generalized method of moments (GMM) 421
13.2.4 NB2 with an endogenous multinomial treatment
variable 422
13.2.5 Endogeneity resulting from measurement error 425
13.3 Sample s election and stratification 428
13.3.1 Negative binomial with endogenous stratification 429
13.3.2 Sample s election models 433
13.3.3 Endogenous switching models 438
13.4 Quantile count models 441
14 Count panel models 447
14.1 Overview of count panel models 447
14.2 Generalized estimating equations: negative binomial 450
14.2.1 The GEE algorithm 450
14.2.2 GEE correlation structures 452
14.2.3 Negative binomial GEE models 455
14.2.4 GEE goodness-of-fit 464
14.2.5 GEE marginal effects 466
14.3 Unconditional fixed-effects negative binomial model 468
14.4 Conditional fixed-effects negative binomial model 474
14.5 Random-effects negative binomial 478
14.6 Mixed-effects negative binomial models 488
14.6.1 Random-intercept negative binomial models 488
14.6.2 Non-parametric r andom-intercept negative binomial 494
14.6.3 Random-coefficient negative binomial models 496
14.7 Multilevel models 500
15 Bayesian negative binomial models 502
15.1 Bayesian versus frequentist methodology 502
15.2 The logic of Bayesian regression estimation 506
15.3 Applications 510

Appendix A: Constructing and interpreting interaction terms 520
Appendix B: Data s ets, commands, functions 530
References and further reading 532
Index 541