Preface

Chapter 1

Varieties of Count Data

Some Points of Discussion

1.1 What Are Counts?

1.2 Understanding a Statistical Count Model

1.2.1 Basic Structure of a Linear Statistical Model

1.2.2 Models and Probability

1.2.3 Count Models

1.2.4 Structure of a Count Model

1.3 Varieties of Count Models

1.4 Estimation – the Modeling Process

1.4.1 Software for Modeling

1.4.2 Maximum Likelihood Estimation

1.4.3 Generalized Linear Models and IRLS Estimation

1.5 Summary

Chapter 2

Poisson Regression

Some Points of Discussion

2.1 Poisson Model Assumptions

2.2 Apparent Overdispersion

2.3 Constructing a "True" Poisson Model

2.4 Poisson Regression: Modeling Real Data

2.5 Interpreting Coefficients and Rate Ratios

2.5.1 How to Interpret a Poisson Coefficient and Associated Statistics

2.5.2 Rate Ratios and Probability

2.6 Exposure: Modeling over Time, Area, and Space

2.7 Prediction

2.8 Poisson Marginal Effects

2.8.1 Marginal Effect at the Mean

2.8.2 Average Marginal Effects

2.8.3 Discrete Change or Partial Effects

2.9 Summary

Chapter 3

Testing Overdispersion

Some Points of Discussion

3.1 Basics of Count Model Fit Statistics

3.2 Overdispersion: What, Why, and How

3.3 Testing Overdispersion

3.3.1 Score Test

3.3.2 Lagrange Multiplier Test

3.3.3 Chi2 Test: Predicted versus Observed Counts

3.4 Methods of Handling Overdispersion

3.4.1 Scaling Standard Errors: Quasi-count Models

3.4.2 Quasi-likelihood Models

3.4.3 Sandwich or Robust Variance Estimators

3.4.4 Bootstrapped Standard Errors

3.5 Summary

Chapter 4

Assessment of Fit

Some Points of Discussion

4.1 Analysis of Residual Statistics

4.2 Likelihood Ratio Test

4.2.1 Standard Likelihood Ratio Test

4.2.2 Boundary Likelihood Ratio Test

4.3 Model Selection Criteria

4.3.1 Akaike Information Criterion

4.3.2. Bayesian Information Criterion

4.4 Setting up and Using a Validation Sample

4.5 Summary and an Overview of the Modeling Process

4.5.1 Summary of What We Have Thus Far Discussed

Chapter 5

Negative Binomial Regression

Some Points of Discussion

5.1 Varieties of Negative Binomial Models

5.2 Negative Binomial Model Assumptions

5.2.1 A Word Regarding Parameterization of the Negative Binomial

5.3 Two Modeling Examples

5.3.1 Example: **rwm1984**

5.3.2 Example: **medpar**

5.4 Additional Tests

5.4.1 General Negative Binomial Fit Tests

5.4.2 Adding a Parameter – NB-P Negative Binomial

5.4.3 Modeling the Dispersion – Heterogeneous Negative Binomial

5.5 Summary

Chapter 6

Poisson Inverse Gaussian Regression

Some Points of Discussion

6.1 Poisson Inverse Gaussian Model Assumptions

6.2 Constructing and Interpreting the PIG Model

6.2.1 Software Considerations

6.2.2 Examples

6.3 Summary – Comparing Poisson, NB, and PIG Models

Chapter 7

Problems with Zeros

Some Points of Discussion

7.1 Counts without Zeros – Zero-Truncated Models

7.1.1 Zero-Truncated Poisson (ZTP)

7.1.2 Zero-Truncated Negative Binomial (ZTNB)

7.1.3 Zero-Truncated Poisson Inverse Gaussian (ZTPIG)

7.1.4 Zero-Truncated NB-P (ZTNBP)

7.1.5 Zero-Truncated Poisson Log-Normal (ZTPLN)

7.1.6 Zero-Truncated Model Summary

7.2 Two-Part Hurdle Models

7.2.1 Poisson and Negative Binomial Logit Hurdle Models

7.2.2 PIG-Logit and Poisson Log-Normal Hurdle Models

7.2.3 PIG-Poisson Hurdle Model

7.3 Zero-Inflated Mixture Models

7.3.1 Overview of Guidelines

7.3.2 Fit Tests for Zero-Inflated Models

7.3.3 Fitting Zero-Inflated Models

7.3.4 Good and Bad Zeros

7.3.5 Zero-Inflated Poisson (ZIP)

7.3.6 Zero-Inflated Negative Binomial (ZINB)

7.3.7 Zero-Inflated Poisson Inverse Gaussian (ZIPIG)

7.4 Summary – Finding the Optimal Model

Chapter 8

Modeling Underdispersed Count Data – Generalized Poisson

Some Points of Discussion

Chapter 9

Complex Data: More Advanced Models

Types of Data and Problems Dealt with in This Chapter

9.1 Small and Unbalanced Data – Exact Poisson Regression

9.2 Modeling Truncated and Censored Counts

9.2.1 Truncated Count Models

9.2.2 Censored Count Models

9.2.3 Poisson-Logit Hurdle at 3 Model

9.3 Counts with Multiple Components – Finite Mixture Models

9.4 Adding Smoothing Terms to a Model – GAM

9.5 When All Else Fails: Quantile Count Models

9.6 A Word about Longitudinal and Clustered Count Models

9.6.1 Generalized Estimating Equations (GEEs)

9.6.2 Mixed-Effects and Multilevel Models

9.7 Three-Parameter Count Models

9.8 Bayesian Count Models – Future Directions of Modeling?

9.9 Summary

Appendix: SAS Code

Bibliography

Index