Contents
Chapter 1 Cox Regression Model
Authors
Hans C. van Houwelingen
Department of Medical Statistics and Bioinformatics
Leiden University Medical Center
Leiden, The Netherlands
email: jcvanhouwelingen@lumc.nl
Theo Stijnen
Department of Medical Statistics and Bioinformatics
Leiden University Medical Center
Leiden, The Netherlands
email: T.Stijnen@lumc.nl
Content
1.1 Basic statistical concepts 5
1.1.1 Survival time and censoring time 5
1.1.2 The Kaplan-Meier estimator 6
1.1.3 The hazard function 7
1.2 The proportional hazards (Cox) model 9
1.3 Fitting the Cox model 9
1.4 Example: NKI breast cancer data 11
1.5 Martingale residuals, model fit 14
1.6 Extensions of the data structure 16
1.6.1 Delayed entry, left truncation 16
1.6.2 Time-dependent covariates 17
1.6.3 Continued example 17
1.7 Beyond proportional hazards assumption 19
1.7.1 Stratified models 19
1.7.2 Time-varying coefficients, Schoenfeld residuals 20
1.7.3 Continued example 21
1.7.4 Final remarks 21
Data
Errata
Chapter 2 Bayesian Analysis of the Cox Model
Authors
Joseph G. Ibrahim
Department of Biostatistics
University of North Carolina
Chapel Hill, NC, USA
email: ibrahim@bios.unc.edu
Ming-Hui Chen
Department of Statistics
University of Connecticut
Storrs, CT, USA
email: ming-hui.chen@uconn.edu
Danjie Zhang
Department of Statistics
University of Connecticut
Storrs, CT, USA
Debajyoti Sinha
Department of Statistics
Florida State University
Tallahassee, FL, USA
email: sinhad@stat.fsu.edu
Content
2.1 Introduction 27
2.2 Fully parametric models 29
2.3 Semiparametric models 32
2.3.1 Piecewise constant hazard model 32
2.3.2 Models using a gamma process 33
2.3.3 Gamma process prior with continuous-data likelihood 33
2.3.4 Relationship to partial likelihood 33
2.3.5 Gamma process on baseline hazard 34
2.3.6 Beta process models 35
2.3.7 Correlated prior processes 37
2.3.8 Dirichlet process models 38
2.4 Prior elicitation 39
2.5 Other topics 40
2.6 A case study: an analysis of melanoma data 40
2.7 Discussion 42
Chapter 3 Alternatives to the Cox Model
Authors
Torben Martinussen
Department of Biostatistics
University of Copenhagen
Copenhagen, Denmark
email: tma@sund.ku.dk
Limin Peng
Department of Biostatistics and Bioinformatics
Emory University
Atlanta, GA, USA
email: lpeng@emory.edu
Content
3.1 Additive hazards regression 49
3.1.1 Model specification and inferential procedures 50
3.1.2 Goodness-of-fit procedures 53
3.1.3 Further results on additive hazard models 54
3.1.3.1 Structural properties of the additive hazard model 54
3.1.3.2 Clustered survival data and additive hazard model 55
3.1.3.3 Additive hazard change point model 55
3.1.3.4 Additive hazard and high-dimensional regressors 56
3.1.3.5 Combining the Cox model and the additive model 56
3.1.3.6 Gastrointestinal tumour data 57
3.2 The accelerated failure time model 58
3.2.1 Parametric models 58
3.2.2 Semiparametric models 59
3.2.2.1 Inference based on hazard specification 59
3.2.2.2 Inference using the additive mean specification 61
3.3 Quantile regression for survival analysis 62
3.3.1 Introduction 62
3.3.2 Estimation under random right censoring with C always known 63
3.3.3 Estimation under covariate-independent random right censoring 63
3.3.4 Estimation under standard random right censoring 64
3.3.4.1 Self-consistent approach 64
3.3.4.2 Martingale-based approach 66
3.3.5 Variance estimation and other inference 67
