Preface to the second edition

1. Introduction

1.1 What is a negative binomial model?

1.2 A brief history of the negative binomial

1.3 Overview of the book

2. The concept of risk

2.1 Risk and 2 × 2 tables

2.2 Risk and 2 × *k* tables

2.3 Risk ratio confidence intervals

2.4 Risk difference

2.5 The relationship of risk to odds ratios

2.6 Marginal probabilities: joint and conditional

3. Overview of count response models

3.1 Varieties of count response model

3.2 Estimation

3.3 Fit considerations

4. Methods of estimation

4.1 Derivation of the IRLS algorithm

4.1.1 Solving for ∂ L or *U* — the gradient

4.1.2 Solving for ∂^{2} L

4.1.3 The IRLS fitting algorithm

4.2 Newton–Raphson algorithms

4.2.1 Derivation of the Newton–Raphson

4.2.2 GLM with OIM

4.2.3 Parameterizing from *μ* to *x*′*Β*

4.2.4 Maximum likelihood estimators

5. Assessment of count models

5.1 Residuals for count response models

5.2 Model fit tests

5.2.1 Traditional fit tests

5.2.2 Information criteria fit tests

5.3 Validation models

6. Poisson regression

6.1 Derivation of the Poisson model

6.1.1 Derivation of the Poisson from the binomial distribution

6.1.2 Derivation of the Poisson model

6.2 Synthetic Poisson models

6.2.1 Construction of synthetic models

6.2.2 Changing response and predictor values

6.2.3 Changing multivariable predictor values

6.3 Example: Poisson model

6.3.1 Coefficient parameterization

6.3.2 Incidence rate ratio parameterization

6.4 Predicted counts

6.5 Effects plots

6.6 Marginal effects, elasticities, and discrete change

6.6.1 Marginal effects for Poisson and negative binomial effects models

6.6.2 Discrete change for Poisson and negative binomial models

6.7 Parameterization as a rate model

6.7.1 Exposure in time and area

6.7.2 Synthetic Poisson with offset

6.7.3 Example

7. Overdispersion

7.1 What is overdispersion?

7.2 Handling apparent overdispersion

7.2.1 Creation of a simulated base Poisson model

7.2.2 Delete a predictor

7.2.3 Outliers in data

7.2.4 Creation of interaction

7.2.5 Testing the predictor scale

7.2.6 Testing the link

7.3 Methods of handling real overdispersion

7.3.1 Scaling of standard errors / quasi-Poisson

7.3.2 Quasi-likelihood variance multipliers

7.3.3 Robust variance estimators

7.3.4 Bootstrapped and jackknifed standard errors

7.4 Tests of overdispersion

7.4.1 Score and Lagrange multiplier tests

7.4.2 Boundary likelihood ratio test

7.4.3 *R*^{2}_{p} and *R*^{2}_{pd} tests for Poisson and negative binomial models

7.5 Negative binomial overdispersion

8. Negative binomial regression

8.1 Varieties of negative binomial

8.2 Derivation of the negative binomial

8.2.1 Poisson–gamma mixture model

8.2.2 Derivation of the GLM negative binomial

8.3 Negative binomial distributions

8.4 Negative binomial algorithms

8.4.1 NB-C: canonical negative binomial

8.4.2 NB2: expected information matrix

8.4.3 NB2: observed information matrix

8.4.4 NB2: R maximum likelihood function

9. Negative binomial regression: modeling

9.1 Poisson versus negative binomial

9.2 Synthetic negative binomial

9.3 Marginal effects and discrete change

9.4 Binomial versus count models

9.5 Examples: negative binomial regression

Example 1: Modeling number of marital affairs

Example 2: Heart procedures

Example 3: Titanic survival data

Example 4: Health reform data

10. Alternative variance parameterizations

10.1 Geometric regression: NB α = 1

10.1.1 Derivation of the geometric

10.1.2 Synthetic geometric models

10.1.3 Using the geometric model

10.1.4 The canonical geometric model

10.2 NB1: The linear negative binomial model

10.2.1 NB1 as QL-Poisson

10.2.2 Derivation of NB1

10.2.3 Modeling with NB1

10.2.4 NB1: R maximum likelihood function

10.3 NB-C: Canonical negative binomial regression

10.3.1 NB-C overview and formulae

10.3.2 Synthetic NB-C models

10.3.3 NB-C models

10.4 NB-H: Heterogeneous negative binomial regression

10.5 The NB-P model: generalized negative binomial

10.6 Generalized Waring regression

10.7 Bivariate negative binomial

10.8 Generalized Poisson regression

10.9 Poisson inverse Gaussian regression (PIG)

10.10 Other count models

11. Problems with zero counts

11.1 Zero-truncated count models

11.2 Hurdle models

11.2.1 Theory and formulae for hurdle models

11.2.2 Synthetic hurdle models

11.2.3 Applications

11.2.4 Marginal effects

11.3 Zero-inflated negative binomial models

11.3.1 Overview of ZIP/ZINB models

11.3.2 ZINB algorithms

11.3.3 Applications

11.3.4 Zero-altered negative binomial

11.3.5 Tests of comparative fit

11.3.6 ZINB marginal effects

11.4 Comparison of models

12. Censored and truncated count models

12.1 Censored and truncated models — econometric parameterization

12.1.1 Truncation

12.1.2 Censored models

12.2 Censored Poisson and NB2 models — survival parameterization

13. Handling endogeneity and latent class models

13.1 Finite mixture models

13.1.1 Basics of finite mixture modeling

13.1.2 Synthetic finite mixture models

13.2 Dealing with endogeneity and latent class models

13.2.1 Problems related to endogeneity

13.2.2 Two-stage instrumental variables approach

13.2.3 Generalized method of moments (GMM)

13.2.4 NB2 with an endogenous multinomial treatment variable

13.2.5 Endogeneity resulting from measurement error

13.3 Sample selection and stratification

13.3.1 Negative binomial with endogenous stratification

13.3.2 Sample selection models

13.3.3 Endogenous switching models

13.4 Quantile count models

14. Count panel models

14.1 Overview of count panel models

14.2 Generalized estimating equations: negative binomial

14.2.1 The GEE algorithm

14.2.2 GEE correlation structures

14.2.3 Negative binomial GEE models

14.2.4 GEE goodness-of-fit

14.2.5 GEE marginal effects

14.3 Unconditional fixed-effects negative binomial model

14.4 Conditional fixed-effects negative binomial model

14.5 Random-effects negative binomial

14.6 Mixed-effects negative binomial models

14.6.1 Random-intercept negative binomial models

14.6.2 Non-parametric random-intercept negative binomial

14.6.3 Random-coefficient negative binomial models

14.7 Multilevel models

15. Bayesian negative binomial models

15.1 Bayesian versus frequentist methodology

15.2 The logic of Bayesian regression estimation

15.3 Applications

Appendix A: Constructing and interpreting interaction terms

Appendix B: Data sets, commands, functions

References and further reading

Index