负二项回归分析指南：第二版

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"负二项回归(第2版) Negative binomial regression，2nd ed" 负二项回归是一种在统计学中广泛使用的分析方法，尤其在处理计数数据时，它被视为广义线性回归（GLM）的一个重要组成部分。GLM包括了多种回归模型，如逻辑斯谛回归（Logistic Regression）和泊松回归（Poisson Regression）。负二项回归适用于那些因变量是非负整数计数，并且存在过度分散（overdispersion）的情况。过度分散是指实际观测到的数据方差大于泊松回归模型预测的方差，这通常意味着模型需要考虑更多的变异性。 Joseph Michael Hilbe的《负二项回归》第二版是对这一主题的深入探讨，它详细阐述了负二项回归模型的基本原理及其各种变体。这本书不仅涵盖了理论基础，还提供了丰富的实际案例，通过Stata和R软件进行详细解释。此外，书中也包含了一些使用SAS和LIMDEP软件的例子，使得不同软件的用户都能找到适用的方法。在书的开篇，作者讨论了风险和风险比的概念，这是理解回归分析的重要基础。接下来，书中对基本的泊松模型进行了详尽的分析，然后转向过度分散的主题，探讨其含义和范围。通过模拟实验和真实数据的分析，书中的例子贯穿始终，旨在帮助读者清晰地理解各种模型的构建、解释以及比较评估。负二项回归的变体包括调整模型以适应不同的数据特性和研究问题，例如，可以引入滞后效应、异方差性或时间序列结构。此外，负二项回归还常用于社会科学的多个领域，如经济学、社会学、公共卫生和生物统计学等，用来研究如疾病发病率、交通事故次数、顾客满意度调查等计数数据。本书的新版增加了许多实用的特性，比如增加了与R软件的结合，这使得它成为研究者和数据分析人员的理想手册，无论他们是在选择模型、构建模型、解读结果，还是进行不同模型之间的比较，都能从中受益匪浅。通过对负二项回归的深入理解和实践，读者能够更好地应对实际研究中的复杂计数数据问题。

xiv Preface to the second edition

In the ﬁrst edition I provided the reader with Stata code to create Poisson,

NB2 and NB1 synthetic count models. We now examine the nature of these

types of synthetic models and describe how they can be used to understand

the relationship of models to data. In addition, synthetic models are provided

to estimate two-part synthetic hurdle models. This code can be useful as a

paradigm for the optimization or maximum likelihood estimation of more

complex count models. Also discussed in this edition are marginal effects and

discrete change, which have a prominent place in econometrics, but which can

also be used with considerable value in other disciplines. I provide details on

how to construct and interpret both marginal effects and discrete change for the

major models discussed in the text.

New models added to the text from the ﬁrst edition include: ﬁnite mix-

ture models, quantile count models, bivariate negative binomial, and various

methods used to model endogeneity. We previously examined negative bino-

mial sample selection models and negative binomial models with endogenous

stratiﬁcation, but we now add instrumental variables, generalized method of

moments, and methods of dealing with predictors having missing values. Stata

now supports these models.

Finally, Stata 11 appeared in late July 2009, offering several capabilities

related to our discussion which were largely unavailable in the version used

with the ﬁrst edition of this text. In particular, Stata’s glm command now

allows maximum likelihood estimation of the negative binomial heterogeneity

parameter, which it did not in earlier versions. R’s glm function and SAS’s

STAT/GENMOD procedure also provide the same capability. This option

allows easier estimation and comparative evaluation of NB2 models.

I created the code and provided explanations for the development and use of

a variety of synthetic regression models used in the text. Most were originally

written using pseudo-random number generators I published in 1995, but a few

were developed a decade later. When Stata offered their own suite of pseudo-

random number generators in 2009, I re-wrote the code for constructing these

synthetic models using Stata’s code. This was largely done in the ﬁrst t wo

months of 2009. The synthetic models appearing in the ﬁrst edition of this

book, which are employed in this edition, now use the new code, as do the

other synthetic models. In fact, Stata code for most of the synthetic models

appearing in this text was published in the Stata Journal (Volume 10:1, pages

104–124). Readers are referred to that source for additional explanation on

constructing synthetic models, including many binomial models that are not in

this book. It should be noted that the synthetic Stata models discussed in the

text have corresponding R scripts provided to duplicate model results.

New subjects discussed in the second edition xv

R data ﬁles, functions, and scripts written for this text are available in the

COUNT package that can be downloaded on CRAN. I recommend Muenchen

and Hilbe (2010), R for Stata Users (Springer) for Stata users who wish to

understand the logic of R functions. For those who wish to learn more about

Stata, and how it can be used for data management and programming, I refer you

to the books published by Stata Press. See www.stata-press.com . The websites

which readers can access to download data ﬁles and user-authored commands

and functions are:

Cambridge University Press: www.cambridge.org/9780521198158

Stata bookstore (Stata ﬁles/commands):

http://www.Stata.com/bookstore/nbr2.html

Errata and a post-publication Negative Binomial Regression Extensions can

be found on the above sites, as well as at: http://www.statistics.com/

hilbe/nbr.php. Additional code, text, graphs and tables for the book are

available in this electronic document.

I should mention that practicing researchers rarely read this type of text from

beginning straight through to the end. Rather, they tend to view it as a type of

handbook where a model they wish to use can be reviewed for modeling details.

The text can indeed be used in such a manner. However, most of the chapters

are based on information contained in earlier chapters. Material explained in

Chapters 2, 5, 6, and 7 is particularly important for mastery of the discussion

of negative binomial models presented in Chapters 8 and 9. Knowledge of the

material given in Chapters 7 and 8 is fundamental to understanding extended

Poisson and negative binomial models. Because of this, I encourage you to make

an exception and to read sequentially though the text, at least through Chapter 9.