3.3.6 Extensions to other survival settings 68
3.3.7 An illustration of quantile regression for survival analysis 68
Chapter 4 Transformation Models
Author
Danyu Lin
Department of Biostatistics
University of North Carolina
Chapel Hill, NC, USA
email: lin@bios.unc.edu
Content
4.1 Introduction 77
4.2 Data, models and likelihoods 78
4.2.1 Transformation models for counting processes 78
4.2.2 Transformation models with random effects for dependent failure times 80
4.2.3 Joint models for repeated measures and failure times 80
4.3 Estimation 81
4.4 Asymptotic properties 82
4.5 Examples 83
4.5.1 Lung cancer study 83
4.5.2 Colon cancer study 84
4.5.3 HIV study 86
4.6 Discussion 88
Chapter 5 High-Dimensional Regression Models
Authors
Jennifer A. Sinnott
Department of Biostatistics
Harvard School of Public Health
Boston, MA, USA
email: jennifer.sinnott@gmail.com
Tianxi Cai
Department of Biostatistics
Harvard School of Public Health
Boston, MA, USA
email: tcai@hsph.harvard.edu
Content
5.1 Introduction 93
5.2 Methods based on feature selection 95
5.2.1 Discrete feature selection 95
5.2.2 Shrinkage methods 96
5.2.3 Methods based on group structure 98
5.2.4 Selection of tuning parameters 99
5.3 Methods based on derived variables 100
5.3.1 Principal components regression 101
5.3.2 Approaches based on partial least squares 102
5.4 Other models 102
5.4.1 Nonparametric hazard model 103
5.4.2 Additive risk model 103
5.4.3 Accelerated failure time model 103
5.4.4 Semiparametric linear transformation models 104
5.5 Data analysis example 104
5.6 Remarks 106
Chapter 6 Cure Models
Authors
Yingwei Peng
Department of Public Health Sciences
Department of Mathematics and Statistics
Division of Cancer Care and Epidemiology, Cancer Research Institute
Queen's University
Kingston, Ontario, Canada
email: yingwei.peng@queensu.ca
Jeremy M. G. Taylor
Department of Biostatistics
University of Michigan
Ann Arbor, MI, USA
email: jmgt@umich.edu
Content
6.1 Introduction 113
6.2 Mixture cure models 114
6.2.1 Model formulation 114
6.2.2 Estimation methods 116
6.2.3 Tonsil cancer example 117
6.2.4 Identifiability 118
6.2.5 Mixture cure model for clustered survival data 120
6.3 Proportional hazards cure model 122
6.3.1 Model formulation 122
6.3.2 Estimation methods 123
6.3.3 Proportional hazards cure model for clustered survival data 124
6.4 Unifying cure models based on transformations 125
6.5 Joint modeling of longitudinal and survival data with a cure fraction 127
6.6 Cure models and relative survival in population studies 128
6.7 Software for cure models 128
Chapter 7 Causal Models
Author
Theis Lange
Department of Biostatistics
University of Copenhagen
Copenhagen, Denmark
email: thlan@sund.ku.dk
Naja H. Rod
Section of Social Medicine
Department of Public Health
University of Copenhagen
Copenhagen, Denmark
email: nahuro@sund.ku.dk
Content
7.1 Introduction 135
7.2 Tools for formalizing cause and effect 136
7.3 Analysis in the absence of feedback 139
7.3.1 Causal interpretation of the classic Cox modeling approach 140
7.3.2 Mimicking an actual randomized trial 141
7.4 Analysis with exposure-confounder feedback 142
7.4.1 The medical background of the HAART example 143
7.4.2 Defining causal effects in the presence of feedback 143
7.4.3 Estimating causal effects from observational data in the presence of feedback 144
7.4.4 Assumptions for drawing causal conclusions in the presence of feedback 145
