Handbook of Statistics Vol 23 - Advances in Survival Analysis
Table of contents
Preface v
Contributors xxi
PART I. GENERAL METHODOLOGY
Ch. 1. Evaluation of the Performance of Survival Analysis Models:
Discrimination and Calibration Measures 1
R.B. D?Agostino and B.-H. Nam
1. Introduction 1
2. Discrimination index 2
3. Calibration measures in survival analysis 6
Appendix A 20
References 25
Ch. 2. Discretizing a Continuous Covariate in Survival Studies 27
J.P. Klein and J.-T. Wu
1. Introduction 27
2. Techniques based on the Cox model with a single covariate 28
3. Extensions of Contal and O?Quigley?s approach 37
4. Discussion 41
Acknowledgements 42
References 42
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viii Table of contents
Ch. 3. On Comparison of Two Classification Methods with Survival Endpoints 43
Y. Lu, H. Jin and J. Mi
1. Introduction 43
2. Degree of separation index 44
3. Estimation and inference procedures 48
4. Distribution property of test statistics under the null hypothesis 52
5. Application examples 55
6. Discussion and conclusion 56
Acknowledgement 58
References 58
Ch. 4. Time-Varying Effects in Survival Analysis 61
T.H. Scheike
1. Time-varying effects in survival analysis 61
2. Estimation for proportional or additive models 67
3. Testing in proportional and additive hazards models 74
4. Survival with malignant melanoma 79
5. Discussion 83
Acknowledgement 83
References 83
Ch. 5. Kaplan?Meier Integrals 87
W. Stute
1. Introduction 87
2. The SLLN 91
3. The CLT 93
4. Bias 95
5. The jackknife 97
6. Censored correlation and regression 100
7. Conclusions 102
References 103
PART II. CENSORED DATA AND INFERENCE
Ch. 6. Statistical Analysis of Doubly Interval-Censored Failure Time Data 105
J. Sun
1. Introduction 105
2. Nonparametric estimation of a distribution function 107
3. Semiparametric regression analysis 114
4. Nonparametric comparison of survival functions 117
5. Discussion and future researches 119
References 121
Table of contents ix
Ch. 7. The Missing Censoring-Indicator Model of Random Censorship 123
S. Subramanian
1. Introduction 123
2. Overview of the estimators of a survival function 127
3. Semiparametric estimation in the MCI model 134
4. Conclusion 139
Acknowledgement 140
References 140
Ch. 8. Estimation of the Bivariate Survival Function with Generalized Bivariate
Right Censored Data Structures 143
S. Kele?s, M.J. van der Laan and J.M. Robins
1. Introduction 143
2. Modeling the censoring mechanism 144
3. Constructing an initial mapping from full data estimating functions to observed data estimating
functions 147
4. Generalized Dabrowska?s estimator 151
5. Orthogonalized estimating function and corresponding estimator 153
6. Simulations 159
7. Discussion 162
Appendix A 162
References 172
Ch. 9. Estimation of Semi-Markov Models with Right-Censored Data 175
O. Pons
1. Introduction 175
2. Definition of the estimators 176
3. Asymptotic distribution of the estimators 181
4. Generalization to models with covariates 189
5. Discussion 193
References 194
PART III. TRUNCATED DATA AND INFERENCE
Ch. 10. Nonparametric Bivariate Estimation with Randomly Truncated
Observations 195
? G?rler
1. Introduction 195
2. Estimation of the bivariate distribution function 197
3. Estimation of bivariate hazard 202
4. Bivariate density estimation 205
References 206
x Table of contents
PART IV. HAZARD RATE ESTIMATION
Ch. 11. Lower Bounds for Estimating a Hazard 209
C. Huber and B. MacGibbon
1. Introduction 209
2. Framework 210
3. Kullback information and Hellinger distances based on hazards 213
4. A general device to derive lower bounds for estimating a function 215
5. Lower bound for the rate of estimation of a hazard function with right censoring 218
6. Rate of convergence for the kernel estimator of the hazard function 222
Acknowledgement 224
Appendix A 224
References 225
Ch. 12. Non-Parametric Hazard Rate Estimation under Progressive
