Applications of Regression Models in Epidemiology

Transcription

1 Applications of Regression Models in Epidemiology

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3 Applications of Regression Models in Epidemiology Erick Suárez, Cynthia M. Pérez, Roberto Rivera, and Melissa N. Martínez

4 Copyright 2017 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) , outside the United States at (317) or fax (317) Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at Library of Congress Cataloging-in-Publication Data: Names: Erick L. Suárez, Erick L., 1953 Title: Applications of Regression Models in Epidemiology / Erick Suarez [and three others]. Description: Hoboken, New Jersey : John Wiley & Sons, Inc., [2017] Includes index. Identifiers: LCCN ISBN (cloth) ISBN (epub) Subjects: LCSH: Medical statistics. Regression analysis. Public health. Classification: LCC RA407.A DDC 610.2/1 dc23 LC record available at Printed in the United States of America

5 To our loved ones To those who have a strong commitment to social justice, human rights, and public health.

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7 vii Table of Contents Preface xv Acknowledgments About the Authors xvii xix 1 Basic Concepts for Statistical Modeling Introduction Parameter Versus Statistic Probability Definition Conditional Probability Concepts of Prevalence and Incidence Random Variables Probability Distributions Centrality and Dispersion Parameters of a Random Variable Independence and Dependence of Random Variables Special Probability Distributions Binomial Distribution Poisson Distribution Normal Distribution Hypothesis Testing Confidence Intervals Clinical Significance Versus Statistical Significance Data Management Study Design Data Collection Data Entry Data Screening What to Do When Detecting a Data Issue Impact of Data Issues and How to Proceed Concept of Causality 21 References 22

8 viii Table of Contents 2 Introduction to Simple Linear Regression Models Introduction Specific Objectives Model Definition Model Assumptions Graphic Representation Geometry of the Simple Regression Model Estimation of Parameters Variance of Estimators Hypothesis Testing About the Slope of the Regression Line Using the Student s t-distribution Using ANOVA Coefficient of Determination R Pearson Correlation Coefficient Estimation of Regression Line Values and Prediction Confidence Interval for the Regression Line Prediction Interval of Actual Values of the Response Example Predictions Predictions with the Database Used by the Model Predictions with Data Not Used to Create the Model Residual Analysis Conclusions 46 Practice Exercise 47 References 48 3 Matrix Representation of the Linear Regression Model Introduction Specific Objectives Definition Matrix Matrix Representation of a SLRM Matrix Arithmetic Addition and Subtraction of Matrices Matrix Multiplication Special Matrices Linear Dependence Rank of a Matrix Inverse Matrix [A 1 ] Application of an Inverse Matrix in a SLRM Estimation of β Parameters in a SLRM Multiple Linear Regression Model (MLRM) 57

9 Table of Contents ix 3.14 Interpretation of the Coefficients in a MLRM ANOVA in a MLRM Using Indicator Variables (Dummy Variables) Polynomial Regression Models Centering Multicollinearity Interaction Terms Conclusion 66 Practice Exercise 66 References 67 4 Evaluation of Partial Tests of Hypotheses in a MLRM Introduction Specific Objectives Definition of Partial Hypothesis Evaluation Process of Partial Hypotheses Special Cases Examples Conclusion 75 Practice Exercise 75 References 75 5 Selection of Variables in a Multiple Linear Regression Model Introduction Specific Objectives Selection of Variables According to the Study Objectives Criteria for Selecting the Best Regression Model Coefficient of Determination, R Adjusted Coefficient of Determination, R 2 A Mean Square Error (MSE) Mallows s C p Akaike Information Criterion Bayesian Information Criterion All Possible Models Stepwise Method in Regression Forward Selection Backward Elimination Stepwise Selection Limitations of Stepwise Methods Conclusion 83 Practice Exercise 84 References 85

10 x Table of Contents 6 Correlation Analysis Introduction Specific Objectives Main Correlation Coefficients Based on SLRM Pearson Correlation Coefficient ρ Relationship Between r and β^ Major Correlation Coefficients Based on MLRM Pearson Correlation Coefficient of Zero Order Multiple Correlation Coefficient Partial Correlation Coefficient Partial Correlation Coefficient of the First Order Partial Correlation Coefficient of the Second Order Semipartial Correlation Coefficient Significance Tests Suggested Correlations Example Conclusion 94 Practice Exercise 95 References 95 7 Strategies for Assessing the Adequacy of the Linear Regression Model Introduction Specific Objectives Residual Definition Initial Exploration Initial Considerations Standardized Residual Jackknife Residuals (R-Student Residuals) Normality of the Errors Correlation of Errors Criteria for Detecting Outliers, Leverage, and Influential Points Leverage Values Cook s Distance COV RATIO DFBETAS DFFITS Summary of the Results Multicollinearity Transformation of Variables Conclusion 114 Practice Exercise 115 References 116

