Handbook of Statistics 23: Advances in Survival Analysis

Preface......Page 1
Table of contents......Page 3
Contributors......Page 16
Measures of predictive accuracy......Page 21
Discrimination index in logistic regression......Page 22
Extension of C statistics to survival analysis......Page 23
A chi-square statistic for the survival time model with censored observations......Page 26
Poisson approximation to survival time model......Page 27
Chi-square statistic based on the Poisson approximation......Page 30
The asymptotic null distribution of K2 and Kt2......Page 37
Algorithms for generating numerical examples......Page 38
Comparison of the proposed statistic (2) and Kt2......Page 39
References......Page 45
Introduction......Page 46
Martingale residuals......Page 47
Estimation of the threshold parameter......Page 49
Inference for beta......Page 51
Comparison of test procedures......Page 53
Accelerated failure time models with a single covariate......Page 56
Adjustment for other covariates......Page 59
Discussion......Page 60
References......Page 61
Introduction......Page 62
Degree of separation index......Page 63
Estimation and inference procedures......Page 67
Distribution property of test statistics under the null hypothesis......Page 71
Application examples......Page 74
Discussion and conclusion......Page 75
References......Page 77
Time-varying effects in survival analysis......Page 79
Estimation for proportional or additive models......Page 85
Cox's proportional regression model......Page 86
Aalen's additive model......Page 89
Additive-multiplicative intensity models......Page 90
Testing in proportional and additive hazards models......Page 92
Non-parametric model and testing......Page 93
Semi-parametric model......Page 94
Test for time-dependent effects in semi-parametric model......Page 96
Survival with malignant melanoma......Page 97
References......Page 101
Introduction......Page 104
The SLLN......Page 108
The CLT......Page 110
Bias......Page 112
The jackknife......Page 114
Censored correlation and regression......Page 117
Conclusions......Page 119
References......Page 120
Introduction......Page 122
Estimation without truncation......Page 124
Estimation with truncation......Page 127
Semiparametric regression analysis......Page 131
Nonparametric comparison of survival functions......Page 134
Discussion and future researches......Page 136
References......Page 138
Introduction......Page 140
The (nonparametric) KME......Page 144
Estimators of S(t) under MCAR......Page 145
Estimators of S(t) under MAR (CAR)......Page 147
Information bound for estimating S(t)......Page 148
Information bound for estimating H1(t)......Page 150
Semiparametric estimation in the MCI model......Page 151
Comparison of asymptotic variances......Page 155
Conclusion......Page 156
References......Page 157
Introduction......Page 159
Modeling the censoring mechanism......Page 160
Estimating functions in the observed data model......Page 163
Initial mappings that correspond with a specified RAL estimator......Page 164
Generalized Dabrowska's estimator......Page 167
Orthogonalized estimating function and corresponding estimator......Page 169
Estimation of Q1(FX,G) by Monte Carlo simulations......Page 173
Simulations......Page 175
Comparison of µn0 with µnDab......Page 176
Comparison of µn0, µnDab and µn1......Page 177
Discussion......Page 178
References......Page 188
Introduction......Page 190
Definition of the estimators......Page 191
Asymptotic distribution of the estimators......Page 196
Generalization to models with covariates......Page 204
Discussion......Page 208
References......Page 209
Introduction......Page 210
Estimation of the bivariate distribution function......Page 212
An alternative estimator......Page 215
Doubly truncated data......Page 216
Estimation of bivariate hazard......Page 217
Reverse hazard with right truncation......Page 219
Bivariate density estimation......Page 220
References......Page 221
Introduction......Page 223
Framework......Page 224
Kullback information and Hellinger distances based on hazards......Page 227
A general device to derive lower bounds for estimating a function......Page 229
Lower bound for F={h>=0: ||h(s)||22Rate of convergence for the kernel estimator of the hazard function......Page 236
Risk......Page 237
Acknowledgement......Page 238
References......Page 239
Introduction......Page 241
Non-parametric maximum likelihood estimator......Page 242
