Statistical Methods for Survival Data Analysis

ELISA T. LEE, PhD, is Regents Professor and George Lynn Cross Research Professor of Biostatistics and Epidemiology and Director of the Center for American Indian Health Research at the University of Oklahoma Health Sciences Center. JOHN Wenyu WANG, PhD, is Professor of Research at the Center for American Indian Health Research at the University of Oklahoma Health Sciences Center.

Preface xi

1 Introduction 1

1.1 Preliminaries 1

1.2 Censored Data 2

1.3 Scope of the Book 5

2 Functions of Survival Time 8

2.1 Definitions 8

2.2 Relationships of the Survival Functions 15

Exercises 16

3 Examples of Survival Data Analysis 19

3.1 Example 3.1: Comparison of Two Treatments and Three Diets 19

3.2 Example 3.2: Comparison of Two Survival Patterns Using Life Tables 26

3.3 Example 3.3: Fitting Survival Distributions to Tumor-Free Times 28

3.4 Example 3.4: Comparing Survival of a Cohort with that of a General Population — Relative Survival 30

3.5 Example 3.5: Identification of Risk Factors for Incident Events 33

3.6 Example 3.6: Identification of Risk Factors for the Prevalence of Age-Related Macular Degeneration 38

3.7 Example 3.7: Identification of Significant Risk Factors for Incident Hypertension Using Related Data (Repeated Measurements) in a Longitudinal Study 46

Exercises 54

4 Nonparametric Methods of Estimating Survival Functions 68

4.1 Product-Limit Estimates of Survivorship Function 69

4.2 N elson–Aalen Estimates of Survivorship Function 82

4.3 Life-Table Analysis 83

4.4 Relative Survival Rates 96

4.5 Standardized Rates and Ratios 98

Exercises 104

5 Nonparametric Methods for Comparing Survival Distributions 108

5.1 Comparison of Two Survival Distributions 108

5.2 The Mantel and Haenszel Test 123

5.3 Comparison of K (K > 2) Samples 128

Exercises 130

6 Some Well-Known Parametric Survival Distributions And Their Applications 133

6.1 Exponential Distribution 133

6.2 Weibull Distribution 138

6.3 Lognormal Distribution 143

6.4 Gamma, Generalized Gamma, and Extended Generalized Gamma Distributions 148

6.5 Log-Logistic Distribution 153

6.6 O ther Survival Distributions 155

Exercises 159

7 Estimation Procedures for Parametric Survival Distributions Without Covariates 161

7.1 General Maximum Likelihood Estimation Procedure 161

7.2 Exponential Distribution 165

7.3 Weibull Distribution 178

7.4 Lognormal Distribution 180

7.5 The Extended Generalized Gamma Distribution 183

7.6 The Log-Logistic Distribution 184

7.7 Gompertz Distribution 185

7.8 Graphical Methods 186

Exercises 203

8 Tests of Goodness-of-Fit and Distribution Selection 206

8.1 Goodness-of-Fit Test Statistics Based on Asymptotic Likelihood Inferences 207

8.2 Tests for Appropriateness of a Family of Distributions 210

8.3 Selection of a Distribution by Using BIC or AIC Procedure 216

8.4 Tests for a Specific Distribution with Known Parameters 217

8.5 Hollander and Proschan’s Test for Appropriateness of a Given Distribution with Known Parameters 220

Exercises 224

9 Parametric Methods for Comparing Two Survival Distributions 226

9.1 Log-Likelihood Ratio Test for Comparing Two Survival Distributions 226

9.2 Comparison of Two Exponential Distributions 229

9.3 Comparison of Two Weibull Distributions 234

9.4 Comparison of Two Gamma Distributions 236

Exercises 237

10 Parametric Methods for Regression Model Fitting and Identification of Prognostic Factors 239

10.1 Preliminary Examination of Data 240

10.2 General Structure of Parametric Regression Models and Their Asymptotic Likelihood Inference 242

10.3 Exponential AFT Model 246

10.4 Weibull AFT Model 255

10.5 Lognormal AFT Model 258

10.6 The Extended Generalized Gamma AFT Model 261

10.7 Log-Logistic AFT Model 264

10.8 O ther Parametric Regression Models 268

10.9 Model Selection Methods 270

Exercises 279

11 Identification of Risk Factors Related to Survival Time: Cox Proportional Hazards Model 282

11.1 The Proportional Hazards Model 282

11.2 The Partial Likelihood Function 285

11.3 Identification of Significant Covariates 302

11.4 Estimation of the Survivorship Function with Covariates 309

11.5 Adequacy Assessment of the Proportional Hazards Model 317

Exercises 334

12 Identification of Prognostic Factors Related to Survival Time: Non-Proportional Hazards Models 337

12.1 Models with Time-Dependent Covariates 337

12.2 Stratified Proportional Hazards Model 346

12.3 Competing Risks Model 350

12.4 Recurrent Event Models 353

12.5 Models for Related Observations 370

Exercises 382

13 Identification of Risk Factors Related to Dichotomous and Polychotomous Outcomes 384

13.1 Univariate Analysis 385

13.2 Logistic and Conditional Logistic Regression Model for Dichotomous Outcomes 392

13.3 Models for Polychotomous Outcomes 421

13.4 Models for Related Observations 432

Exercises 440

Appendix 443

References 466

Index 477