Applied Logistic Regression

Preface to the Third Edition xiii

1 Introduction to the Logistic Regression Model 1

1.1 Introduction 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