Table of contents for Analysis of multivariate survival data / Philip Hougaard.


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1  Introduction                                                    1
   1.1  What is survival data? .................                 2
   1.2  Univariate data ..........         .....      ....        6
         1.2.1 Population data ..      .......      ....          6
         1.2.2 Survival of diabetics................             7
   1.3  Interval censored data  ...................7
         1.3.1  Diabetic nephropathy ................8
   1.4  Multivariate data structures ................9
         1.4.1 Parallel data..................               .     9
         1.4.2 Longitudinal data .    .........                   10
   1.5  Several individuals ............                          11
         1.5.1 Danish twins .... ......               .....12
         1.5.2 Adoption study . .....      .........              13
         1.5.3  Litter-matched tumorigenesis experiment .....     13
         1.5.4 Length of leukemia remission ...... . . ...        15
   1.6  Similar organs ..........           .............         16
         1.6.1 Diabetic retinopathy ... .........        ...16
         1.6.2 Amalgam fillings ...........                       17
   1.7  Recurrent events... . . . . . ..... .... .                17
         1.7.1 Mammary tumors ...........                         18
         1.7.2 Epileptic seizures...............                  19
         1.7.3  Infections in catheters for patients on dialysis .  21
         1.7.4 Insurance claims .............                     21
   1.8  Repeated measurements    ........ .                       22



        1.8.1 Watermaze ......................                 23
        1.8.2 Exercise test times...........     ....     .     23
  1.9  Different events................      .........    .     24
        1.9.1 Albuminuria ......    .    ....... ..             25
        1.9.2 Colon cancer .......... .                   ..26
        1.9.3  Survival after myocardial infarction ........    26
        1.9.4  Stanford heart transplant data ...........       26
  1.10 Competing risks ....       ........      ..       . 27
        1.10.1 Radiation-exposed mice . .. .  . .. . .  .       28
        1.10.2 Malignant melanoma  ..    . . . ....      .      29
   1.11 Types of censoring  ................             ..      29
   1.12 Types of truncation .   . . .. . .. ..   .. .  . ..      30
   1.13 Explanatory variables  .............. .                  31
   1.14  Purpose of multivariate survival studies . . .....      32
   1.15 Chapter summary ..                           ..33
   1.16 Bibliographic comments .................                 34
   1.17 Exercises ..................           ......            35

2 Univariate survival data                                       36
   2.1  Special features of survival data... .. .......          37
        2.1.1 Censored data . . .. . .. .. . .. . .. .. . ..    38
        2.1.2  Conditioning. ..........39
        2.1.3  Choice of parametric/non-parametric models . .40
        2.1.4  Multiple events  .............42
   2.2  Parametric methods..............             .           43
        2.2.1  The exponential distribution . ...........45
        2.2.2 The Weibull distribution. .. . . . . .            47
        2.2.3 The Gompertz distribution ....     . . ..         54
        2.2.4  Piecewise constant hazards    .......56
        2.2.5  Hidden causes of death ...58
        2.2.6 Truncated distributions.                          58
        2.2.7  Mixture distributions ..59
        2.2.8 Comparison of parametric models . . .. .. . .     65
   2.3  Simple non-parametric methods . . . .. . .. . .. ..      69
        2.3.1 The Kaplan-Meier estimate . . . . . .. .. . .     70
        2.3.2 The Nelson-Aalen estimate .. . .. .. .            74
   2.4  Regression models .......                                76
        2.4.1  The proportional hazards probability model . .77
        2.4.2  The partial likelihood estimation method . .79
        2.4.3 Estimation of absolute risk.                      83
        2.4.4  Truncation and time-dependent covariates ..84
        2.4.5  Residuals and robust variance estimates ..89
        2.4.6  Neglected covariates      ........90
        2.4.7  Computational aspects .....92
        2.4.8  Applications . .........93



         2.4.9  Summary of the Cox model .     ...........98
   2.5  Accelerated failure times... . .. .......... .99
        2.5.1 Comparison of accelerated failure time models and
               proportional hazards models.. ........          100
        2.5.2  Neglected covariates  ......101
        2.5.3 Applications ..................                  103
   2.6  Counting processes ................. .                  104
   2.7  Chapter summary .     .....................            105
   2.8  Bibliographic comments  ............                   106
   2.9  Exercises ...............       . .......107

3  Dependence structures                                       112
   3.1  Probability mechanisms .  .. ......       .. . ..113
        3.1.1 Common events . . . . . ......      .....        114
        3.1.2 Common risks models   . .. . .. . .. ..          114
        3.1.3 Event-related dependence .. . . ......           116
        3.1.4 Relevance of the mechanisms. .. . ....           117
   3.2  Dependence time frame . .    .....                     117
        3.2.1 Early/late dependence    .........               119
        3.2.2 Instantaneous dependence .. . . ......           120
        3.2.3 Short-term dependence    ...............         120
        3.2.4 Long-term dependence .. . . . ...     ....121
   3.3  Examples ....               ...     .   .......        122
        3.3.1 Birth histories            ....................  122
        3.3.2 Married couples .    ..... . . .....        .    123
        3.3.3 Twin dependence .........                        123
        3.3.4 Adoptive children     .......         ...123
        3.3.5 Traffic deaths.    . .. .. . .. . .  .  . ..     123
        3.3.6 Epileptic seizures.          .. ......124
        3.3.7 Bleeding patterns............                    124
        3.3.8  Amalgam fillings .   .........125
        3.3.9 Competing risks .          ........125
   3.4  Chapter summary ....                        .. ..125
   3.5  Bibliographic comments   ...........126
   3.6  Exercises     ...........................              126