7.4.5 Implementing the mini trials approach146
7.5 Appendix: R code 149
Chapter 8 Classical Regression Models for Competing Risks
Authors
Jan Beyersmann
Institute of Statistics
Ulm University
Ulm, Germany
email: jan.beyersmann@uni-ulm.de
Thomas H. Scheike
Department of Biostatistics
University of Copenhagen
Copenhagen, Denmark
email: thsc@sund.ku.dk
Content
8.1 Introduction 157
8.2 The competing risks multistate model 158
8.2.1 The multistate model 158
8.2.2 Advantages over the latent failure time model 159
8.3 Nonparametric estimation 160
8.4 Data example (I) 162
8.5 Regression models for the cause-specific hazards 163
8.5.1 Cox's proportional hazards model 165
8.5.2 Aalen's additive hazards model 166
8.6 Data example (II) 167
8.7 Regression models for the cumulative incidence functions 168
8.7.1 Subdistribution hazard 169
8.8 Data example (III) 170
8.9 Other regression approaches 171
8.10 Further remarks 173
Chapter 9 Bayesian Regression Models for Competing Risks
Authors
Ming-Hui Chen
Department of Statistics
University of Connecticut
Storrs, CT, USA
email: ming-hui.chen@uconn.edu
Mario de Castro
Instituto de Ciencias Matematicas e de Computacao
Universidade de Sao Paulo
Sao Carlos-SP, Brazil
email: mcastro@icmc.usp.br
Miaomiao Ge
Clinical Bio Statistics
Boehringer Ingelheim Pharmaceuticals, Inc.
Ridgefield, CT, USA
email miaomiao1107@gmail.com
Yuanye Zhang
Novartis Institutes for BioMedical Research Inc.
Cambridge, MA, USA
email: yuanyevickiezhang@gmail.com
Content
9.1 Introduction 180
9.2 The models for competing risks survival data 181
9.2.1 Multivariate time to failure model 181
9.2.2 Cause-specific hazards model 183
9.2.3 Mixture model 184
9.2.4 Subdistribution model 185
9.2.5 Connections between the CS, M, and S models 186
9.2.6 Fully specified subdistribution model 187
9.3 Bayesian inference 188
9.3.1 Priors and posteriors 189
9.3.2 Computational development 190
9.3.3 Bayesian model comparison 191
9.4 Application to an AIDS study 192
9.5 Discussion 195
Chapter 10 Pseudo-Value Regression Models
Authors
Brent R. Logan
Division of Biostatistics
Medical College of Wisconsin
Milwaukee, WI, USA
email: blogan@mcw.edu
Tao Wang
Division of Biostatistics
Medical College of Wisconsin
Milwaukee, WI, USA
email: TaoWang@mcw.edu
Content
10.1 Introduction 199
10.2 Applications 201
10.2.1 Survival data 201
10.2.2 Cumulative incidence for competing risks 202
10.2.3 Multi-state models 204
10.2.4 Quality adjusted survival 206
10.3 Generalized linear models based on pseudo-values 207
10.3.1 Estimation 207
10.3.2 Assumptions and formal justification 208
10.3.3 Covariate-dependent censoring 208
10.3.4 Clustered data 209
10.4 Model diagnosis 209
10.4.1 Graphical assessment 210
10.4.2 Tests of model fit 211
10.5 Software 211
10.6 Examples 212
10.6.1 Example 1: Survival and cumulative incidence 212
10.6.2 Example 2: Multi-state model 215
10.7 Conclusions 217
Chapter 11 Binomial Regression Models
Authors
Randi Gron
Department of Biostatistics
University of Copenhagen
Copenhagen, Denmark
email: ragr@biostat.ku.dk
Thomas A. Gerds
Department of Biostatistics
University of Copenhagen
Copenhagen, Denmark
email: tag@biostat.ku.dk
Content
11.1 Introduction 222
11.1.1 Choice of time horizons 222
11.1.2 Modeling options 222
11.1.3 Right-censored data 223
11.1.4 Interval-censored data 223
11.1.5 Variance estimation 224
