Applications of Regression Models in Epidemiology

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.

• Preface xvAcknowledgments xviiAbout the Authors xix1 Basic Concepts for Statistical Modeling 11.1 Introduction 11.2 Parameter Versus Statistic 21.3 Probability Definition 31.4 Conditional Probability 31.5 Concepts of Prevalence and Incidence 41.6 Random Variables 41.7 Probability Distributions 41.8 Centrality and Dispersion Parameters of a Random Variable 61.9 Independence and Dependence of Random Variables 71.10 Special Probability Distributions 71.10.1 Binomial Distribution 71.10.2 Poisson Distribution 81.10.3 Normal Distribution 91.11 Hypothesis Testing 111.12 Confidence Intervals 141.13 Clinical Significance Versus Statistical Significance 141.14 Data Management 151.14.1 Study Design 151.14.2 Data Collection 161.14.3 Data Entry 171.14.4 Data Screening 181.14.5 What to Do When Detecting a Data Issue 191.14.6 Impact of Data Issues and How to Proceed 201.15 Concept of Causality 21References 222 Introduction to Simple Linear Regression Models 252.1 Introduction 252.2 Specific Objectives 262.3 Model Definition 262.4 Model Assumptions 282.5 Graphic Representation 292.6 Geometry of the Simple Regression Model 292.7 Estimation of Parameters 302.8 Variance of Estimators 312.9 Hypothesis Testing About the Slope of the Regression Line 322.9.1 Using the Student’s t-Distribution 322.9.2 Using ANOVA 322.10 Coefficient of Determination R2 342.11 Pearson Correlation Coefficient 342.12 Estimation of Regression Line Values and Prediction 352.12.1 Confidence Interval for the Regression Line 352.12.2 Prediction Interval of Actual Values of the Response 362.13 Example 362.14 Predictions 392.14.1 Predictions with the Database Used by the Model 402.14.2 Predictions with Data Not Used to Create the Model 422.14.3 Residual Analysis 442.15 Conclusions 46Practice Exercise 47References 483 Matrix Representation of the Linear Regression Model 493.1 Introduction 493.2 Specific Objectives 493.3 Definition 503.3.1 Matrix 503.4 Matrix Representation of a SLRM 503.5 Matrix Arithmetic 513.5.1 Addition and Subtraction of Matrices 513.6 Matrix Multiplication 523.7 Special Matrices 533.8 Linear Dependence 543.9 Rank of a Matrix 543.10 Inverse Matrix [A−1] 543.11 Application of an Inverse Matrix in a SLRM 563.12 Estimation of β Parameters in a SLRM 563.13 Multiple Linear Regression Model (MLRM) 573.14 Interpretation of the Coefficients in a MLRM 583.15 ANOVA in a MLRM 583.16 Using Indicator Variables (Dummy Variables) 603.17 Polynomial Regression Models 633.18 Centering 643.19 Multicollinearity 653.20 Interaction Terms 653.21 Conclusion 66Practice Exercise 66References 674 Evaluation of Partial Tests of Hypotheses in a MLRM 694.1 Introduction 694.2 Specific Objectives 694.3 Definition of Partial Hypothesis 704.4 Evaluation Process of Partial Hypotheses 714.5 Special Cases 714.6 Examples 724.7 Conclusion 75Practice Exercise 75References 755 Selection of Variables in a Multiple Linear Regression Model 775.1 Introduction 775.2 Specific Objectives 775.3 Selection of Variables According to the Study Objectives 775.4 Criteria for Selecting the Best Regression Model 785.4.1 Coefficient of Determination, R2 785.4.2 Adjusted Coefficient of Determination, R2A 785.4.3 Mean Square Error (MSE) 795.4.4 Mallows’s Cp 795.4.5 Akaike Information Criterion 795.4.6 Bayesian Information Criterion 805.4.7 All Possible Models 805.5 Stepwise Method in Regression 805.5.1 Forward Selection 815.5.2 Backward Elimination 825.5.3 Stepwise Selection 825.6 Limitations of Stepwise Methods 835.7 Conclusion 83Practice Exercise 84References 856 Correlation Analysis 876.1 Introduction 876.2 Specific Objectives 876.3 Main Correlation Coefficients Based on SLRM 876.3.1 Pearson Correlation Coefficient ρ 886.3.2 Relationship Between r and ^β1 896.4 Major Correlation Coefficients Based on MLRM 896.4.1 Pearson Correlation Coefficient of Zero Order 896.4.2 Multiple Correlation Coefficient 906.5 Partial Correlation Coefficient 906.5.1 