Smoothing the NPMLE of Lambda......Page 243
Assumptions and preliminary results......Page 244
Consistency......Page 247
Optimal bandwidth......Page 249
Asymptotic normality......Page 251
Two methods to choose an optimal bandwidth......Page 252
A simulation study......Page 253
Technical results for the mean......Page 258
Technical results for the variance......Page 260
References......Page 262
Introduction......Page 264
Underlying alternative hypothesis and assumptions......Page 265
An overview of available tests......Page 269
Hypothesis testing and statistical computer packages......Page 272
Suggested guidelines......Page 273
References......Page 274
Introduction......Page 276
Within pair difference tests.......Page 278
Pooled sample tests.......Page 279
Implementing nonparametric tests with paired data.......Page 280
Applications to survival data.......Page 281
Within-pair difference tests with censored data......Page 282
Pooled sample tests with censored data......Page 285
Testing H0 when there are missing data......Page 286
Overview......Page 287
References......Page 288
Introduction......Page 289
The modified Kolmogorov-Smirnov test......Page 293
A Levene-type test......Page 294
Linear rank tests......Page 299
References......Page 301
Introduction......Page 302
Basic quantities......Page 303
Univariate estimation......Page 304
Inference based on the crude hazard rates......Page 309
Tests based on the cumulative incidence function......Page 311
Regression techniques based on the cumulative hazard function......Page 316
Discussion......Page 320
References......Page 321
Background......Page 323
Identifiability......Page 324
Competing risks analyses based on estimable quantities......Page 325
Regression models for hazards......Page 326
The log-rank test applied to competing risk data......Page 327
A test for equality of all cause-specific hazards across multiple groups......Page 328
Competing risks analysis based on cumulative incidence functions......Page 329
A partially parametric cumulative incidence function estimator......Page 330
Tests for comparing cumulative incidence functions......Page 332
A trial to study tamoxifen and radiotherapy for the treatment of women with small (<=1 cm) tumors......Page 333
Analysis of cause-specific hazards......Page 334
Cumulative incidence estimates and test......Page 335
References......Page 337
Introduction......Page 340
Model: Description and notation......Page 342
Estimation......Page 343
Approximate confidence intervals......Page 348
Fisher information matrix......Page 349
Bootstrap confidence intervals......Page 350
Bayesian analysis......Page 351
Simulation study......Page 352
Numerical example......Page 354
Unknown causes of failure......Page 355
Conclusions......Page 356
References......Page 357
Overview......Page 358
Time to event data and competing risks......Page 359
Rate functions for point processes......Page 361
Joint models......Page 363
Connections with competing risks methodology......Page 365
Bone metastases and skeletal related events......Page 366
References......Page 369
Introduction......Page 371
The standard and joint discrete-time proportional hazards models......Page 372
Likelihood function......Page 373
Maximum likelihood estimation......Page 374
Discretizing continuous auxiliary data......Page 375
A binary measurement at baseline......Page 376
Polychotomous measurement at baseline......Page 378
Two binary repeated measurements......Page 380
Joint models: Recurrent events predicting survival......Page 382
Other scenarios for censoring and survival......Page 385
Discussion......Page 386
Acknowledgements......Page 387
References......Page 390
The Hosmer and Lemeshow type test statistics......Page 391
Necessity for time-dependent indicator variables......Page 394
Examples......Page 397
Summary......Page 399
References......Page 401
Introduction......Page 403
Notation and statistics......Page 404
Conditions for valid tests......Page 405
Efficiency considerations......Page 406
Bias correction......Page 407
Discussion......Page 410
MATLAB code for computing statistical tests......Page 412
References......Page 416
Introduction......Page 418
The Cox or the proportional hazards model......Page 419
Accelerated failure time model......Page 420
Definitions......Page 422
Relations with the linear transformations and frailty models......Page 423
The first GPH model......Page 424
The third GPH model......Page 425
