COUNT DATA ANALYSIS USING POISSON REGRESSION...

COUNT DATA ANALYSIS USING POISSON REGRESSION

AND HANDLING OF OVERDISPERSION

RAIHANA BINTI ZAINORDIN

UNIVERSITI TEKNOLOGI MALAYSIA

COUNT DATA ANALYSIS USING POISSON REGRESSION AND HANDLING OF OVERDISPERSION

RAIHANA BINTI ZAINORDIN

A dissertation submitted in partial fulfillment of the requirements for the award of the degree of

Master of Science (Mathematics)

Faculty of Science Universiti Teknologi Malaysia

NOVEMBER 2009

iii

To Mak and Abah

because I could not have made it without you both.

And to sunflower

because life should be meaningful.

iv

ACKNOWLEDGEMENT

First and foremost, all praise be to Allah, the Almighty, the Benevolent for His

blessings and guidance for giving me the inspiration to embark on this project and

instilling in me the strength to see that this thesis becomes a reality.

I would like to say a big thank you to my amazing supervisor, P.M. Dr. Robiah

binti Adnan for believing in me to finish up this dissertation. Her advice, encouragement

and help mean a lot to me. So I can never thank her enough.

I would also like to thank my friend Ehsan who is never tired of helping and

teaching me things that I do not know.

As always, I am forever grateful to my parents for all their love and sacrifice. To

my sisters (Nordiana, Noraiman Amalina, Nurul Aina), thanks for giving me the reason

to laugh and cry.

A heartfelt thanks goes to all my friends who are always willing to help and

support me. And finally to my best friend, Ikhwan, thank you so much for being there.

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ABSTRACT

Count data is very common in various fields such as in biomedical science,

public health and marketing. Poisson regression is widely used to analyze count data. It

is also appropriate for analyzing rate data. Poisson regression is a part of class of models

in generalized linear models (GLM). It uses natural log as the link function and models

the expected value of response variable. The natural log in the model ensures that the

predicted values of response variable will never be negative. The response variable in

Poisson regression is assumed to follow Poisson distribution. One requirement of the

Poisson distribution is that the mean equals the variance. In real-life application,

however, count data often exhibits overdispersion. Overdipersion occurs when the

variance is significantly larger than the mean. When this happens, the data is said to be

overdispersed. Overdispersion can cause underestimation of standard errors which

consequently leads to wrong inference. Besides that, test of significance result may also

be overstated. Overdispersion can be handled by using quasi-likelihood method as well

as negative binomial regression. The simulation study has been done to see the

performance of Poisson regression and negative binomial regression in analyzing data

that has no overdispersion as well as data that has overdispersion. The results show that

Poisson regression is most appropriate for data that has no overdispersion while negative

binomial regression is most appropriate for data that has overdispersion.

vi

ABSTRAK

Data bilangan adalah sangat lazim dalam pelbagai bidang, contohnya bidang

sains bioperubatan, kesihatan awam dan bidang pemasaran. Regresi Poisson digunakan

secara meluas untuk menganalisis data bilangan. Regresi Poisson juga sesuai untuk

menganalisis data kadaran. Regresi Poisson merupakan sebahagian daripada model kelas

model linear teritlak. Regresi ini menggunakan logaritma asli sebagai fungsi hubungan.

Regresi ini memodelkan nilai jangkaan bagi pembolehubah maklum balas. Logaritma

asli digunakan untuk memastikan supaya nilai ramalan bagi pembolehubah maklumbalas

tidak akan berbentuk negatif. Pembolehubah maklumbalas dalam regresi Poisson

dianggap mengikut taburan Poisson. Salah satu ciri taburan Poisson ialah nilai min

pembolehubah adalah sama dengan nilai varians. Walaubagaimanapun, dalam aplikasi

sebenar, data bilangan sering mempamerkan masalah lebih serakan. Masalah lebih

serakan terjadi apabila nilai varians melebihi nilai min. Apabila ini terjadi, sesebuah data

itu dikatakan terlebih serak. Masalah lebih serakan boleh menyebabkan kurang anggaran

terhadap sisihan piawai yang kemudiannya memberi inferens yang salah. Selain

daripada itu, keputusan ujian signifikan pula akan terlebih anggar. Masalah lebih serakan

boleh diatasi dengan menggunakan kaedah kebolehjadian quasi dan juga regresi

binomial negatif. Kajian simulasi telah dibuat untuk melihat keputusan regresi Poisson

dan regresi binomial negatif dalam menganalisis data yang tidak mempunyai masalah

lebih serakan dan juga data yang mempunyai masalah lebih serakan. Keputusan

menunjukkan bahawa regresi Poisson adalah paling sesuai untuk data yang tidak

mempunyai masalah lebih serakan manakala regresi binomial negatif adalah paling

sesuai untuk data yang mempunyai masalah lebih serakan.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