Partial Correlation Coefficient of the First Order 916.5.2 Partial Correlation Coefficient of the Second Order 916.5.3 Semipartial Correlation Coefficient 916.6 Significance Tests 926.7 Suggested Correlations 926.8 Example 926.9 Conclusion 94Practice Exercise 95References 957 Strategies for Assessing the Adequacy of the Linear Regression Model 977.1 Introduction 977.2 Specific Objectives 987.3 Residual Definition 987.4 Initial Exploration 987.5 Initial Considerations 1027.6 Standardized Residual 1027.7 Jackknife Residuals (R-Student Residuals) 1047.8 Normality of the Errors 1057.9 Correlation of Errors 1067.10 Criteria for Detecting Outliers, Leverage, and Influential Points 1077.11 Leverage Values 1087.12 Cook’s Distance 1087.13 COV Ratio 1097.14 DFBETAS 1107.15 DFFITS 1107.16 Summary of the Results 1117.17 Multicollinearity 1117.18 Transformation of Variables 1147.19 Conclusion 114Practice Exercise 115References 1168 Weighted Least-Squares Linear Regression 1178.1 Introduction 1178.2 Specific Objectives 1178.3 Regression Model with Transformation into the Original Scale of Y 1178.4 Matrix Notation of the Weighted Linear Regression Model 1198.5 Application of the WLS Model with Unequal Number of Subjects 1208.5.1 Design without Intercept 1218.5.2 Model with Intercept and Weighting Factor 1228.6 Applications of the WLS Model When Variance Increases 1238.6.1 First Alternative 1238.6.2 Second Alternative 1248.7 Conclusions 125Practice Exercise 126References 1279 Generalized Linear Models 1299.1 Introduction 1299.2 Specific Objectives 1299.3 Exponential Family of Probability Distributions 1309.3.1 Binomial Distribution 1309.3.2 Poisson Distribution 1319.4 Exponential Family of Probability Distributions with Dispersion 1319.5 Mean and Variance in EF and EDF 1329.6 Definition of a Generalized Linear Model 1339.7 Estimation Methods 1349.8 Deviance Calculation 1359.9 Hypothesis Evaluation 1369.10 Analysis of Residuals 1389.11 Model Selection 1399.12 Bayesian Models 1399.13 Conclusions 140References 14010 Poisson Regression Models for Cohort Studies 14110.1 Introduction 14110.2 Specific Objectives 14210.3 Incidence Measures 14210.3.1 Incidence Density 14210.3.2 Cumulative Incidence 14510.4 Confounding Variable 14610.5 Stratified Analysis 14710.6 Poisson Regression Model 14810.7 Definition of Adjusted Relative Risk 14910.8 Interaction Assessment 15010.9 Relative Risk Estimation 15110.10 Implementation of the Poisson Regression Model 15210.11 Conclusion 161Practice Exercise 162References 16211 Logistic Regression in Case–Control Studies 16511.1 Introduction 16511.2 Specific Objectives 16611.3 Graphical Representation 16611.4 Definition of the Odds Ratio 16711.5 Confounding Assessment 16811.6 Effect Modification 16811.7 Stratified Analysis 16911.8 Unconditional Logistic Regression Model 17011.9 Types of Logistic Regression Models 17111.9.1 Binary Case 17211.9.2 Binomial Case 17211.10 Computing the ORcrude 17311.11 Computing the Adjusted OR 17311.12 Inference on OR 17411.13 Example of the Application of ULR Model: Binomial Case 17511.14 Conditional Logistic Regression Model 17811.15 Conclusions 183Practice Exercise 183References 18812 Regression Models in a Cross-Sectional Study 19112.1 Introduction 19112.2 Specific Objectives 19212.3 Prevalence Estimation Using the Normal Approach 19212.4 Definition of the Magnitude of the Association 19812.5 POR Estimation 20012.5.1 Woolf’s Method 20012.5.2 Exact Method 20212.6 Prevalence Ratio 20412.7 Stratified Analysis 20412.8 Logistic Regression Model 20712.8.1 Modeling Prevalence Odds Ratio 20712.8.2 Modeling Prevalence Ratio 20912.9 Conclusions 210Practice Exercise 210References 21113 Solutions to Practice Exercises 213Chapter 2 Practice Exercise 213Chapter 3 Practice Exercise 216Chapter 4 Practice Exercise 220Chapter 5 Practice Exercise 221Chapter 6 Practice Exercise 223Chapter 7 Practice Exercise 225Chapter 8 Practice Exercise 228Chapter 10 Practice Exercise 230Chapter 11 Practice Exercise 233Chapter 12 Practice Exercise 240Index 245