Regression Analysis of Count Data

Students in both the natural and social sciences often seek regression models to explain the frequency of events, such as visits to a doctor, auto accidents or job hiring. This analysis provides the most comprehensive and up-to-date account of models and methods to interpret such data. The authors combine theory and practice to make sophisticated methods of analysis accessible to practitioners working with widely different types of data and software. The treatment will be useful to researchers in areas such as applied statistics, econometrics, operations research, actuarial studies, demography, biostatistics, and quantitatively-oriented sociology and political science. The book may be used as a reference work on count models or by students seeking an authoritative overview. The analysis is complemented by template programs available on the Internet through the authors' homepages.

List of Figures

xi

(1)

List of Tables

xii

(3)

Preface

xv

1 Introduction

1

(18)

1.1 Poisson Distribution

3

(5)

1.2 Poisson Regression

8

(2)

1.3 Examples

10

(5)

1.4 Overview of Major Issues

15

(2)

1.5 Bibliographic Notes

17

(2)

2 Model Specification and Estimation

19

(40)

2.1 Introduction

19

(1)

2.2 Example and Definitions

20

(2)

2.3 Likelihood-Based Models

22

(5)

2.4 Generalized Linear Models

27

(10)

2.5 Moment-Based Models

37

(7)

2.6 Testing

44

(6)

2.7 Derivations

50

(7)

2.8 Bibliographic Notes

57

(1)

2.9 Exercises

57

(2)

3 Basic Count Regression

59

(37)

3.1 Introduction

59

(2)

3.2 Poisson MLE, PMLE, and GLM

61

(9)

3.3 Negative binomial MLE and QGPMLE

70

(7)

3.4 Overdispersion Tests

77

(2)

3.5 Use of Regression Results

79

(6)

3.6 Ordered and Other Discrete-Choice Models

85

(3)

3.7 Other Models

88

(5)

3.8 Iteratively Reweighted Least Squares

93

(1)

3.9 Bibliographic Notes

94

(1)

3.10 Exercises

95

(1)

4 Generalized Count Regression

96

(43)

4.1 Introduction

96

(1)

4.2 Mixture Models for Unobserved Heterogeneity

97

(9)

4.3 Models Based on Waiting-Time Distributions

106

(6)

4.4 Katz, Double-Poisson, and Generalized Poisson

112

(5)

4.5 Truncated Counts

117

(4)

4.6 Censored Counts

121

(2)

4.7 Hurdle and Zero-Inflated Models

123

(5)

4.8 Finite Mixtures and Latent Class Analysis

128

(6)

4.9 Estimation by Simulation

134

(1)

4.10 Derivations

135

(1)

4.11 Bibliographic Notes

136

(1)

4.12 Exercises

137

(2)

5 Model Evaluation and Testing

139

(50)

5.1 Introduction

139

(1)

5.2 Residual Analysis

140

(11)

5.3 Goodness of Fit

151

(7)

5.4 Hypothesis Tests

158

(5)

5.5 Inference with Finite Sample Corrections

163

(5)

5.6 Conditional Moment Specification Tests

168

(14)

5.7 Discriminating among Nonnested Models

182

(3)

5.8 Derivations

185

(2)

5.9 Bibliographic Notes

187

(1)

5.10 Exercises

188

(1)

6 Empirical Illustrations

189

(32)

6.1 Introduction

189

(1)

6.2 Background

190

(2)

6.3 Analysis of Demand for Health Services

192

(14)

6.4 Analysis of Recreational Trips

206

(10)

6.5 LR Test: A Digression

216

(2)

6.6 Concluding Remarks

218

(1)

6.7 Bibliographic Notes

219

(1)

6.8 Exercises

220

(1)

7 Time Series Data

221

(30)

7.1 Introduction

221

(1)

7.2 Models for Time Series Data

222

(4)

7.3 Static Regression

226

(8)

7.4 Integer-Valued ARMA Models

234

(4)

7.5 Autoregressive Models

238

(2)

7.6 Serially Correlated Error Models

240

(2)

7.7 State-Space Models

242

(2)

7.8 Hidden Markov Models

244

(1)

7.9 Discrete ARMA Models

245

(1)

7.10 Application

246

(2)

7.11 Derivations: Tests of Serial Correlation

248

(2)

7.12 Bibliographic Notes

250

(1)

7.13 Exercises

250

(1)

8 Multivariate Data

251

(24)

8.1 Introduction

251

(1)

8.2 Characterizing Dependence

252

(4)

8.3 Parametric Models

256

(4)

8.4 Moment-Based Estimation

260

(3)

8.5 Orthogonal Polynomial Series Expansions

263

(6)

8.6 Mixed Multivariate Models

269

(3)

8.7 Derivations

272

(1)

8.8 Bibliographic Notes

273

(2)

9 Longitudinal Data

275

(26)

9.1 Introduction

275

(1)

9.2 Models for Longitudinal Data

276

(4)

9.3 Fixed Effects Models

280

(7)

9.4 Random Effects Models

287

(3)

9.5 Discussion

290

(3)

9.6 Specification Tests

293

(1)

9.7 Dynamic and Transition Models

294

(5)

9.8 Derivations

299

(1)

9.9 Bibliographic Notes

300

(1)

9.10 Exercises

300

(1)

10 Measurement Errors

301

(25)

10.1 Introduction

301

(1)

10.2 Measurement Errors in Exposure

302

(5)

10.3 Measurement Errors in Regressors

307

(2)

10.4 Measurement Errors in Counts

309

(4)

10.5 Underreported Counts

313

(10)

10.6 Derivations

323

(1)

10.7 Bibliographic Notes

324

(1)

10.8 Exercises

325

(1)

11 Nonrandom Samples and Simultaneity

326

(18)

11.1 Introduction

326

(1)

11.2 Alternative Sampling Frames

326

(5)

11.3 Simultaneity

331

(5)

11.4 Sample Selection

336

(7)

11.5 Bibliographic Notes

343

(1)

12 Flexible Methods for Counts

344

(27)

12.1 Introduction

344

(1)

12.2 Efficient Moment-Based Estimation

345

(5)

12.3 Flexible Distributions Using Series Expansions

350

(6)

12.4 Flexible Models of Conditional Mean

356

(2)

12.5 Flexible Models of Conditional Variance

358

(6)

12.6 Example and Model Comparison

364

(3)

12.7 Derivations

367

(1)

12.8 Count Models: Retrospect and Prospect

367

(2)

12.9 Bibliographic Notes

369

(2)

Appendices:

371

(8)

A Notation and Acronyms

371

(3)

B Functions, Distributions, and Moments

374

(4)

B.1 Gamma Function

374

(1)

B.2 Some Distributions

375

(1)

B.3 Moments of Truncated Poisson

376

(2)

C Software

378

(1)

References

379

(20)

Author Index

399

(5)

Subject Index

404

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Regression Analysis of Count Data

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