11.1.6 Time-varying covariates 224
11.1.7 Comparison with cause-specific modeling 224
11.2 Modeling 225
11.2.1 Logistic link 225
11.2.2 Log link 225
11.2.3 Complementary log-log link 226
11.2.4 Constant and time-varying regression coefficients 226
11.3 Estimation 227
11.3.1 Weighted response 227
11.3.2 Working censoring model 227
11.3.3 Weighted estimating equations 228
11.4 Variance estimation 228
11.4.1 Asymptotic variance estimate 228
11.4.2 Bootstrap confidence limits 229
11.5 Software implementation 230
11.6 Example 232
11.6.1 Melanoma data 232
11.6.2 Choice of link function 233
11.6.3 Effect of choice of time points 233
11.6.4 Compare confidence limits 234
11.7 Simulations 235
11.7.1 Competing risks model 235
11.7.2 Misspecified censoring model 235
11.7.3 Compare confidence limits 237
11.7.4 Effect of choice of time points 238
11.8 Final remarks 239
Chapter 12 Regression Models in Bone Marrow Transplantation -- A Case Study
Authors
Mei-Jie Zhang
Division of Biostatistics
Medical College of Wisconsin
Milwaukee, WI, USA
email: meijie@mcw.edu
Marcelo C. Pasquini
Division of Biostatistics
Medical College of Wisconsin
Milwaukee, WI, USA
email: pasquini@mcw.edu
Kwang Woo Ahn
Division of Biostatistics
Medical College of Wisconsin
Milwaukee, WI, USA
email: kwooahn@mcw.edu
Content
12.1 Introduction 243
12.2 Data 245
12.3 Survival analysis 245
12.3.1 Fitting Cox proportional hazards model 245
12.3.2 Adjusted survival curves based on a Cox regression model 248
12.4 Competing risks data analysis 252
12.4.1 Common approaches for analyzing competing risks data 252
12.4.2 Adjusted cumulative incidence curves based on a stratified Fine-Gray model 258
12.5 Summary 260
Chapter 13 Classical Model Selection
Authors
Florence H. Yong
Department of Biostatistics
Harvard School of Public Health
Boston, MA, USA
email: florenceyong04@hotmail.com
Tianxi Cai
Department of Biostatistics
Harvard School of Public Health
Boston, MA, USA
email: tcai@hsph.harvard.edu
L.J. Wei
Department of Biostatistics
Harvard School of Public Health
Boston, MA, USA
email: wei@hsph.harvard.edu
Lu Tian
Department of Health Research and Policy
Stanford University School of Medicine
Stanford, CA, USA
email: lutian@stanford.edu
Content
13.1 Introduction 265
13.2 Mayo Clinic primary biliary cirrhosis (PBC) data 266
13.3 Model building procedures and evaluation 267
13.3.1 Variable selection methods 268
13.3.2 Model evaluation based on prediction capability 269
13.4 Application of conventional model development and \penalty -\@M inferences 270
13.41 Model building 270
13.4.2 Selecting procedure using C-statistics 270
13.4.3 Making statistical inferences for the selected model 270
13.5 Challenges and a proposal 272
13.5.1 Over-fitting issue 273
13.5.2 Noise variables become significant risk factors 273
13.5.3 Utilizing cross-validation in model selection process 274
13.5.4 3-in-1 dataset modeling proposal 275
13.6 Establishing a prediction model 275
13.6.1 Evaluating model's generalizability 277
13.6.2 Reducing over-fitting via 3-in-1 proposal 278
13.7 Remarks 279
Errata
Chapter 14 Bayesian Model Selection
Author
Purushottam W. Laud
Division of Biostatistics
Medical College of Wisconsin
Milwaukee, WI, USA
email: laud@mcw.edu
Content
14.1 Introduction 285
14.2 Posterior model probabilities and Bayes factor 286
14.3 Criterion-based model selection 287
14.3.1 Information criteria 287
14.3.1.1 BIC 288