Second CE-model......Page 426
Changing shape and scale models......Page 427
PH model with time-dependent regression coefficients......Page 428
Additive hazards model and its generalizations......Page 429
Remarks on parametric and semi-parametric estimation......Page 430
References......Page 435
Introduction......Page 437
Estimation......Page 438
Asymptotic results......Page 439
Efficiency consideration......Page 440
Kolmogorov-Smirnov test......Page 441
Gill-Schumacher test......Page 442
General model I......Page 443
Implementation and application......Page 444
Some remarks......Page 446
References......Page 447
Introduction......Page 448
The likelihood and estimating equations......Page 449
A Gibbs-like estimation procedure......Page 453
Brownian bridge statistic......Page 454
Partial residuals......Page 455
The unconstrained bootstrap......Page 456
The constrained bootstrap......Page 457
AML......Page 458
Stanford heart transplant data......Page 461
Bladder cancer......Page 463
References......Page 464
26. Cumulative Damage Approaches Leading to Inverse Gaussian Accelerated Test Models......Page 465
Inverse Gaussian as a lifetime or strength model......Page 466
Inverse Gaussian accelerated test models......Page 467
Estimation for the inverse Gaussian accelerated test models......Page 473
Application of the inverse Gaussian accelerated test models to chloroprene exposure data......Page 478
Conclusion......Page 480
References......Page 481
The failure time Gamma model......Page 482
The maximum likelihood equations......Page 483
The problem......Page 484
The hybrid approximation......Page 485
Pediatric cancer data......Page 490
Leukemia data......Page 491
Concluding remarks......Page 492
Fisher information matrix......Page 493
References......Page 496
Introduction......Page 497
Inference for the shared frailty model......Page 500
The EM algorithm......Page 502
The gamma frailty model......Page 503
The positive stable frailty model......Page 505
The shared frailty model for recurrent events......Page 506
Seizure data and its analysis......Page 509
References......Page 518
The state space models and the generalized Bayesian approach......Page 520
A general Bayesian procedure for estimating unknown parameters and state variables via state space models......Page 521
Stochastic modeling of the birth-death-immigration-illness-cure processes......Page 522
The stochastic differential equations for the state variables......Page 523
The stochastic system model and the probability distribution of the state variables......Page 525
The observation model......Page 526
The multi-level Gibbs sampling procedures for the birth-death-immigration-illness-cure processes......Page 527
The survival probabilities of normal and sick people......Page 529
Some illustrative examples......Page 530
References......Page 535
The basic first hitting time model......Page 537
A Wiener process with an inverse Gaussian first hitting time......Page 538
A two-dimensional Wiener model for a marker and first hitting time......Page 539
Additional first hitting time models......Page 541
Other literature sources......Page 542
References......Page 543
Introduction......Page 544
The model......Page 545
Simulation study......Page 548
Illustration......Page 551
Conclusions and discussion......Page 554
References......Page 555
Introduction......Page 557
``Explainable'' dependent censoring in survival analysis......Page 558
Multistage models: Stage occupation probabilities and marginal transition hazards......Page 560
Estimation of marginal waiting time distributions......Page 562
Regression models for waiting time distributions......Page 564
Modeling the censoring hazard using Aalen's linear hazards model......Page 568
References......Page 571
Introduction......Page 573
Cox proportional hazards model......Page 575
Multivariate survival methods......Page 576
Bivariate matrix valued counting process......Page 577
Matrix valued counting process framework and the cox proportional hazards model......Page 580
Multivariate matrix valued counting process framework......Page 581
Matrix valued counting process framework with repeated measures data......Page 583
Multivariate matrix valued counting process model and the Cox proportional hazards model......Page 585
Parameterizations and interpretations......Page 586
Estimation......Page 590
Methods......Page 591
Results......Page 592
Discussion......Page 593