COVER i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES xi

LIST OF FIGURES xii

LIST OF SYMBOLS xiv

LIST OF APPENDICES xv

1 INTRODUCTION 1

1.1 Count Data 1

1.2 Statement of the Problem 3

1.3 Objectives of the Study 4

1.4 Scope of the Study 4

1.5 Significance of the Study 4

1.6 Outline of the Study 5

1.7 Analysis Flow Chart 7

2 LITERATURE REVIEW 8

2.1 Generalized Linear Models 8

viii

2.1.1 Random component 8

2.1.2 Systematic component 9

2.1.3 Link 9

2.2 Principles of Statistical Modelling 10

2.2.1 Exploratory Data Analysis 10

2.2.2 Model Fitting 10

2.3 Poisson Distribution 12

2.4 Poisson Regression 14

2.5 Problems in Poisson Regression 16

2.5.1 Truncation and Censoring 16

2.5.2 Excess Zero 17

2.5.3 Overdispersion 19

2.6 Alternative Count Models 20

2.7 Negative Binomial Regression 21

3 POISSON REGRESSION ANALYSIS 23

3.1 The Model 23

3.2 Estimation of Parameters Using Maximum

Likelihood Estimation (MLE)

24

3.3 Standard Errors for Regression Coefficients 31

3.4 Interpretation of Coefficients 31

3.5 Elasticity 34

3.6 Model Checking Using Pearson Chi-Squares

and Deviance

35

3.6.1 Pearson Chi-Squares 36

3.6.2 Deviance 36

3.7 Model Residuals 37

3.8 Inference 38

3.8.1 Test of Significance 38

3.8.2 Confidence Intervals 39

ix

3.9 Handling Overdispersion 39

3.9.1 Quasi-Likelihood Method 39

3.9.1.1 Estimating the Overdispersion

Parameter

43

3.9.1.2 Testing for Overdispersion 44

3.9.2 Negative Binomial Regression Analysis 45

3.10 Example 46

4 ANALYSIS OF POISSON REGRESSION USING

SAS

58

4.1 Introduction 58

4.2 Nursing Home Data 58

4.3 Choosing the Right Model 65

4.4 Results and Discussion 67

4.5 Negative Binomial Regression 76

4.6 Results and Discussion 77

5 SIMULATION STUDY 85

5.1 Data Simulation 85

5.2 Analysis of Data with No Overdispersion 86

5.2.1 Results and Discussion 93

5.2.1.1 Goodness-of-fit 93

5.2.1.2 Significance, Confidence

Intervals, and Standard Errors

94

5.3 Analysis of Data with Overdispersion 94

5.3.1 Results and Discussion 101

5.3.1.1 Goodness-of-fit 101

5.3.1.2 Significance, Confidence

Intervals, and Standard Errors

102

5.4 Conclusions 103

x

5.5 The Simulation Codes 103

5.5.1 Codes for the simulation of non-

overdispersed data

104

5.5.2 Codes for the simulation of

overdispersed data

105

6 SUMMARY AND CONCLUSIONS 106

6.1 Summary 106

6.2 Conclusions 107

6.3 Recommendations 107

REFERENCES 108

APPENDICES 114

xi

LIST OF TABLES

TABLE NO. TITLE PAGE

3.1 Elephant’s mating success regarding age 47

3.2 Iterative reweighted least squares results 49

3.3 Residuals for elephant’s mating success data 53

3.4 Adjusted standard errors 57

4.1 Nursing home data 59

4.2 Log likelihood and deviance for model M1, M2, and M3 65

4.3 Elasticities of the explanatory variables in nursing home

data for model M3

69

4.4 Pearson residuals and adjusted residuals for nursing home

data

70

4.5 Comparison among standard errors for ordinary Poisson

regression and corrected Poisson regression

75

4.6 Elasticities of the explanatory variables in nursing home

data for negative binomial regression model

80

4.7 Residuals for negative binomial regression 81

5.1 Pearson chi-square and deviance for Poisson regression and

negative binomial regression obtained from data that has no

overdispersion

93

5.2 Pearson chi-square and deviance for Poisson regression and

negative binomial regression obtained from overdispersed

data

101

xii

LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 Steps in model fitting 11

3.1 Steps in quasi-likelihood approach 40

3.2 SAS result for analysis of elephant’s mating success data 50

4.1 SAS output for model (M1) 62



4.4 SAS output for corrected Poisson regression 74

4.5 SAS output for negative binomial regression 79

5.1 Poisson regression SAS output of non-overdispersed data for 10=µ

87

5.2 Poisson regression SAS output of non-overdispersed data

for 20=µ

88

5.3 Poisson regression SAS output of non-overdispersed data for 50=µ

89

5.4 Negative binomial regression SAS output of non-overdispersed data for 10=µ

90


91


92

5.7 Poisson regression SAS output of overdispersed data for 10=µ

95


96


97

5.10 Negative binomial regression SAS output of overdispersed data for 10=µ

98

xiii


99


100

xiv

LIST OF SYMBOLS

Y Response variable

X Predictor variable

β Regression coefficient

η Link Function

I Information matrix

