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