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### More About This Title Applications of Regression Models in Epidemiology

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**A one-stop guide for public health students and practitioners learning the applications of classical regression models in epidemiology**

This book is written for public health professionals and students interested in applying regression models in the field of epidemiology. The academic material is usually covered in public health courses including (i) Applied Regression Analysis, (ii) Advanced Epidemiology, and (iii) Statistical Computing. The book is composed of 13 chapters, including an introduction chapter that covers basic concepts of statistics and probability. Among the topics covered are linear regression model, polynomial regression model, weighted least squares, methods for selecting the best regression equation, and generalized linear models and their applications to different epidemiological study designs. An example is provided in each chapter that applies the theoretical aspects presented in that chapter. In addition, exercises are included and the final chapter is devoted to the solutions of these academic exercises with answers in all of the major statistical software packages, including STATA, SAS, SPSS, and R. It is assumed that readers of this book have a basic course in biostatistics, epidemiology, and introductory calculus. The book will be of interest to anyone looking to understand the statistical fundamentals to support quantitative research in public health.

In addition, this book:

• Is based on the authors’ course notes from 20 years teaching regression modeling in public health courses

• Provides exercises at the end of each chapter

• Contains a solutions chapter with answers in STATA, SAS, SPSS, and R

• Provides real-world public health applications of the theoretical aspects contained in the chapters

*Applications of Regression Models in Epidemiology *is a reference for graduate students in public health and public health practitioners.

**ERICK SUÁREZ **is a Professor of the Department of Biostatistics and Epidemiology at the University of Puerto Rico School of Public Health. He received a Ph.D. degree in Medical Statistics from the London School of Hygiene and Tropical Medicine. He has 29 years of experience teaching biostatistics.

**CYNTHIA M. PÉREZ **is a Professor of the Department of Biostatistics and Epidemiology at the University of Puerto Rico School of Public Health. She received an M.S. degree in Statistics and a Ph.D. degree in Epidemiology from Purdue University. She has 22 years of experience teaching epidemiology and biostatistics.

**ROBERTO RIVERA **is an Associate Professor at the College of Business at the University of Puerto Rico at Mayaguez. He received a Ph.D. degree in Statistics from the University of California in Santa Barbara. He has more than five years of experience teaching statistics courses at the undergraduate and graduate levels.

**MELISSA N. MARTÍNEZ **is an Account Supervisor at Havas Media International. She holds an MPH in Biostatistics from the University of Puerto Rico and an MSBA from the National University in San Diego, California. For the past seven years, she has been performing analyses for the biomedical research and media advertising fields.

- English

### English

**ERICK SUÁREZ **is a Professor of the Department of Biostatistics and Epidemiology at the University of Puerto Rico School of Public Health. He received a Ph.D. degree in Medical Statistics from the London School of Hygiene and Tropical Medicine. He has 29 years of experience teaching biostatistics.

**CYNTHIA M. PÉREZ **is a Professor of the Department of Biostatistics and Epidemiology at the University of Puerto Rico School of Public Health. She received an M.S. degree in Statistics and a Ph.D. degree in Epidemiology from Purdue University. She has 22 years of experience teaching epidemiology and biostatistics.

**ROBERTO RIVERA **is an Associate Professor at the College of Business at the University of Puerto Rico at Mayaguez. He received a Ph.D. degree in Statistics from the University of California in Santa Barbara. He has more than five years of experience teaching statistics courses at the undergraduate and graduate levels.

**MELISSA N. MARTÍNEZ **is an Account Supervisor at Havas Media International. She holds an MPH in Biostatistics from the University of Puerto Rico and an MSBA from the National University in San Diego, California. For the past seven years, she has been performing analyses for the biomedical research and media advertising fields.