U Score fuction

W Weight matrix

E Elasticity

2X Pearson chi-squares statistic

D Deviance statistic

R Pearson residual

Z Wald statistic

φ Dispersion parameter

xv

LIST OF APPENDICES

APPENDIX TITLE PAGE

A SAS Codes for Elephant’s Mating Success Data 114

B The Values of iµ̂ for Elephant’s Mating Success Data 115

C SAS Codes for Nursing Home Data 117

D SAS Output of Residual Analysis for Poisson Regression

in Nursing Home Data

119

E The Values of iµ̂ for Nursing Home Data 124

F SAS Output of Residual Analysis for Negative Binomial

Regression in Nursing Home Data

126

G Simulated Data 131

CHAPTER 1

INTRODUCTION

1.1 Count Data

An event count refers to the number of times an event occurs, for example the

number of individuals arriving at a serving station (e.g.: bank teller, gas station, cash

register, etc.) within a fixed interval, the number of failures of electronic components per

unit of time, the number of homicides per year, or the number of patents applied for and

received. In many fields such as in social, behavioral and biomedical sciences, as well as

in public health, marketing, education, biological and agricultural sciences and industrial

quality control, the response variable of interest is often measured as a nonnegative

integer or count.

Significant early developments in count models, however, took place in actuarial

science, biostatistics, and demography. In recent years these models have also been used

extensively in economics, political science, and sociology. The special features of data

in their respective fields of application have fueled developments that have enlarged the

scope of these models. An important milestone in the development of count data

regression model was the emergence of the generalized linear models, of which the

Poisson regression is a special case.

2

In another case, an event may be thought of as the realization of a point process

governed by some specified rate of occurrence of the event. The number of events may

be characterized as the total number of such realizations over some unit of time. The

dual of the event count is the inter-arrival time, defined as the length of the period

between events. Count data regression is useful in studying the occurrence rate per unit

of time.

The approach taken to the analysis of count data sometimes depends on how the

counts are assumed to arise. Count data can arise from two common ways:

i) Counts arise from a direct observation of a point process.

ii) Counts arise from discretization of continuous latent data.

In the first case, examples are the number of telephone calls arriving at central

telephone exchange, the number of monthly absences at workplace, the number of

airline accidents, the number of hospital admissions, and so forth. The data may also

consist of inter-arrival times for events.

In the second case, consider the following example. Credit rating of agencies

may be stated as AAA, AAB, AA, A, BBB, B, and so forth, where AAA indicates the

greatest credit. Suppose one codes these as my ,...,1,0= . These are pseudocounts that

can be analyzed using a count regression. But one may also regard this as an ordinal

ranking that can be modeled using a suitable latent variable model such as ordered

probit.

Typically, the characteristic of count data is that the counts occur over some

fixed area or observation period and that the things that people count are often rare.

Count data, even though numeric, can create some problems if it is analyzed using the

regular linear regression because of the limited range of most of the values and because

only nonnegative integer values can occur. Thus, count data can potentially result in a

highly skewed distribution, one that cut off at zero. Therefore, it is often unreasonable to

3

assume that the response variable and the resulting errors have a normal distribution,

making linear regression a less appropriate option for analysis. A suitable way to deal

with count data is to use Poisson distribution and log link function in the analysis. The

regression model that uses these kinds of options is called Poisson regression or Poisson

log-linear regression model.

Basically, the most popular methods to model count data are Poisson and

negative binomial regression models. But Poisson regression is the more popular of the

two and is applied to various fields.