Type-II Censoring 227
N. Balakrishnan and L. Bordes
1. Introduction 227
2. Smoothing cumulative hazard rate estimator 228
3. Asymptotics 230
4. Simulation study 238
Appendix A: Technical results for the mean 244
Appendix B: Technical results for the variance 246
References 248
PART V. COMPARISON OF SURVIVAL CURVES
Ch. 13. Statistical Tests of the Equality of Survival Curves:
Reconsidering the Options 251
G.P. Suciu, S. Lemeshow and M. Moeschberger
1. Introduction 251
2. Underlying alternative hypothesis and assumptions 252
3. An overview of available tests 256
4. Hypothesis testing and statistical computer packages 259
5. Applications to papers from major medical journal 260
6. Suggested guidelines 260
7. Discussion and conclusions 261
References 261
Table of contents xi
Ch. 14. Testing Equality of Survival Functions with Bivariate Censored Data:
A Review 263
P.V. Rao
1. Introduction 263
2. Testing H0 with uncensored paired data 265
3. Within-pair difference tests with censored data 269
4. Pooled sample tests with censored data 272
5. Testing H0 when there are missing data 273
6. Overview 274
References 275
Ch. 15. Statistical Methods for the Comparison of Crossing Survival Curves 277
C.T. Le
1. Introduction 277
2. The modified Kolmogorov?Smirnov test 281
3. A Levene-type test 282
4. Linear rank tests 287
References 289
PART VI. COMPETING RISKS AND ANALYSIS
Ch. 16. Inference for Competing Risks 291
J.P. Klein and R. Bajorunaite
1. Introduction 291
2. Basic quantities 292
3. Univariate estimation 293
4. Inference based on the crude hazard rates 298
5. Tests based on the cumulative incidence function 300
6. Regression techniques based on the cumulative hazard function 305
7. Discussion 309
Acknowledgement 310
References 310
Ch. 17. Analysis of Cause-Specific Events in Competing Risks Survival Data 313
J. Dignam, J. Bryant and H.S. Wieand
1. Introduction 313
2. Competing risks analysis based on cause-specific hazard functions 316
3. Competing risks analysis based on cumulative incidence functions 319
4. Examples: Competing risks analysis of events after breast cancer treatment 323
5. Summary 327
Acknowledgements 327
References 327
xii Table of contents
Ch. 18. Analysis of Progressively Censored Competing
Risks Data 331
D. Kundu, N. Kannan and N. Balakrishnan
1. Introduction 331
2. Model: Description and notation 333
3. Estimation 334
4. Confidence intervals 339
5. Bayesian analysis 342
6. Simulation study 343
7. Numerical example 345
8. Some generalizations and extensions 346
9. Conclusions 347
References 348
Ch. 19. Marginal Analysis of Point Processes
with Competing Risks 349
R.J. Cook, B. Chen and P. Major
1. Introduction 349
2. Rate functions for point processes 352
3. Point processes with terminal events 354
4. Application to a breast cancer trial 357
5. Discussion 360
Acknowledgements 360
References 360
PART VII. PROPORTIONAL HAZARDS MODEL AND ANALYSIS
Ch. 20. Categorical Auxiliary Data in the Discrete Time Proportional
Hazards Model 363
P. Slasor and N. Laird
1. Introduction 363
2. The standard and joint discrete-time proportional hazards models 364
3. Specification of the survival model and censoring 367
4. Discretizing continuous auxiliary data 367
5. Joint models: Recurrent events predicting survival 374
6. Other scenarios for censoring and survival 377
7. Discussion 378
Acknowledgements 379
Appendix A 379
References 382
Table of contents xiii
Ch. 21. Hosmer and Lemeshow type Goodness-of-Fit Statistics
for the Cox Proportional Hazards Model 383
S. May and D.W. Hosmer
1. Introduction 383
2. The Hosmer and Lemeshow type test statistics 383
3. Necessity for time-dependent indicator variables 386
4. Examples 389
5. Summary 391
Appendix A 391
Appendix B 391
Appendix C 392
References 393
Ch. 22. The Effects of Misspecifying Cox?s Regression Model
on Randomized Treatment Group Comparisons 395
A.G. DiRienzo and S.W. Lagakos