11 Table of Contents xi 8 Weighted Least-Squares Linear Regression Introduction Specific Objectives Regression Model with Transformation into the Original Scale of Y Matrix Notation of the Weighted Linear Regression Model Application of the WLS Model with Unequal Number of Subjects Design without Intercept Model with Intercept and Weighting Factor Applications of the WLS Model When Variance Increases First Alternative Second Alternative Conclusions 125 Practice Exercise 126 References Generalized Linear Models Introduction Specific Objectives Exponential Family of Probability Distributions Binomial Distribution Poisson Distribution Exponential Family of Probability Distributions with Dispersion Mean and Variance in EF and EDF Definition of a Generalized Linear Model Estimation Methods Deviance Calculation Hypothesis Evaluation Analysis of Residuals Model Selection Bayesian Models Conclusions 140 References Poisson Regression Models for Cohort Studies Introduction Specific Objectives Incidence Measures Incidence Density Cumulative Incidence Confounding Variable 146

12 xii Table of Contents 10.5 Stratified Analysis Poisson Regression Model Definition of Adjusted Relative Risk Interaction Assessment Relative Risk Estimation Implementation of the Poisson Regression Model Conclusion 161 Practice Exercise 162 References Logistic Regression in Case Control Studies Introduction Specific Objectives Graphical Representation Definition of the Odds Ratio Confounding Assessment Effect Modification Stratified Analysis Unconditional Logistic Regression Model Types of Logistic Regression Models Binary Case Binomial Case Computing the OR crude Computing the Adjusted OR Inference on OR Example of the Application of ULR Model: Binomial Case Conditional Logistic Regression Model Conclusions 183 Practice Exercise 183 References Regression Models in a Cross-Sectional Study Introduction Specific Objectives Prevalence Estimation Using the Normal Approach Definition of the Magnitude of the Association POR Estimation Woolf s Method Exact Method Prevalence Ratio Stratified Analysis Logistic Regression Model Modeling Prevalence Odds Ratio 207

13 Table of Contents xiii Modeling Prevalence Ratio Conclusions 210 Practice Exercise 210 References 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

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15 xv Preface This book is intended to serve as a guide for statistical modeling in epidemiologic research. Our motivation for writing this book lies in our years of experience teaching biostatistics and epidemiology for different academic and professional programs at the University of Puerto Rico Medical Sciences Campus. This subject matter is usually covered in biostatistics courses at the master s and doctoral levels at schools of public health. The main focus of this book is statistical models and their analytical foundations for data collected from basic epidemiological study designs. This 13-chapter book can serve equally well as a textbook or as a source for consultation. Readers will be exposed to the following topics: linear and multiple regression models, matrix notation in regression models, correlation analysis, strategies for selecting the best model, partial hypothesis testing, weighted least-squares linear regression, generalized linear models, conditional and unconditional logistic regression models, Poisson regression, and programming codes in STATA, SAS, R, and SPSS for different practice exercises. We have started with the assumption that the readers of this book have taken at least a basic course in biostatistics and epidemiology. However, the first chapter describes the basic concepts needed for the rest of the book. Erick Suárez University of Puerto Rico, Medical Sciences Campus Cynthia M. Pérez University of Puerto Rico, Medical Sciences Campus Roberto Rivera University of Puerto Rico, Mayagüez Campus Melissa N. Martínez Havas Media International Company

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17 xvii Acknowledgments We wish to express our gratitude to our departmental colleagues for their continued support in the writing of this book. We are grateful to our colleagues and students for helping us to develop the programming for some of the examples and exercises: Heidi Venegas, Israel Almódovar, Oscar Castrillón, Marievelisse Soto, Linnette Rodríguez, José Rivera, Jorge Albarracín, and Glorimar Meléndez. We would also like to thank Sheila Ward for providing editorial advice. This book has been made possible by financial support received from grant CA096297/CA from the National Cancer Institute and award number 2U54MD from the National Institute on Minority Health and Health Disparities, both parts of the U.S. National Institutes of Health. Finally, we would like to thank our families for encouraging us throughout the development of this book.

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19 xix About the Authors Erick Suárez is Professor of Biostatistics at the Department of Biostatistics and Epidemiology of the University of Puerto Rico Graduate School of Public Health. He received a Ph.D. degree in Medical Statistics from the London School of Hygiene and Tropical Medicine. With more than 29 years of experience teaching biostatistics at the graduate level, he has also directed in mentoring and training efforts for public health students at the University of Puerto Rico. His research interests include HIV, HPV, cancer, diabetes, and genetical statistics. Cynthia M. Pérez is a Professor of Epidemiology at the Department of Biostatistics and Epidemiology of the University of Puerto Rico Graduate School of Public Health. She received an M.S. degree in Statistics and a Ph. D. degree in Epidemiology from Purdue University. Since 1994, she has taught epidemiology and biostatistics. She has directed mentoring and training efforts for public health and medical students at the University of Puerto Rico. Her research interests include diabetes, cardiovascular disease, periodontal disease, viral hepatitis, and HPV infection. Roberto Rivera is an Associate Professor at the College of Business of the University of Puerto Rico at Mayaguez. He received an M.A. and 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 and his research interests include asthma, periodontal disease, marine sciences, and environmental statistics. Melissa N. Martínez is a statistical analyst at the Havas Media International Company, located in Miami, FL. She has an MPH in Biostatistics from the University of Puerto Rico, Medical Sciences Campus and currently graduated from the Master of Business Analytics program at National University, San Diego, CA. For the past 7 years, she has been performing statistical analyses in the biomedical research, healthcare, and media advertising fields. She has assisted with the design of clinical trials, performing sample size calculations and writing the clinical trial reports.