14.3.1.2 DIC 288
14.3.2 Predictive criteria 289
14.3.2.1 Cross-valid prediction 289
14.3.2.2 Replicate experiment prediction 290
14.4 Search-based variable selection 291
14.4.1 Stochastic search variable selection 292
14.4.2 Reversible jump MCMC 293
14.5 Discussion 294
Chapter 15 Model Selection for High-Dimensional Models
Authors
Rosa J. Meijer
Department of Medical Statistics and Bioinformatics
Leiden University Medical Center
Leiden, The Netherlands
email: R.J.Meijer@lumc.nl
Jelle J. Goeman
Department of Epidemiology and Biostatistics
Radboud University Medical Center
email: Jelle.Goeman@radboudumc.nl
Content
15.1 Introduction 301
15.2 Selecting variables by fitting a prediction model 302
15.2.1 Screening by penalized methods 303
15.2.2 Screening by univariate selection 306
15.2.3 Practical usefulness of methods possessing screening properties 307
15.2.4 The Van de Vijver dataset 309
15.3 Selecting variables by testing individual covariates 310
15.3.1 Methods for FWER control 312
15.3.2 Methods for FDR control 315
15.3.3 Confidence intervals for the number of true discoveries 316
15.4 Reducing the number of variables beforehand and incorporating background knowledge 318
15.5 Discussion 319
Chapter 16 Robustness of Proportional Hazards Regression
Authors
John O'Quigley
Universite Pierre et Marie Curie - Paris VI
Paris, France
email: john.oquigley@upmc.fr
Ronghui Xu
University of California School of Medicine
Division of Biostatsitics and Bioinformatics
San Diego, CA, USA
email: rxu@ucsd.edu
Content
16.1 Introduction 323
16.2 Impact of censoring on estimating equation 325
16.2.1 Model-based expectations 325
16.2.2 More robust estimating equations 326
16.3 Robust estimator of average regression effect 329
16.3.1 The robust estimator 329
16.3.2 Interpretation of average regression effect 330
16.3.3 Simulations 331
16.4 When some covariates have non-PH 333
16.5 Proportional hazards regression for correlated data 335
Chapter 17 Nested Case-Control and Case-Cohort Studies
Authors
Ornulf Borgan
Department of Mathematics
University of Oslo
Oslo, Norway
email: borgan@math.uio.no
Sven Ove Samuelsen
Department of Mathematics
University of Oslo
Oslo, Norway
email: osamuels@math.uio.no
Content
17.1 Introduction 343
17.2 Cox regression for cohort data 344
17.3 Nested case-control studies 345
17.3.1 Sampling of controls 345
17.3.2 Estimation and relative efficiency 346
17.3.3 Example: radiation and breast cancer 347
17.3.4 A note on additional matching 348
17.4 Case-cohort studies 348
17.4.1 Sampling of the subcohort 348
17.4.2 Prentice's estimator 349
17.4.3 IPW estimators 350
17.4.4 Stratified sampling of the subcohort 351
17.4.5 Example: radiation and breast cancer 351
17.4.6 Post-stratification and calibration 352
17.5 Comparison of the cohort sampling designs 353
17.5.1 Statistical efficiency and analysis 353
17.5.2 Study workflow and multiple endpoints 353
17.5.3 Simple or stratified sampling 354
17.6 Re-use of controls in nested case-control studies 354
17.7 Theoretical considerations 355
17.7.1 Nested case-control data 355
17.7.2 Case-cohort data 357
17.8 Nested case-control: stratified sampling and absolute risk estimation 358
17.8.1 Counter-matching 358
17.8.2 Estimation of absolute risk 359
17.9 Case-cohort and IPW-estimators: absolute risk and alternative models 360
17.10 Maximum likelihood estimation 361
17.11 Closing remarks 362