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### English

Preface xv

Acknowledgments xvii

About the Authors xix

**1 Basic Concepts for Statistical Modeling 1**

1.1 Introduction 1

1.2 Parameter Versus Statistic 2

1.3 Probability Definition 3

1.4 Conditional Probability 3

1.5 Concepts of Prevalence and Incidence 4

1.6 Random Variables 4

1.7 Probability Distributions 4

1.8 Centrality and Dispersion Parameters of a Random Variable 6

1.9 Independence and Dependence of Random Variables 7

1.10 Special Probability Distributions 7

1.11 Hypothesis Testing 11

1.12 Confidence Intervals 14

1.13 Clinical Significance Versus Statistical Significance 14

1.14 Data Management 15

1.15 Concept of Causality 21

References 22

**2 Introduction to Simple Linear Regression Models 25**

2.1 Introduction 25

2.2 Specific Objectives 26

2.3 Model Definition 26

2.4 Model Assumptions 28

2.5 Graphic Representation 29

2.6 Geometry of the Simple Regression Model 29

2.7 Estimation of Parameters 30

2.8 Variance of Estimators 31

2.9 Hypothesis Testing About the Slope of the Regression Line 32

2.10 Coefficient of Determination R2 34

2.11 Pearson Correlation Coefficient 34

2.12 Estimation of Regression Line Values and Prediction 35

2.13 Example 36

2.14 Predictions 39

2.15 Conclusions 46

Practice Exercise 47

References 48

**3 Matrix Representation of the Linear Regression Model 49**

3.1 Introduction 49

3.2 Specific Objectives 49

3.3 Definition 50

3.3.1 Matrix 50

3.4 Matrix Representation of a SLRM 50

3.5 Matrix Arithmetic 51

3.6 Matrix Multiplication 52

3.7 Special Matrices 53

3.8 Linear Dependence 54

3.9 Rank of a Matrix 54

3.10 Inverse Matrix [A 54

3.11 Application of an Inverse Matrix in a SLRM 56

3.12 Estimation of β Parameters in a SLRM 56

3.13 Multiple Linear Regression Model (MLRM) 57

3.14 Interpretation of the Coefficients in a MLRM 58

3.15 ANOVA in a MLRM 58

3.16 Using Indicator Variables (Dummy Variables) 60

3.17 Polynomial Regression Models 63

3.18 Centering 64

3.19 Multicollinearity 65

3.20 Interaction Terms 65

3.21 Conclusion 66

Practice Exercise 66

References 67

**4 Evaluation of Partial Tests of Hypotheses in a MLRM 69**

4.1 Introduction 69

4.2 Specific Objectives 69

4.3 Definition of Partial Hypothesis 70

4.4 Evaluation Process of Partial Hypotheses 71

4.5 Special Cases 71

4.6 Examples 72

4.7 Conclusion 75

Practice Exercise 75

References 75

**5 Selection of Variables in a Multiple Linear Regression Model 77**

5.1 Introduction 77

5.2 Specific Objectives 77

5.3 Selection of Variables According to the Study Objectives 77

5.4 Criteria for Selecting the Best Regression Model 78

5.5 Stepwise Method in Regression 80

5.6 Limitations of Stepwise Methods 83

5.7 Conclusion 83

Practice Exercise 84

References 85

**6 Correlation Analysis 87**

6.1 Introduction 87

6.2 Specific Objectives 87

6.3 Main Correlation Coefficients Based on SLRM 87

6.4 Major Correlation Coefficients Based on MLRM 89

6.5 Partial Correlation Coefficient 90

6.6 Significance Tests 92

6.7 Suggested Correlations 92

6.8 Example 92

6.9 Conclusion 94

Practice Exercise 95

References 95

**7 Strategies for Assessing the Adequacy of the Linear Regression Model 97**

7.1 Introduction 97

7.2 Specific Objectives 98

7.3 Residual Definition 98

7.4 Initial Exploration 98

7.5 Initial Considerations 102

7.6 Standardized Residual 102

7.7 Jackknife Residuals (R-Student Residuals) 104

7.8 Normality of the Errors 105

7.9 Correlation of Errors 106

7.10 Criteria for Detecting Outliers, Leverage, and Influential Points 107

7.11 Leverage Values 108

7.12 Cook’s Distance 108

7.13 COV RATIO 109

7.14 DFBETAS 110

7.15 DFFITS 110

7.16 Summary of the Results 111

7.17 Multicollinearity 111

7.18 Transformation of Variables 114

7.19 Conclusion 114

Practice Exercise 115

References 116

**8 Weighted Least-Squares Linear Regression 117**

8.1 Introduction 117

8.2 Specific Objectives 117

8.3 Regression Model with Transformation into the Original Scale of Y 117

8.4 Matrix Notation of the Weighted Linear Regression Model 119

8.5 Application of the WLS Model with Unequal Number of Subjects 120

8.6 Applications of the WLS Model When Variance Increases 123

8.7 Conclusions 125

Practice Exercise 126

References 127

**9 Generalized Linear Models 129**

9.1 Introduction 129

9.2 Specific Objectives 129

9.3 Exponential Family of Probability Distributions 130

9.4 Exponential Family of Probability Distributions with Dispersion 131

9.5 Mean and Variance in EF and EDF 132

9.6 Definition of a Generalized Linear Model 133

9.7 Estimation Methods 134

9.8 Deviance Calculation 135

9.9 Hypothesis Evaluation 136

9.10 Analysis of Residuals 138

9.11 Model Selection 139

9.12 Bayesian Models 139

9.13 Conclusions 140

References 140

**10 Poisson Regression Models for Cohort Studies 141**

10.1 Introduction 141

10.2 Specific Objectives 142

10.3 Incidence Measures 142

10.4 Confounding Variable 146

10.5 Stratified Analysis 147

10.6 Poisson Regression Model 148

10.7 Definition of Adjusted Relative Risk 149

10.8 Interaction Assessment 150

10.9 Relative Risk Estimation 151

10.10 Implementation of the Poisson Regression Model 152

10.11 Conclusion 161

Practice Exercise 162

References 162

**11 Logistic Regression in Case–Control Studies 165**

11.1 Introduction 165

11.2 Specific Objectives 166

11.3 Graphical Representation 166

11.4 Definition of the Odds Ratio 167

11.5 Confounding Assessment 168

11.6 Effect Modification 168

11.7 Stratified Analysis 169

11.8 Unconditional Logistic Regression Model 170

11.9 Types of Logistic Regression Models 171

11.10 Computing the ORcrude 173

11.11 Computing the Adjusted OR 173

11.12 Inference on OR 174

11.13 Example of the Application of ULR Model: Binomial Case 175

11.14 Conditional Logistic Regression Model 178

11.15 Conclusions 183

Practice Exercise 183

References 188

**12 Regression Models in a Cross-Sectional Study 191**

12.1 Introduction 191

12.2 Specific Objectives 192

12.3 Prevalence Estimation Using the Normal Approach 192

12.4 Definition of the Magnitude of the Association 198

12.5 POR Estimation 200

12.6 Prevalence Ratio 204

12.7 Stratified Analysis 204

12.8 Logistic Regression Model 207

12.9 Conclusions 210

Practice Exercise 210

References 211

**13 Solutions to Practice Exercises 213**

Chapter 2 Practice Exercise 213

Chapter 3 Practice Exercise 216

Chapter 4 Practice Exercise 220

Chapter 5 Practice Exercise 221

Chapter 6 Practice Exercise 223

Chapter 7 Practice Exercise 225

Chapter 8 Practice Exercise 228

Chapter 10 Practice Exercise 230

Chapter 11 Practice Exercise 233

Chapter 12 Practice Exercise 240

Index 245