1. Introduction 395
2. Notation and statistics 396
3. Conditions for valid tests 397
4. Bias correction 399
5. Discussion 402
Acknowledgement 404
Appendix A: MATLAB code for computing statistical tests 404
References 408
Ch. 23. Statistical Modeling in Survival Analysis and Its Influence
on the Duration Analysis 411
V. Bagdonavi?cius and M. Nikulin
1. Introduction 411
2. The Cox or the proportional hazards model 412
3. Accelerated failure time model 413
4. Generalized proportional hazards model 415
5. Regression models with cross-effects of survival functions 419
6. Changing shape and scale models 420
7. Models with time-dependent regression coefficients 421
8. Additive hazards model and its generalizations 422
9. Remarks on parametric and semi-parametric estimation 423
References 428
xiv Table of contents
PART VIII. ACCELERATED MODELS AND ANALYSIS
Ch. 24. Accelerated Hazards Model: Method, Theory and Applications 431
Y.Q. Chen, N.P. Jewell and J. Yang
1. Introduction 431
2. Estimation 432
3. Asymptotic results 433
4. Efficiency consideration 434
5. Model adequacy 435
6. Extensions 437
7. Implementation and application 438
8. Some remarks 440
References 441
Ch. 25. Diagnostics for the Accelerated Life Time Model of Survival Data 443
D. Zelterman and H. Lin
1. Introduction 443
2. The likelihood and estimating equations 444
3. A Gibbs-like estimation procedure 448
4. Diagnostic measures 449
5. The bootstrap procedure 451
6. Numerical examples 453
Acknowledgement 459
References 459
Ch. 26. Cumulative Damage Approaches Leading
to Inverse Gaussian Accelerated Test Models 461
A. Onar and W.J. Padgett
1. Inverse Gaussian as a lifetime or strength model 462
2. Inverse Gaussian accelerated test models 463
3. Estimation for the inverse Gaussian accelerated test models 469
4. Application of the inverse Gaussian accelerated test models to chloroprene exposure data 474
5. Conclusion 476
Acknowledgement 477
References 477
Ch. 27. On Estimating the Gamma Accelerated Failure-Time Models 479
K.M. Koti
1. The failure time Gamma model 479
2. The maximum likelihood equations 480
3. The problem 481
4. The hybrid approximation 482
Table of contents xv
5. The SAS/IML subroutine NLPTR 487
6. Pediatric cancer data 487
7. Leukemia data 488
8. Concluding remarks 489
Acknowledgements 490
Appendix A: Fisher information matrix 490
References 493
PART IX. FRAILTY MODELS AND APPLICATIONS
Ch. 28. Frailty Model and its Application to Seizure Data 495
N. Ebrahimi, X. Zhang, A. Berg and S. Shinnar
1. Introduction 495
2. Inference for the shared frailty model 498
3. The shared frailty model for recurrent events 504
4. Seizure data and its analysis 507
5. Concluding remarks 516
Acknowledgements 516
References 516
PART X. MODELS AND APPLICATIONS
Ch. 29. State Space Models for Survival Analysis 519
W.Y. Tan and W. Ke
1. Introduction 519
2. The state space models and the generalized Bayesian approach 519
3. Stochastic modeling of the birth?death?immigration?illness?cure processes 521
4. A state space model for the birth?death?immigration?illness?cure processes 524
5. The multi-level Gibbs sampling procedures for the birth?death?immigration?illness?cure processes 526
6. The survival probabilities of normal and sick people 528
7. Some illustrative examples 529
8. Conclusions 534
References 534
Ch. 30. First Hitting Time Models for Lifetime Data 537
M.-L.T. Lee and G.A. Whitmore
1. Introduction 537
2. The basic first hitting time model 537
3. Data for model estimation 538
4. A Wiener process with an inverse Gaussian first hitting time 538
5. A two-dimensional Wiener model for a marker and first hitting time 539
xvi Table of contents
6. Longitudinal data 541
7. Additional first hitting time models 541
8. Other literature sources 542
Acknowledgements 543
References 543
Ch. 31. An Increasing Hazard Cure Model 545
Y. Peng and K.B.G. Dear
1. Introduction 545
2. The model 546
3. Simulation study 549
