Applied Logistic Regression

1 Applied Logistic RegressionApplied Logistic RegressionThird EditionDAVID W. HOSMER, of Biostatistics (Emeritus)Division of Biostatistics and EpidemiologyDepartment of Public HealthSchool of Public Health and Health SciencesUniversity of MassachusettsAmherst, MassachusettsSTANLEY LEMESHOWDean, College of Public HealthProfessor of BiostatisticsCollege of Public HealthThe Ohio State UniversityColumbus, OhioRODNEY X. STURDIVANTC olonel, ArmyAcademy and Associate ProfessorDepartment of Mathematical SciencesUnited States Military AcademyWest Point, New YorkCopyright 2013 by John Wiley & Sons, Inc. All rights by John Wiley & Sons, Inc., Hoboken, New simultaneously in part of this publication may be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except aspermitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the priorwritten permission of the Publisher, or authorization through payment of the appropriate per-copy feeto the Copyright Clearance Center, Inc.

2 , 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400,fax (978) 750-4470, or on the web at Requests to the Publisher for permissionshould be addressed to the Permissions Department, John Wiley &Sons, Inc., 111 River Street,Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online of Liability/Disclaimer of Warranty: While the publisher and author have used their best effortsin preparing this book, they make no representations or warranties with respect to the accuracy orcompleteness of the contents of this book and specifically disclaim any implied warranties ofmerchantability or fitness for a particular purpose. No warranty may be created or extended by salesrepresentatives or written sales materials. The advice and strategies contained herein may not besuitable for your situation. You should consult with a professional where appropriate.

3 Neither thepublisher nor author shall be liable for any loss of profit or any other commercial damages, includingbut not limited to special, incidental, consequential, or other general information on our other products and services or for technical support, please contact ourCustomer Care Department within the United States at (800) 762-2974, outside the United States at(317) 572-3993 or fax (317) also publishes its books in a variety of electronic formats. Some content that appears in printmay not be available in electronic formats. For more information about Wiley products, visit our website at of Congress Cataloging-in-Publication Data Is AvailableHosmer, David Logistic Regression / David W. Hosmer, Jr., Stanley Lemeshow, Rodney X. Sturdivant. -3rd bibliographic references and 978-0-470-58247-3 (cloth)

4 Printed in the United States of America10987654321To our wives, Trina, Elaine, and Mandy,and our sons, daughters,and grandchildrenContentsPreface to the Third Editionxiii1 Introduction to the Logistic Regression Introduction, Fitting the Logistic Regression Model, Testing for the Significance of the Coefficients, Confidence Interval Estimation, Other Estimation Methods, Data Sets Used in Examples and Exercises, The ICU Study, The Low Birth Weight Study, The Global Longitudinal Study of Osteoporosisin Women, The Adolescent Placement Study, The Burn Injury Study, The Myopia Study, The NHANES Study, The Polypharmacy Study, 31 Exercises, 322 The Multiple Logistic Regression Introduction, The Multiple Logistic Regression Model, Fitting the Multiple Logistic Regression Model, Testing for the Significance of the Model, Confidence Interval Estimation, Other Estimation Methods, 45 Exercises, 46viiviiicontents3 Interpretation of the Fitted Logistic Regression Introduction, Dichotomous Independent Variable, Polychotomous Independent Variable, Continuous Independent Variable, Multivariable Models, Presentation and Interpretation of the Fitted Values, A Comparison of Logistic Regression and Stratified Analysisfor 2 2 Tables, 82 Exercises, 874 Model-Building Strategies and Methods for Logistic Introduction, Purposeful Selection of Covariates, Methods to Examine the Scale of a ContinuousCovariate in the Logit, Examples of Purposeful Selection, Other Methods for Selecting Covariates.

5 Stepwise Selection of Covariates, Best Subsets Logistic Regression , Selecting Covariates and Checking their ScaleUsing Multivariable Fractional Polynomials, Numerical Problems, 145 Exercises, 1505 Assessing the Fit of the Introduction, Summary Measures of Goodness of Fit, Pearson Chi-Square Statistic, Deviance,and Sum-of-Squares, The Hosmer Lemeshow Tests, Classification Tables, Area Under the Receiver Operating CharacteristicCurve, Other Summary Measures, Logistic Regression Diagnostics, Assessment of Fit via External Validation, Interpretation and Presentation of the Results from a FittedLogistic Regression Model, 212 Exercises, 2236 Application of Logistic Regression with Different Introduction, Cohort studies , Case-Control studies , Fitting Logistic Regression Models to Data from ComplexSample Surveys, 233 Exercises, 2427 Logistic Regression for Matched Case-Control Introduction, Methods For Assessment of Fit in a 1 MMatchedStudy, An Example Using the Logistic Regression Model in a 1 1 Matched Study, An Example Using the Logistic Regression Model in a 1 MMatched Study, 260 Exercises, 2678 Logistic Regression Models for Multinomial and The Multinomial Logistic Regression Model, Introduction to the Model and Estimation of ModelParameters, Interpreting and Assessing the Significance of theEstimated Coefficients, Model-Building Strategies for Multinomial LogisticRegression, Assessment of Fit and Diagnostic Statistics for theMultinomial Logistic Regression Model, Ordinal Logistic Regression Models.