For those who have no interest at all in the derivation of count models, or of the

algorithms by which these models are estimated, I suggest that you only skim

through Chapter 4. Analogical to driving a car and understanding automotive

mechanics, one can employ a sophisticated negative binomial model without

understanding exactly how estimation is achieved. But, again as with a car,

if you wish to re-parameterize a model or to construct an entirely new count

model, understanding the discussion in Chapter 4 is essential. However, I have

reiterated certain important concepts and relationships throughout the text in

order to provide background for those who have not been able to read earlier

chapters, and to emphasize them for those who may ﬁnd reminding to be

helpful.

xvi Preface to the second edition

To those who made this a better text

I wish to acknowledge several individuals who have contributed speciﬁcally

to the second edition of Negative Binomial Regression. Chris Dorger of Intel

kindly read through the manuscript, identifying typos and errors, and re-derived

the derivations and equations presented in the text in order to make certain that

they are expressed correctly. He primarily checked Chapters 2 through 10.

Dr. Dorger provided assistance as the book was just forming, and at the end,

prior to submission of the manuscript to the publisher. He spent a substan-

tial amount of time working on this project. Dr. Tad Hogg, a physicist with

the Institute of Molecular Manufacturing, spent many hours doing the same,

and provided excellent advice regarding the clariﬁcation of particular subjects.

James Hardin, with whom I have co-authored two texts on statistical modeling

and whom I consider a very good friend, read over the manuscript close to the

end, providing valuable insight and help, particularly with respect to Chapters

13 and 14. Andrew Robinson of the University of Melbourne provided invalu-

able help in setting up the COUNT package on CRAN, which has all of the R

functions and scripts for the book, as well as data in the form of data frames.

He also helped in re-designing some of the functions I developed to run more

efﬁciently. We are now writing a book together on the Methods of Statistical

Model Estimation (Chapman & Hall/CRC). Gordon Johnston of SAS Institute,

author of the GENMOD procedure and longtime friend, provided support on

SAS capabilities related to GEE and Bayesian models. He is responsible for

developing the SAS programs used for the examples in Section 15.3, and read

over Sections 15.1 and 15.2, offering insightful advice and making certain that

the concepts expressed were correct. Allan Medwick, who recently earned a

Ph.D. in statistics at the University of Pennsylvania read carefully through the

entire manuscript near the end, checking for readability and typos; he also

helped running the SAS models in Chapter 14. The help of these statisticians

in ﬁne-tuning the text has been invaluable, and I deeply appreciate their assis-

tance. I am alone responsible for any remaining errors, but readers should be

aware that every attempt has been made to assure accuracy.

I also thank Professor James Albert of Bowling Green University for his

insightful assistance in helping develop R code for Monte Carlo simulations.

Robert Muenchen, co-author with me of R for Stata Users (2010), assisted me

in various places to streamline R code, and at times to make sure my code did

exactly as I had intended. The efforts of these two individuals also contributed

much to the book’s value.

Again, as in the ﬁrst edition, I express my gratitude and friendship to the late

John Nelder, with whom I have discussed these topics for some 20 years. In fact,

To those who made this a better text xvii

I ﬁrst came to the idea of including the negative binomial family in standard

generalized linear models software in 1992 while John and I hiked down and

back up the Grand Canyon Angel Trail together. We had been discussing his

newly developed kk add-ons to GenStat software, which included rudimentary

negative binomial code, when it became clear to me that the negative binomial,

with its heterogeneity parameter as a constant, is a full-ﬂedged member of the

single parameter exponential family of distributions, and as such warranted

inclusion as a GLM family member. It was easy to then parameterize the

negative binomial probability distribution into exponential family form, and

abstract the requisite GLM link, mean, variance, and inverse link functions –

as any other GLM family member. It was also clear that this was not the

traditional negative binomial, but that the latter could be developed by simple

transformations. His visits to Arizona over the years, and our long discussions,

have led to many of the approaches I now take to GLM–related models, even

where a few of my ﬁndings have run counter to some of the positions he initially

maintained.

I must also acknowledge the expertise of Robert Rigby and Mikis

Stasinopoulos, authors of R’s gamlss suite of functions, who upon my request

rewrote part of the software in such a manner that it can now be used to

estimate censored and truncated count models, which were not previously

available using R. These capabilities are now part of gamlss on CRAN. Also

to be acknowledged is Masakazu Iwasaki, Clinical Statistics Group Manager

for Research & Development at Schering-Plough K.K., Tokyo, who re-worked

code he had earlier developed into a viable R function for estimating bivari-

ate negative binomial regression. He amended it expressely for this text; it is

currently the only bivariate negative binomial function of which I am aware,

and is available in the COUNT package. Malcolm Faddy and David Smith pro-

vided code and direction regarding the implementation of Faddy’s namesake

distribution for the development of a new count model for both under- and

overdispersion. Robert LaBudde, President of Least Cost Formulations, Ltd.,

and adjunct professor of statistics at Old Dominion University, provided very

useful advice related to R programming, for which I am most grateful. The

above statisticians have helped add new capabilities to R, and have made this

text a more valuable resource as a result.

The various participants in my Statistics.com web course, Modeling Count

Data, should be recognized. Nearly all of them are professors teaching statistics

courses in various disciplines, or researchers working in the corporate world or

for a governmental agency. Their many questions and comments have helped

shape this second edition. Elizabeth Kelly of the Los Alamos National Labo-

ratories in New Mexico and Kim Dietrich, University of Washington, provided

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