References......Page 598
Further reading......Page 599
Notation and basic functions of interest......Page 601
Intensity function......Page 602
Semiparametric models for recurrent event data......Page 603
Andersen-Gill (AG) proportional intensity model......Page 604
Prentice-Williams-Peterson (PWP) models......Page 606
Models for recurrence time hazards......Page 609
Wei-Lin-Weissfeld (WLW) marginal hazards model......Page 610
Pepe and Cai rate models......Page 612
Marginal means/rates models......Page 613
Joint distribution function......Page 616
Recurrence time survival function......Page 618
References......Page 619
Introduction......Page 622
Motivating examples......Page 623
Simple current status data......Page 624
Epidemiological applications-calculation of the relative risk......Page 625
Regression models......Page 626
Different sampling schemes......Page 627
Case-control sampling......Page 628
Doubly censored current status data......Page 629
Competing risk outcomes......Page 631
Bivariate current status data......Page 633
Outcomes with intermediate stage......Page 635
Counting processes......Page 636
Conclusion......Page 637
References......Page 638
Data......Page 640
Background......Page 641
Inference for progressive Markov models......Page 642
A goodness-of-fit statistic......Page 644
Distribution of the goodness-of-fit statistic......Page 645
The fit of the Markov model......Page 647
Some general remarks on table construction......Page 650
Background......Page 651
Background to mixture models......Page 652
Description of the model......Page 655
Significance testing with respect to a subpopulation at no risk......Page 657
Description of the model......Page 659
A goodness-of-fit statistic......Page 661
Application to the PsA data......Page 662
The fit of the negative binomial model......Page 664
Discussion......Page 667
Formulas for the estimated transition probabilities......Page 668
References......Page 669
The measurement of gene expression levels......Page 671
The statistical analysis of genes......Page 672
Data reduction and model fitting......Page 674
Assessing predictive accuracy......Page 675
Simulations......Page 676
Results......Page 677
Stepwise......Page 678
Clustering......Page 679
Discussion......Page 680
References......Page 682
Introduction......Page 685
Dropout process......Page 686
Quality of life instruments and data......Page 687
Clinical studies......Page 688
Results......Page 689
Study of data missingness......Page 690
Analysis of variance......Page 691
Time to event definition......Page 692
Results......Page 693
Preliminaries......Page 694
Definitions and notations......Page 695
The semi-Markovian model......Page 696
Likelihood function......Page 697
Estimation and tests......Page 699
Asymptotic results......Page 703
Joint distribution of QoL and survival-dropout processes......Page 705
Likelihood function......Page 707
The EM-algorithm......Page 709
Score and information for beta.......Page 710
Estimation of standard errors......Page 711
Numerical comparison with the DK method......Page 712
Specificity and interest of the time to QoL deterioration approach......Page 716
Specificity and interest of the joint modelling approach......Page 719
Extension of previous analyses to the latent nature of quality of life......Page 721
References......Page 722
39. Modelling Survival Data using Flowgraph Models......Page 725
Series flowgraph model: HIV blood transfusion data......Page 727
Data analysis of HIV/AIDS data......Page 728
Converting flowgraph MGFs to densities......Page 730
Parametric assumptions......Page 733
Constructed likelihood: Incomplete data......Page 734
Loop flowgraph models......Page 736
A systematic procedure for solving flowgraphs......Page 738
Data analysis for diabetic retinopathy data......Page 739
Summary......Page 741
References......Page 742
Introduction......Page 743
Brown-Proschan (1983) model......Page 744
Kijima's (1989) models......Page 745
Dorado-Hollander-Sethuraman (1997) model......Page 746
Last-Szekli (1998) model......Page 747
Estimation in the DHS model......Page 748
Estimation in the BBS model......Page 751
A two-sample test in the BBS model......Page 754
Goodness-of-fit tests in the BBS model......Page 755
Testing the minimal repair assumption in the BBS model......Page 757
References......Page 759
Subject Index......Page 760
Handbook of Statistics: Contents of Previous Volumes......Page 767