Cure Models: Methods, Applications, and Implementation
Yingwei Peng, Binbing Yu
Contents
Preface xiii
Glossary xv
1 Introduction 1
1.1 A Brief Review of Cure Models . . . . . . . . . . . . . . . . . 1
1.1.1 Time-to-Event Data and Cured Subjects . . . . . . . . 1
1.1.2 Survival Models and Cure Models . . . . . . . . . . . 2
1.2 Aim and Scope of the Book . . . . . . . . . . . . . . . . . . . 4
2 The Parametric Cure Model 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Parametric Mixture Cure Models . . . . . . . . . . . . . . . 7
2.2.1 Parametric Incidence Submodel . . . . . . . . . . . . . 8
2.2.2 Parametric Latency Submodel . . . . . . . . . . . . . 9
2.2.2.1 Parametric PH Latency Submodel . . . . . . 10
2.2.2.2 Parametric AFT Latency Submodel . . . . . 11
2.2.2.3 Other Parametric Latency Submodels . . . . 12
2.3 Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Direct Maximization of Observed Likelihood Function 13
2.3.2 Estimation via EM Algorithm . . . . . . . . . . . . . . 14
2.4 Non-Mixture Cure Models . . . . . . . . . . . . . . . . . . . 16
2.4.1 Proportional Hazards Cure Model . . . . . . . . . . . 16
2.4.2 Cure Models Based on Tumor Activation Scheme . . . 19
2.4.3 Cure Models Based on Frailty Models . . . . . . . . . 20
2.4.4 Cure Models Based on Box-Cox Transformation . . . 21
2.5 Model Assessment . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.1 Choosing an Appropriate Parametric Distribution . . 23
2.5.2 Mixture vs Non-Mixture Cure Models . . . . . . . . . 24
2.5.3 Goodness of Fit by Residuals . . . . . . . . . . . . . . 25
2.6 Software and Applications . . . . . . . . . . . . . . . . . . . 26
2.6.1 R Package gfcure . . . . . . . . . . . . . . . . . . . . . 27
2.6.2 R Package mixcure . . . . . . . . . . . . . . . . . . . . 31
2.6.3 R Package flexsurvcure . . . . . . . . . . . . . . . . . . 35
2.6.4 SAS Macro PSPMCM . . . . . . . . . . . . . . . . . . 37
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 The Semiparametric and Nonparametric Cure Models 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Semiparametric Mixture Cure Models . . . . . . . . . . . . . 41
3.2.1 Semiparametric PH Latency Submodel . . . . . . . . . 42
3.2.1.1 Restrictions on the Upper Tail of the Baseline
Distribution . . . . . . . . . . . . . . . . . . 43
3.2.1.2 Time-Dependent Covariates in the Latency
Submodel . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Semiparametric AFT Latency Submodel . . . . . . . . 44
3.2.2.1 Linear Rank Method . . . . . . . . . . . . . 45
3.2.2.2 M-Estimation Method . . . . . . . . . . . . . 46
3.2.2.3 Kernel Smoothing Method . . . . . . . . . . 46
3.2.3 Semiparametric AH Latency Submodel . . . . . . . . 47
3.2.3.1 Linear Rank Method . . . . . . . . . . . . . 47
3.2.3.2 Kernel Smoothing Method . . . . . . . . . . 48
3.2.4 Semiparametric Transformation Latency Submodels . 49
3.2.5 Semiparametric Incidence Submodel . . . . . . . . . . 51
3.2.6 Semiparametric Spline-Based Cure Models . . . . . . 52
3.3 Nonparametric Mixture Cure Models . . . . . . . . . . . . . 54
3.3.1 Nonparametric Incidence Submodels . . . . . . . . . . 54
3.3.1.1 Kaplan-Meier Estimator . . . . . . . . . . . . 54
3.3.1.2 Generalized Kaplan-Meier Estimator . . . . . 55
3.3.2 Nonparametric Latency Submodels . . . . . . . . . . . 57
3.4 Semiparametric Non-Mixture Cure Models . . . . . . . . . . 58
3.4.1 Semiparametric PHC Model . . . . . . . . . . . . . . . 58
3.4.2 General Non-Mixture Cure Models . . . . . . . . . . . 59
3.5 Model Assessment . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.1 Residuals for Overall Model Fitting . . . . . . . . . . 62
3.5.2 Residuals for Latency Submodels . . . . . . . . . . . . 63
3.5.3 Assessing Cure Rate Prediction . . . . . . . . . . . . . 65
3.5.4 Concordance Measures for Cure Models . . . . . . . . 67
3.5.5 Testing Goodness-of-Fit of Parametric Cure Rate
Estimation . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.6 Variable Selection . . . . . . . . . . . . . . . . . . . . 70
3.6 Software and Applications . . . . . . . . . . . . . . . . . . . 72
3.6.1 R Package mixcure . . . . . . . . . . . . . . . . . . . . 72
3.6.2 R Package smcure . . . . . . . . . . . . . . . . . . . . 73
3.6.3 SAS Macro PSPMCM . . . . . . . . . . . . . . . . . . 74
3.6.4 R Package Survival . . . . . . . . . . . . . . . . . . . . 75
3.6.5 R Package npcure . . . . . . . . . . . . . . . . . . . . 77
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4 Cure Models for Multivariate Survival Data and Competing
Risks 83
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Marginal Cure Models . . . . . . . . . . . . . . . . . . . . . . 84
4.2.1 Marginal Models with Working Independence . . . . . 84
4.2.2 Marginal Models with Specified Correlation Structures 86
4.3 Cure Models with Random Effects . . . . . . . . . . . . . . . 89