6 Introduction to the Models, Methods for Fitting, andInterpretation of Model Parameters, Model Building Strategies for Ordinal LogisticRegression Models, 305 Exercises, 310xcontents9 Logistic Regression Models for the Analysis of Correlated Introduction, Logistic Regression Models for the Analysis of CorrelatedData, Estimation Methods for Correlated Data Logistic RegressionModels, Interpretation of Coefficients from Logistic RegressionModels for the Analysis of Correlated Data, Population Average Model, Cluster-Specific Model, Alternative Estimation Methods for theCluster-Specific Model, Comparison of Population Average andCluster-Specific Model, An Example of Logistic Regression Modeling withCorrelated Data, Choice of Model for Correlated Data Analysis, Population Average Model, Cluster-Specific Model, Additional Points to Consider when Fitting LogisticRegression Models to Correlated Data, Assessment of Model Fit, Assessment of Population Average Model Fit, Assessment of Cluster-Specific Model Fit, Conclusions, 374 Exercises, 37510 Special Introduction, Application of Propensity Score Methods in LogisticRegression Modeling, Exact Methods for Logistic Regression Models, Missing Data, Sample Size Issues when Fitting Logistic RegressionModels, Bayesian Methods for Logistic Regression , The Bayesian Logistic Regression Model, MCMC Simulation, An Example of a Bayesian Analysis and ItsInterpretation, Other Link Functions for Binary Regression Models, Mediation, Distinguishing Mediators from Confounders.

7 Implications for the Interpretation of an AdjustedLogistic Regression Coefficient, Why Adjust for a Mediator? Using Logistic Regression to Assess Mediation:Assumptions, More About Statistical Interaction, Additive versus Multiplicative Scale RiskDifference versus Odds Ratios, Estimating and Testing Additive Interaction, 451 Exercises, 456 References459 Index479 Preface to the Third EditionThis third edition ofApplied Logistic Regressioncomes 12 years after the 2000publication of the second edition. During this interval there has been considerableeffort researching statistical aspects of the Logistic Regression model particularlywhen the outcomes are correlated. At the same time, capabilities of computer soft-ware packages to fit models grew impressively to the point where they now provideaccess to nearly every aspect of model development a researcher might need.

8 As iswell-recognized in the statistical community, the inherent danger of this easy-to-usesoftware is that investigators have at their disposal powerful computational tools,about which they may have only limited understanding. It is our hope that this thirdedition will help bridge the gap between the outstanding theoretical developmentsand the need to apply these methods to diverse fields of was the case in the first two editions, the primary objective of the third editionis to provide an introduction to the underlying theory of the Logistic regressionmodel, with a major focus on the application, using real data sets, of the availablemethods to explore the relationship between a categorical outcome variable and aset of covariates. The materials in this book have evolved over the past 12 yearsas a result of our teaching and consulting experiences.

9 We have used this book toteach parts of graduate level survey courses, quarter- or semester-long courses, aswell as focused short courses to working professionals. We assume that studentshave a solid foundation in linear Regression methodology and contingency tableanalysis. The positive feedback we have received from students or professionalstaking courses using this book or using it for self-learning or reference, providesus with some assurance that the approach we used in the first two editions workedreasonably well; therefore, we have followed that approach in this new approach we take is to develop the Logistic Regression model from a regres-sion analysis point of view. This is accomplished by approaching Logistic regressionin a manner analogous to what would be considered good statistical practice forlinear Regression .

10 This differs from the approach used by other authors who havebegun their discussion from a contingency table point of view. While the contin-gency table approach may facilitate the interpretation of the results, we believethat it obscures the Regression aspects of the analysis. Thus, discussion of the inter-pretation of the model is deferred until the Regression approach to the analysis isfirmly to the third editionTo a large extent, there are no major differences between the many softwarepackages that include Logistic Regression modeling. When a particular approachis available in a limited number of packages, it will be noted in this text. Ingeneral, analyses in this book have been performed using STATA [Stata Corp.(2011)]. This easy-to-use package combines excellent graphics and analysis rou-tines; is fast; is compatible across Macintosh, Windows and UNIX platforms; andinteracts well with Microsoft Word.