4. Illustration 552
5. Conclusions and discussion 555
Acknowledgements 556
References 556
PART XI. MULTIVARIATE SURVIVAL DATA ANALYSIS
Ch. 32. Marginal Analyses of Multistage Data 559
G.A. Satten and S. Datta
1. Introduction 559
2. ?Explainable?dependent censoring in survival analysis 560
3. Multistage models: Stage occupation probabilities and marginal transition hazards 562
4. Estimation of marginal waiting time distributions 564
5. Regression models for waiting time distributions 566
Appendix A: Modeling the censoring hazard using Aalen?s linear hazards model 570
References 573
Ch. 33. The Matrix-Valued Counting Process Model with Proportional Hazards
for Sequential Survival Data 575
K.L. Kesler and P.K. Sen
1. Introduction 575
2. Introduction to multivariate survival methods 577
3. The matrix valued counting process framework 579
4. Matrix valued counting process framework with repeated measures data 585
5. Estimation 592
6. Example 593
7. Discussion 595
Appendix A 596
References 600
Further reading 601
Table of contents xvii
PART XII. RECURRENT EVENT DATA ANALYSIS
Ch. 34. Analysis of Recurrent Event Data 603
J. Cai and D.E. Schaubel
1. Introduction 603
2. Notation and basic functions of interest 603
3. Semiparametric models for recurrent event data 605
4. Nonparametric estimation of the recurrent event survival and distribution functions 618
5. Conclusion 621
Acknowledgement 621
References 621
PART XIII. CURRENT STATUS DATA ANALYSIS
Ch. 35. Current Status Data: Review, Recent Developments
and Open Problems 625
N.P. Jewell and M. van der Laan
1. Introduction 625
2. Motivating examples 626
3. Simple current status data 627
4. Different sampling schemes 630
5. Complex outcome processes 634
6. Conclusion 640
References 641
PART XIV. DISEASE PROGRESSION ANALYSIS
Ch. 36. Appraisal of Models for the Study of Disease Progression
in Psoriatic Arthritis 643
R. Aguirre-Hern?ndez and V.T. Farewell
1. Introduction 643
2. Data 643
3. Markov models 644
4. Poisson and negative binomial models 654
5. Discussion 670
Appendix A: Formulas for the estimated transition probabilities 671
References 672
xviii Table of contents
PART XV. GENE EXPRESSIONS AND ANALYSIS
Ch. 37. Survival Analysis with Gene Expression Arrays 675
D.K. Pauler, J. Hardin, J.R. Faulkner, M. LeBlanc and J.J. Crowley
1. Introduction 675
2. Methods 678
3. Results 681
4. Discussion 684
Appendix A 685
References 686
PART XVI. QUALITY OF LIFE ANALYSIS
Ch. 38. Joint Analysis of Longitudinal Quality of Life and Survival Processes 689
M. Mesbah, J.-F. Dupuy, N. Heutte and L. Awad
1. Introduction 689
2. Presentation of the clinical trial: QoL instruments and data 691
3. Preliminary analysis 693
4. Time to QoL deterioration 696
5. Semi-Markovian multi-state model 698
6. Joint distribution of QoL and survival?dropout processes 709
7. Discussion 720
References 726
PART XVII. FLOWGRAPH MODELS AND APPLICATIONS
Ch. 39. Modelling Survival Data using Flowgraph Models 729
A.V. Huzurbazar
1. Series flowgraph model: HIV blood transfusion data 731
2. Data analysis of HIV/AIDS data 732
3. Converting flowgraph MGFs to densities 734
4. Likelihood construction in flowgraph models 737
5. Parametric assumptions 737
6. Parallel flowgraph models 740
7. Loop flowgraph models 740
8. A systematic procedure for solving flowgraphs 742
9. Data analysis for diabetic retinopathy data 743
10. Summary 745
References 746
Table of contents xix
PART XVIII. REPAIR MODELS AND ANALYSIS
Ch. 40. Nonparametric Methods for Repair Models 747
M. Hollander and J. Sethuraman
1. Introduction 747
2. General repair models 748
3. Estimation in the DHS model 752
4. Estimation in the BBS model 755
5. A two-sample test in the BBS model 758
6. Goodness-of-fit tests in the BBS model 759
7. Testing the minimal repair assumption in the BBS model 761
Acknowledgement 763
References 763
Subject Index 765
Contents of Previous Volumes 773