1.2 Statement of the Problem

Count data often have variance exceeding the mean. In other words, count data

usually shows greater variability in the response counts than one would expect if the

response distribution truly were Poisson. This violates the Poisson regression

assumption which strictly states that the mean is equal to the variance (equidispersion).

The phenomenon where the variance is greater than the mean is called overdispersion. A

statistical test of overdispersion is highly desirable after running a Poisson regression.

Ignoring overdispersion in the analysis would lead to underestimation of standard errors

and consequent of significance in hypothesis testing. The overdispersion must be

accounted for by the analysis methods appropriate to the data. Poisson regression is not

adequate for analyzing overdispersed data. Therefore, to overcome overdispersion,

quasi-likelihood method will be used as well as negative binomial regression. Negative

binomial regression is more adequate for overdispersed data. This is because negative

binomial regression allows for overdispersion since its variance is naturally greater than

its mean.

4

1.3 Objectives of the Study The objectives of this study are:

i) To study the analysis of Poisson regression.

ii) To illustrate Poisson regression by analyzing count data manually and by using

SAS 9.1.

iii) To demonstrate how to handle overdispersion in Poisson regression using quasi

likelihood approach as well as negative binomial regression approach.

iv) To see the performance of Poisson regression and the performance of negative

binomial regression in analyzing data that has no overdispersion as well as data

that has overdispersion from simulation study.

1.4 Scope of the Study

This study will focus on the analysis of Poisson regression. This study will also

focus on the overdispersion problem that exists when dealing with real life count data.

Overdispersion happens when the variance is greater than the mean which violates the

equidispersion property in Poisson distribution and thus need to be taken care of. In

accordance to overdispersion problem, the performance of Poisson regression and

negative binomial regression in analyzing data that has no overdispersion as well as data

that has overdispersion will be examined from simulation study. The analyses in this

study include manual analysis and analysis by using statistical package. Statistical

package that is used in this study is SAS 9.1.

1.5 Significance of the Study

This study will help the scientists to realize the use of Poisson regression in

analyzing count data. Besides focusing on parameter estimation, this study will also help

5

to highlight about the interpretation of coefficients. This study will also help to

overcome overdispersion problem that occurs in Poisson regression which, if ignored,

may cause underestimation of standard errors and which consequently gives misleading

inference about the regression parameters. Clearly, this study is imperative and will give

much benefit.

1.6 Outline of the Study This dissertation consists of 6 chapters.

Chapter 1 gives rough idea about the study. It begins with the explanation on

count data. This includes the characteristic of count data which is very important

throughout the study. Chapter 1 also explains how the idea about the study came about.

Furthermore, it also explains about the purpose of the study, the scope and the

importance of the study.

Chapter 2 discusses the basic idea that is important in Poisson regression

analysis. This chapter also discusses about common problems in Poisson regression as

well as negative binomial regression other than previous studies done by previous

researchers.

Poisson regression analysis can be found in Chapter 3. This chapter gives clear

descriptions on formulation of Poisson regression model, manual computation of

maximum likelihood estimates, and how to interpret coefficients in Poisson regression.

It also includes other important analyses such as goodness of fit test, residual analysis

and inference. Other than that, this chapter also discusses about the methods to handle

overdispersion. To illustrate Poisson regression, an example is presented here. The

analysis of this example is done manually.

6

Chapter 4 deals with the analysis of Poisson regression using SAS 9.1. A bigger

data is used and more factors are considered. The data is a count data in the form of rate

and it involves overdispersion. SAS codes are provided for convenience.

Chapter 5 presents the simulation study. Data is simulated using R 2.9.2 software

and is analyzed by using SAS 9.1. The performance of Poisson regression and the

performance of negative binomial regression in analyzing data that has no

overdispersion as well as data that has overdispersion are presented in this chapter.

Lastly, the conclusions of the study are discussed in Chapter 6. This chapter

summarizes the whole study. Some recommendations for further research are also made

here.

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1.7 Analysis Flow Chart

No

Yes

Estimate the overdispersion parameter and test for

overdispersion

Run the Analysis

1. Formulation of the model 2. Estimation of parameter using

MLE 3. Interpretation of coefficients 4. Elasticity 5. Model Checking 6. Residual Analysis 7. Inference

POISSON REGRESSION

Does overdispersion

exist?

Run quasi-likelihood method

Run negative binomial regression

Conclude that Poisson regression is

adequate

1. Estimate the dispersion parameter

2. Adjust standard errors

Run the same analysis as in Poisson

regression analysis

108

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