Data
Errata
Chapter 18 Interval Censoring
Authors
Jianguo Sun
Department of Statistics
University of Missouri
Columbia, MO, USA
email: SunJ@Missouri.edu
Junlong Li
Department of Biostatistics
Harvard University
Boston, MA, USA
email: junlong.li@mail.mizzou.edu
Content
18.1 Introduction 369
18.2 Likelihood function and an example 371
18.3 Current status data 373
18.4 Univariate interval-censored data 374
18.5 Multivariate interval-censored data 378
18.6 Competing risks interval-censored data 380
18.7 Informatively interval-censored data 381
18.8 Other types of interval-censored data 382
18.9 Software and concluding remarks 383
Chapter 19 Current Status Data: An Illustration with Data on Avalanche Victims
Authors
Nicholas P. Jewell
University of California
Berkeley, CA, USA
jewell@berkeley.edu
Ruth Emerson
University of California
Berkeley, CA, USA}
Content
19.1 Introduction 391
19.2 Estimation of a single distribution function 393
19.2.1 Inference 395
19.3 Regression methods 398
19.4 Competing risks 404
19.5 Sampling and measurement issues 406
19.6 Other topics 406
19.7 Discussion 407
Erratum
Chapter 20 Multistate Models
Authors
Per Kragh Andersen
Department of Biostatistics
University of Copenhagen
Copenhagen, Denmark
email: P.K.Andersen@biostat.ku.dk
Maja Pohar Perme
Department of Biostatistics and Medical Informatics
University of Ljubljana
Ljubljana, Slovenia
email: maja.pohar@mf.uni-lj.si
Content
20.1 Introduction 417
20.2 Models and inference for transition intensities 418
20.2.1 Models for homogeneous populations 419
20.2.2 Regression models 419
20.2.3 Inference for transition intensities 420
20.2.4 Inference for marginal rate functions 423
20.2.5 Example 424
20.3 Models for transition and state occupation probabilities 429
20.3.1 Plug-in models based on intensities 429
20.3.2 Direct models for probabilities 432
20.3.3 Example 433
20.4 Comments 436
Chapter 21 Landmarking
Author
Hein Putter
Department of Medical Statistics and Bioinformatics
Leiden University Medical Center
Leiden, The Netherlands
email: h.putter@lumc.nl
Content
21.1 Landmarking 441
21.1.1 Immortal time bias 442
21.1.2 Landmarking 442
21.2 Landmarking and dynamic prediction 445
21.2.1 Dynamic prediction 445
21.2.2 The AHEAD data 446
21.2.3 Dynamic prediction and landmarking 447
21.2.4 Landmark super models 449
21.2.5 Application to the AHEAD data 449
21.3 Discussion 453
21.3.1 Implementation of landmarking 453
21.3.2 When to use landmarking 454
Chapter 22 Frailty Models
Author
Philip Hougaard
Biometric Division, Lundbeck
Valby, Denmark
email: phou@lundbeck.com
Content
22.1 Introduction 458
22.2 Purpose of a frailty model 459
22.2.1 Multivariate data examples where a frailty model is useful 459
22.2.2 Multivariate data examples where a frailty model is less useful 459
22.2.3 Univariate data examples 460
22.3 Models for univariate data 460
22.4 Shared frailty models for multivariate data 462
22.5 Frailty models for recurrent events data 463
22.6 Specific frailty distributions 463
22.6.1 Gamma 463
22.6.2 Positive stable 464
22.6.3 PVF 465
22.6.4 Lognormal 465
22.6.5 Differences between the models 465
22.7 Estimation 466
22.8 Asymptotics 466
22.8.1 The parametric case 467
22.8.2 The non-parametric case 467
22.8.3 The semi-parametric case 467
22.9 Extensions 468
22.10 Goodness-of-fit 468
22.10.1 Goodness-of-fit of models without frailty 469
