Applied Logistic Regression

Stanley Lemeshow, PhD, is Dean of the College of Public Health at The Ohio State University. He is also currently the Director of the Summer Program in Applied Biostatistical and Epidemiological Methods, which is held each year at OSU. Dr. Lemeshow is a Fellow of both the American Association for the Advancement of Science and the American Statistical Association.

Rodney X. Sturdivant, PhD, is Founding Director of the Center for Data Analysis and Statistics at the United States Military Academy at West Point, NY. He is also Associate and Academy Professor in the Department of Mathematical Sciences. Colonel Sturdivant received his PhD in biostatistics from the University of Massachusetts at Amherst in 2005.

1 Introduction to the Logistic Regression Model 1

1.1Introduction, 1

1.2 Fitting the Logistic Regression Model, 8

1.3 Testing for the Significance of the Coefficients, 10

1.4 Confidence Interval Estimation, 15

1.5 Other Estimation Methods, 20

1.6 Data Sets Used in Examples and Exercises, 22

1.6.1 The ICU Study, 22

1.6.2 The Low Birth Weight Study, 24

1.6.3 The Global Longitudinal Study of Osteoporosis in Women, 24

1.6.4 The Adolescent Placement Study, 26

1.6.5 The Burn Injury Study, 27

1.6.6 The Myopia Study, 29

1.6.7 The NHANES Study, 31

1.6.8 The Polypharmacy Study, 31

Exercises, 32

2 The Multiple Logistic Regression Model 35

2.1 Introduction, 35

2.2 The Multiple Logistic Regression Model, 35

2.3 Fitting the Multiple Logistic Regression Model, 37

2.4 Testing for the Significance of the Model, 39

2.5 Confidence Interval Estimation, 42

2.6 Other Estimation Methods, 45

Exercises, 46

3 Interpretation of the Fitted Logistic Regression Model 49

3.1 Introduction, 49

3.2 Dichotomous Independent Variable, 50

3.3 Polychotomous Independent Variable, 56

3.4 Continuous Independent Variable, 62

3.5 Multivariable Models, 64

3.6 Presentation and Interpretation of the Fitted Values, 77

3.7 A Comparison of Logistic Regression and Stratified Analysis for 2 × 2 Tables, 82

Exercises, 87

4 Model-Building Strategies and Methods for Logistic Regression 89

4.1 Introduction, 89

4.2 Purposeful Selection of Covariates, 89

4.2.1 Methods to Examine the Scale of a Continuous Covariate in the Logit, 94

4.2.2 Examples of Purposeful Selection, 107

4.3 Other Methods for Selecting Covariates, 124

4.3.1 Stepwise Selection of Covariates, 125

4.3.2 Best Subsets Logistic Regression, 133

4.3.3 Selecting Covariates and Checking their Scale Using Multivariable Fractional Polynomials, 139

4.4 Numerical Problems, 145

Exercises, 150

5 Assessing the Fit of the Model 153

5.1 Introduction, 153

5.2 Summary Measures of Goodness of Fit, 154

5.2.1 Pearson Chi-Square Statistic, Deviance, and Sum-of-Squares, 155

5.2.2 The Hosmer–Lemeshow Tests, 157

5.2.3 Classification Tables, 169

5.2.4 Area Under the Receiver Operating Characteristic Curve, 173

5.2.5 Other Summary Measures, 182

5.3 Logistic Regression Diagnostics, 186

5.4 Assessment of Fit via External Validation, 202

5.5 Interpretation and Presentation of the Results from a Fitted Logistic Regression Model, 212

Exercises, 223

6 Application of Logistic Regression with Different Sampling Models 227

6.1 Introduction, 227

6.2 Cohort Studies, 227

6.3 Case-Control Studies, 229

6.4 Fitting Logistic Regression Models to Data from Complex Sample Surveys, 233

Exercises, 242

7 Logistic Regression for Matched Case-Control Studies 243

7.1 Introduction, 243

7.2 Methods For Assessment of Fit in a 1–M Matched Study, 248

7.3 An Example Using the Logistic Regression Model in a 1–1 Matched Study, 251

7.4 An Example Using the Logistic Regression Model in a 1–M Matched Study, 260

Exercises, 267

8 Logistic Regression Models for Multinomial and Ordinal Outcomes 269

8.1 The Multinomial Logistic Regression Model, 269

8.1.1 Introduction to the Model and Estimation of Model Parameters, 269

8.1.2 Interpreting and Assessing the Significance of the Estimated Coefficients, 272

8.1.3 Model-Building Strategies for Multinomial Logistic Regression, 278

8.1.4 Assessment of Fit and Diagnostic Statistics for the Multinomial Logistic Regression Model, 283

8.2 Ordinal Logistic Regression Models, 289

8.2.1 Introduction to the Models, Methods for Fitting, and Interpretation of Model Parameters, 289

8.2.2 Model Building Strategies for Ordinal Logistic Regression Models, 305

Exercises, 310

9 Logistic Regression Models for the Analysis of Correlated Data 313

9.1 Introduction, 313

9.2 Logistic Regression Models for the Analysis of Correlated Data, 315

9.3 Estimation Methods for Correlated Data Logistic Regression Models, 318

9.4 Interpretation of Coefficients from Logistic Regression Models for the Analysis of Correlated Data, 323

9.4.1 Population Average Model, 324

9.4.2 Cluster-Specific Model, 326

9.4.3 Alternative Estimation Methods for the Cluster-Specific Model, 333

9.4.4 Comparison of Population Average and Cluster-Specific Model, 334

9.5 An Example of Logistic Regression Modeling with Correlated Data, 337

9.5.1 Choice of Model for Correlated Data Analysis, 338

9.5.2 Population Average Model, 339

9.5.3 Cluster-Specific Model, 344

9.5.4 Additional Points to Consider when Fitting Logistic Regression Models to Correlated Data, 351

9.6 Assessment of Model Fit, 354

9.6.1 Assessment of Population Average Model Fit, 354

9.6.2 Assessment of Cluster-Specific Model Fit, 365

9.6.3 Conclusions, 374

Exercises, 375

10 Special Topics 377

10.1 Introduction, 377

10.2 Application of Propensity Score Methods in Logistic Regression Modeling, 377

10.3 Exact Methods for Logistic Regression Models, 387

10.4 Missing Data, 395

10.5 Sample Size Issues when Fitting Logistic Regression Models, 401

10.6 Bayesian Methods for Logistic Regression, 408

10.6.1 The Bayesian Logistic Regression Model, 410

10.6.2 MCMC Simulation, 411

10.6.3 An Example of a Bayesian Analysis and Its Interpretation, 419

10.7 Other Link Functions for Binary Regression Models, 434

10.8 Mediation, 441

10.8.1 Distinguishing Mediators from Confounders, 441

10.8.2 Implications for the Interpretation of an Adjusted Logistic Regression Coefficient, 443

10.8.3 Why Adjust for a Mediator? 444

10.8.4 Using Logistic Regression to Assess Mediation: Assumptions, 445

10.9 More About Statistical Interaction, 448

10.9.1 Additive versus Multiplicative Scale–Risk Difference versus Odds Ratios, 448

10.9.2 Estimating and Testing Additive Interaction, 451

Exercises, 456

References 459

Index 479

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