4.3.1 Mixture Cure Models with Frailties . . . . . . . . . . . 89
4.3.2 Non-Nixture Cure Model with Frailties . . . . . . . . . 92
4.4 Cure Models for Recurrent Event Data . . . . . . . . . . . . 93
4.5 Cure Models for Competing-Risks Survival Data . . . . . . . 96
4.5.1 Classical Approach . . . . . . . . . . . . . . . . . . . . 97
4.5.2 Vertical Approach . . . . . . . . . . . . . . . . . . . . 101
4.6 Software and Applications . . . . . . . . . . . . . . . . . . . 102
4.6.1 R Package geecure . . . . . . . . . . . . . . . . . . . . 102
4.6.2 R Package intcure . . . . . . . . . . . . . . . . . . . . 105
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Joint Modeling of Longitudinal and Survival Data with a
Cure Fraction 111
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2 Longitudinal and Survival Data with a Cured Fraction . . . 111
5.3 Joint Modeling Longitudinal and Survival Data with Shared
Random Effects . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4 Modeling Longitudinal Proportional Data in Joint Modeling 116
5.5 Joint Modeling by Including Longitudinal Effects in Cure
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6 Testing the Existence of Cured Subjects and Sufficient
Follow-up 127
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Tests for Existence of Cured Subjects . . . . . . . . . . . . . 128
6.2.1 Without Covariates . . . . . . . . . . . . . . . . . . . 128
6.2.1.1 Likelihood Ratio Test . . . . . . . . . . . . . 128
6.2.1.2 Score Test . . . . . . . . . . . . . . . . . . . 129
6.2.2 With Covariates . . . . . . . . . . . . . . . . . . . . . 130
6.3 Testing for Sufficient Follow-up . . . . . . . . . . . . . . . . . 131
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7 Bayesian Cure Model 135
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Flexible Cure Model with Latent Activation Schemes . . . . 135
7.2.1 Model Formulation and Inference . . . . . . . . . . . . 136
7.2.2 Bayesian Cure Model with Negative Binomial
Distribution . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2.3 Application . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3 Bayesian Cure Models with Generalized Modified Weibull
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.3.1 Model Formulation and Inference . . . . . . . . . . . . 144
7.3.2 Application . . . . . . . . . . . . . . . . . . . . . . . . 146
7.4 Bayesian Mixture Cure Model with Spatially Correlated
Frailties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.4.1 Spatial Mixture Cure Model . . . . . . . . . . . . . . . 149
7.4.2 Application . . . . . . . . . . . . . . . . . . . . . . . . 152
7.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8 Analysis of Population-Based Cancer Survival Data 159
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.2 Population-Based Cancer Registry and Survival Data . . . . 160
8.3 Parametric Cure Models for Net Survival . . . . . . . . . . . 165
8.3.1 Flexible Parametric Survival Model . . . . . . . . . . 166
8.3.2 Flexible Parametric Cure Model . . . . . . . . . . . . 167
8.3.3 Software Implementations . . . . . . . . . . . . . . . . 168
8.4 Testing the Existence of Statistical Cure . . . . . . . . . . . 171
8.4.1 Testing Hypothesis of Non-Inferiority of Survival . . . 172
8.4.2 A Minimum Version of One-Sample Log-Rank Test . . 172
8.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.5.1 Weibull Mixture Cure Model for Grouped Survival
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.5.2 Analysis of Individually-Listed Colorectal Cancer
Relative Survival Data . . . . . . . . . . . . . . . . . . 178
8.5.3 Testing the Existence of Cure for Colorectal Cancer
Patients . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9 Design and Analysis of Cancer Clinical Trials 187
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9.2 Testing Treatment Effects in the Presence of Cure . . . . . . 189
9.2.1 Comparison of Log-Rank Type Tests . . . . . . . . . . 191
9.2.2 Sample Size for the Weighted Log-Rank Test under the
Proportional Hazards Cure Model . . . . . . . . . . . 194
9.2.3 Power and Sample Size in the Presence of Delayed Onset
of Treatment Effect and Cure . . . . . . . . . . . . . . 199
9.3 Some Design Issues in Clinical Trials with Cure . . . . . . . 204
9.3.1 Cure Modeling in Real-Time Prediction . . . . . . . . 204
9.3.2 Futility Analysis of Survival Data with Cure . . . . . 206
9.3.2.1 Conditional Power for Mixture Cure Models 207
9.3.2.2 Conditional Power for Non-Mixture Cure
Models . . . . . . . . . . . . . . . . . . . . . 210
9.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.4.1 Sample Size Calculation for Trial Design . . . . . . . . 216
9.4.2 Predicting Future Number of Events . . . . . . . . . . 218
9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Bibliography 221
Index 247