22.10.2 Goodness-of-fit of models with frailty 469
22.10.3 Alternative models 469
22.11 Applications 470
22.12 Key aspects of using frailty models 471
22.13 Software 471
22.14 Literature 472
22.15 Summary 472
Chapter 23 Bayesian Analysis of Frailty Models
Author
Paul Gustafson
Department of Statistics
University of British Columbia
Vancouver, BC, Canada
email: gustaf@stat.ubc.ca
Content
23.1 Background 475
23.2 A basic frailty model 476
23.2.1 Modeling the baseline hazard 476
23.2.2 Modeling the frailties 477
23.2.3 Example 479
23.2.4 Model comparison 480
23.3 Recent developments 482
23.4 Final thoughts 484
Chapter 24 Copula Models
Author
Joanna H. Shih
National Cancer Institute
Bethesda, MD, USA
email: jshih@mail.nih.gov
Content
24.1 Introduction 489
24.2 Copula 491
24.2.1 Definition 491
24.2.2 Archimedean copula 491
24.2.3 Bivariate association measures 491
24.2.4 Examples 493
24.3 Estimation 496
24.4 Model assessment 499
24.5 Example 503
24.5.1 Data 503
24.5.2 Analysis 504
24.5.3 Goodness-of-fit 505
24.6 Summary 506
Chapter 25 Clustered Competing Risks
Authors
Guoqing Diao
Department of Statistics
George Mason University
Fairfax, VA, USA
email: gdiao@gmu.edu
Donglin Zeng
Department of Biostatistics
University of North Carolina
Chapel Hill, NC, USA
email: dzeng@bios.unc.edu
Content
25.1 Introduction 511
25.2 Notation and definitions 512
25.3 Estimation of multivariate CSHs and CIFs 513
25.4 Association analysis 514
25.5 Regression analysis 516
25.5.1 Fine and Gray model 516
25.5.2 Conditional approach using Fine and Gray model 517
25.5.3 Marginal approach using Fine and Gray model 518
25.5.4 Mixture model with random effects 518
25.5.5 Alternative approaches 519
25.6 Example 519
25.7 Discussion and future research 520
Chapter 26 Joint Models of Longitudinal and Survival Data
Authors
Wen Ye
Department of Biostatistics
University of Michigan
Ann Arbor, MI, USA
email: wye@umich.edu
Menggang Yu
Department of Biostatistics and Medical Informatics
University of Wisconsin
Madison, WI, USA
email: meyu@biostat.wisc.edu
Content
26.1 Introduction 523
26.2 The basic joint model 525
26.2.1 Survival submodel 526
26.2.2 Longitudinal submodel 526
26.2.3 Joint likelihood formulation and assumptions 527
26.2.4 Estimation 528
26.2.4.1 Maximum likelihood estimation 528
26.2.4.2 Bayesian methods 529
26.2.5 Asymptotic inference for MLEs 529
26.2.6 Example: an AIDS clinical trial 530
26.3 Joint model extension 532
26.3.1 Extension of survival submodel 532
26.3.1.1 Competing risks 532
26.3.1.2 Recurrent event data 533
26.3.1.3 Nonproportional hazards model 533
26.3.2 Extension of longitudinal submodel 534
26.3.2.1 Joint models with discrete longitudinal outcomes 534
26.3.2.2 Joint models with multiple longitudinal biomarkers 534
26.3.3 Variations of the link between survival and longitudinal\newline submodels 535
26.3.4 Joint latent class models 536
26.4 Prediction in joint models 537
26.4.1 Prediction of future longitudinal outcome 537
26.4.2 Prediction of survival distribution 538
26.4.3 Performance of prediction accuracy 538
26.5 Joint model diagnostics 539
26.6 Joint model software 540
Chapter 27 Familial Studies
Author
Karen Bandeen-Roche
Department of Biostatistics
Johns Hopkins Bloomberg School of Public Health
Baltimore, MD, USA
email: kbandeen@jhsph.edu
Content
27.1 Overview 549
27.2 Notation 550
