Applications of Regression Models in Epidemiology

A one–stop guide for public health students and practitioners learning regression analysis and statistical methods

This book is written for public health professionals and students interested in applying regression models in the field of public health. The academic material is usually covered in the following courses: (i) Applied Regression Analysis, (ii) Advanced Epidemiology, and (iii) Statistical Computing for Applying Statistical Modeling. 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 linear regression, methods for selecting the best regression equation, logistic regression model, and Poisson regression model. 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 Public Health 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 5 years of experience teaching statistics courses at the undergraduate and graduate levels.

Dedication

About the Author

Preface

Acknowledgements 

Chapter 1: Basic concepts for Statistical Modeling

1.1 Introduction

1.2 Parameter versus statistic

1.3 Probability definition

1.4 Conditional probability

1.5 Concepts of prevalence and incidence

1.6 Random variables

1.7 Probability distributions

1.8 Centrality and dispersion parameters of a random variable

1.9 Independence and dependence of random variables

1.10 Special probability distribution

1.11 Hypothesis testing

1.12 Confidence intervals

1.13 Clinical significance versus statistical significance

1.14 Data management        

1.15 What to do when detecting a data issue

1.16 Impact of data issues and how to proceed

1.17 Concept of causality

References

Chapter 2: Introduction to Simple Linear Regression Models

2.1 Introduction

2.2 Specific objectives

2.3 Model definition

2.4 Model assumptions

2.5 Graphic representation

2.6 Geometry of the simple regression model

2.7 Estimation of parameters

2.8 Variance of estimators

2.9 Hypothesis testing about the slope of the regression line 

2.10 Coefficient of determination R2

2.11 Pearson correlation coefficient

2.12 Estimation of regression line values and prediction  

2.13 Example

2.14 Predictions  

2.15 Conclusions

2.16 Practice Exercise

Reference

Chapter 3 : Matrix Representation of the Linear Regression Model

3.1 Introduction

3.2 Specific objectives

3.3 Definition

3.4 Matrix representation of a SLRM

3.5 Matrix arithmetic

3.6 Matrix multiplication

3.7 Special matrices

3.8 Linear dependence

3.9 Rank of a matrix

3.10 Inverse matrix [A–1]

3.11 Application of an inverse matrix in a SLRM

3.12 Estimation of Ò parameters in a SLRM

3.13 Multiple linear regression model (MLRM)

3.14 Interpretation of the coefficients in a MLRM

3.15 ANOVA in a MLRM

3.16 Using indicator variables (Dummy Variables)

3.17 Polynomial regression models

3.18 Centering

3.19 Multicollinearity

3.20 Interaction terms

3.21 Conclusion

3.22 Practice Exercise

Reference

Chapter 4 : Evaluation of Partial Tests of Hypotheses in a MLRM

4.1 Introduction

4.2 Specific objectives

4.3 Definition of partial hypothesis

4.4 Evaluation process of partial hypotheses

4.5 Special situations

4.6 Examples

4.7 Conclusion

4.8 Practice exercise

Reference

Chapter 5 : Selection of Variables in a Multiple Linear Regression Model

5.1 Introduction

5.2 Specific Objectives

5.3 Selection of variables according to the study objectives

5.4 Criteria for selecting the best regression model

5.5 Stepwise method in regression

5.6 Limitations of stepwise methods

5.7 Conclusion

5.8 Practice exercise

References

Chapter 6 : Correlation Analysis

6.1 Introduction

6.2 Specific objectives

6.3 Main correlation coefficients based on SLRM

6.4 Major correlation coefficients based on MLRM

6.5 Partial correlation coefficient

6.6 Significance Tests

6.7 Suggested Correlations

6.8 Example

6.9 Conclusion

6.10 Practice Exercise

Reference

Chapter 7 : Strategies for assessing the adequacy of the Linear Regression Model

7.1 Introduction

7.2 Specific objectives

7.3 Residual definition

7.4 Initial exploration

7.5 Initial considerations

7.6 Standardized residual

7.7 Jackknife residuals (R–Student residuals)

7.8 Normality of the errors

7.9 Correlation of Errors

7.10 Criteria for detecting outliers, leverage, and influential points

7.11 Leverage values

7.12 Cook s distance

7.13 COV RATIO

7.14 DFBETAS

7.15 DFFITS

7.16 Summary of the results

7.17 Multicollinearity

7.18 Transformation of variables

7.19 Conclusion

7.20 Practice Exercise

Reference

Chapter 8 : Weighted Least Squares Linear Regression

8.1 Introduction

8.2 Specific objectives

8.3 Regression model with transformation into the original scale of Y

8.4 Matrix Notation of the Weighted Linear Regression Model

8.5 Application of the WLS model with unequal number of subjects

8.6 Applications of the WLS model when variance increases

8.7 Conclusions

8.8 Practice Exercise

Reference

Chapter 9 : Generalized Linear Models 

9.1 Introduction

9.2 Specific objectives

9.3 Exponential Family of Probability Distributions  

9.4 Exponential Family of Probability Distributions with Dispersion

9.5 Mean and Variance in EF and EDF

9.6 Definition of a Generalized Linear Model 

9.7 Estimation Methods 

9.8 Deviance calculation

9.9 Hypothesis Evaluation

9.10 Analysis of Residuals

9.11 Model Selection

9.12 Bayesian Models

9.13 Conclusions

Reference

Chapter 10 : Poisson Regression Models for Cohort Studies

10.1 Introduction

10.2 Specific Objectives

10.3 Incidence Measures

10.4 Confounding variable

10.5 Stratified analysis

10.6 Poisson regression model

10.7 Definition of Adjusted Relative Risk

10.8 Interaction assessment

10.9 Relative Risk Estimation

10.10 Implementation of the Poisson regression model

10.11 Conclusion

10.12 Practice Exercise

Reference

Chapter 11 : Logistic Regression in Case–Control Studies

11.1 Introduction

11.2 Specific Objectives

11.3 Graphical Representation

11.4 Definition of the Odds Ratio

11.5 Confounding assessment

11.6 Effect Modification

11.7 Stratified analysis

11.8 Unconditional Logistic Regression Model

11.9 Types of logistic regression models

11.10 Computing the ORcrude

11.11 Computing the adjusted OR

11.12 Inference on OR

11.13 Example of the application of ULR model–binomial case

11.14 Conditional logistic regression model

11.15 Conclusions

11.16 Practice Exercise

Reference

Chapter 12 : Regression models in a cross–sectional study

12.1 Introduction

12.2 Specific Objectives

12.3 Prevalence estimation using the normal approach

12.4 Definition of the magnitude of the association

12.5 POR Estimation

12.6 Prevalence Ratio

12.7 Stratified analysis

12.8 Logistic Regression Model       

12.9 Conclusions

12.10 Practice Exercise

Reference

Chapter 13 :  Solutions to Practice Exercises

Chapter II: Practice exercise

Chapter III: Practice exercise

Chapter IV: Practice exercise

Chapter V: Practice exercise

Chapter VI: Practice exercise

Chapter VII: Practice exercise

Chapter VIII: Practice exercise

Chapter X: Practice exercise

Chapter XI: Practice exercise

Chapter XII: Practice exercise

Index

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 5 years of experience teaching statistics courses at the undergraduate and graduate levels.