27.3 Analyses aimed exclusively at determining relationships of individuals' failure times to predictor variables 551
27.4 Characterizing familial associations 552
27.4.1 Summary measures of dependence 552
27.4.2 Association through frailty modeling 553
27.4.3 Association through copula modeling and relation to frailty modeling 553
27.4.4 Association modeling specific to familial data: Simple random family sampling 555
27.4.5 Association modeling specific to familial data: Case-control designs 557
27.5 Age- and time-dependence of failure time associations 558
27.5.1 Checking the fit of parametric copula models for association 558
27.5.2 Nonparametric estimation of the conditional hazard ratio as a function of time 559
27.6Competing risks 559
27.6.1 Approaches generalizing the conditional hazard ratio function 560
27.6.2 Alternative approaches to describing and estimating failure time associations subject to competing risks 562
Chapter 28 Sample Size Calculations for Clinical Trials
Authors
Kristin Ohneberg
Institute of Medical Biometry and Medical Informatics
University Medical Center Freiburg
email: ohneberg@imbi.uni-freiburg
Freiburg, Germany
Martin Schumacher
email: ms@imbi.uni-freiburg.de
Content
28.1 Clinical trials and time-to-event data 571
28.1.1 Binomial sample size formula 572
28.1.2 Noncensored time-to-event endpoints 573
28.1.3 Exponential model 573
28.2 Basic formulas 574
28.2.1 Schoenfeld's formula 575
28.2.2 Alternative formula by Freedman 576
28.3 Sample size 577
28.3.1 Parametric estimation 577
28.3.2 Nonparametric approximation 578
28.3.3 Competing risks 579
28.4 Data example 581
28.4.1 4D trial 581
28.4.1.1 Two-state model 581
28.4.1.2 Competing risks analysis 583
28.5 Extensions 586
28.5.1 Multi-arm survival trials 586
28.5.2 Test for non-inferiority/superiority and equivalence 587
28.5.3 Prognostic factors and/or non-randomized comparisons 588
28.5.4 Left truncation 589
28.5.5 Proportional subdistribution modeling 589
28.5.6 Cluster-randomized trials 590
28.5.7 Cox regression with a time-varying covariate 591
28.6 Summary 591
Chapter 29 Group Sequential Designs for Survival Data
Authors
Chris Jennison
Department of Mathematical Sciences
University of Bath
Bath, UK
email: C.Jennison@bath.ac.uk
Bruce Turnbull
Cornell University
Ithaca, NY, USA
email: bwt2@cornell.edu
Content
29.1 Introduction 595
29.2 Canonical joint distribution of test statistics based on accumulating data 596
29.3 Group sequential boundaries and error spending 599
29.4 The group sequential log-rank test 604
29.5 Example: A clinical trial for carcinoma of the oropharynx 605
29.6 Monitoring a hazard ratio with adjustment for strata and covariates 608
29.7 Further work 609
29.8 Concluding remarks 611
Chapter 30 Inference for Paired Survival Data
Authors
Jennifer Le-Rademacher
Division of Biostatistics
Medical College of Wisconsin
Milwaukee, WI, USA
email: jlerade@mcw.edu
Ruta Brazauskas
Division of Biostatistics}{Medical College of Wisconsin
Milwaukee, WI, USA
email: ruta@mcw.edu
Content
30.1 Introduction 615
30.2 Example 616
30.3 Notation 617
30.4 Tests for paired data 618
30.4.1 Rank-based tests 618
30.4.2 Within-pair comparison 620
30.4.3 Weighted Kaplan-Meier comparison 622
30.5 Regression models for paired data 624
30.5.1 Stratified Cox models 624
30.5.2 Marginal Cox models 625
30.5.3 Shared frailty models 626
30.6 Comparing survival probabilities at a fixed time point 627
30.7 Discussion 630