4  Bivariate dependence measures                               128
   4.1  Correlation coefficients.............                  129
   4.2  Kendall's coefficient of concordance .. . . .......    131
        4.2.1 Applications .....................               133
   4.3  Spearman's correlation coefficient......       ...     134
   4.4  Median concordance..............                       135
   4.5  Integrated hazard correlation.... .. . .. . .. .. .    136
   4.6  Local dependence measures .....                        136
   4.7  Chapter summary .......... ...137



   4.8  Bibliographic comments         .................        137
   4.9  Exercises ....................                          138

5 Probability aspects of multi-state models                     139
   5.1  Transition probabilities...                             144
   5.2  Observations and censoring patterns . ...145
   5.3  Selection of state structure...... . . .                145
        5.3.1 The fertility model.           .. . .            146
        5.3.2  The disability model ........ .....151
        5.3.3  Bivariate parallel data .......152
   5.4  Progressive models ..                                   155
        5.4.1 Competing risks ....         . ..                156
        5.4.2 Bivariate parallel data                          157
        5.4.3  The disability model .......... .157
        5.4.4  Recurrent events ...159
        5.4.5  General progressive models . . . .159
   5.5  Homogeneous Markov processes .......159
         5.5.1  The disability model . . . .....161
         5.5.2  Recurrent events ...162
         5.5.3  The Freund model .   ........163
         5.5.4  The Marshall-Olkin model ......165
         5.5.5 The combined model                               166
         5.5.6  The alternating state model ......167
   5.6  General Markov processes   .  ........167
   5.7  Markov extension models .                               168
   5.8  General models  ...... ........169
   5.9  Conditionally simple processes.. .. .. . .. . .. ..     170
   5.10 Hidden states ......... . . . . . . . . ..    .170
   5.11 Chapter summary .. . . . . . . . .....     ......       173
   5.12 Bibliographic comments            .........173
   5.13 Exercises  .. ..        . . .. . .. .. . .. .    .. .   173

6 Statistical inference for multi-state models                  177
   6.1  Statistical models ...........178
         6.1.1  The disability model ........179
         6.1.2  Bivariate parallel data .......179
         6.1.3 Recurrent events                                 180
         6.1.4  Alternating states .  ...180
         6.1.5  General models .........180
   6.2  Likelihood evaluation. ......      . .    ...181
         6.2.1 Markov likelihood functions. . .. . .. .. . .    181
         6.2.2 General likelihood functions.. ..   .. .. . .    182
   6.3  Estimation of transition hazards... .. . .. ..183
         6.3.1  Separate and common constant hazards ..183
         6.3.2 Parametric models ............                   184



         6.3.3  Non-parametric models for the disability model . 185
         6.3.4  Non-parametric models for bivariate parallel data  186
   6.4  Estimation of transition probabilities......             188
   6.5  Censoring patterns and asymptotics .. .  . . .. .. .     190
         6.5.1  Estimation of variance .......191
   6.6  Model checking . . . . . . .. .....      ........        192
   6.7  Applications . . .. . ............      . .........      192
         6.7.1 Heart transplant data.                            193
         6.7.2 Danish twins ......            .....194
         6.7.3  Myocardial infarctions     .......201
         6.7.4 Albuminuria progression . . .. .........          202
         6.7.5 Other applications          .................     206
   6.8  Chapter summary .     .......      ......                206
   6.9  Bibliographic comments        ...........                207
   6.10 Exercises ...                         . ..      .207

7  Shared frailty models                                        215
   7.1  Unspecified frailty.................             ..      218
   7.2  General probability results.. .. . .... . . .            220
         7.2.1  The conditional parametrization . . . . .. . .  221
         7.2.2 The marginal parametrization .....                223
         7.2.3  Updating  ............ .. .....224
         7.2.4  Weibull conditional distributions . . . . . . . .  227
         7.2.5 Quantification of dependence . . .. . .. . .. .   228
         7.2.6 Derived quantities.......                         230
         7.2.7 Regression models ..    ..    .   ......          231
   7.3  Gamma frailty models ............                        233
         7.3.1 Updating  ........      . .....   ....234
         7.3.2 Weibull models .....       ........     ...       235
         7.3.3  Extension to negative dependence . ...236
   7.4  Positive stable frailty distributions  ....      . ..    237
         7.4.1 Updating  .............          .     . .238
         7.4.2 Regression models .    .....   ........   ...     238
         7.4.3 The stable-Weibull model . . . . .. . .. . .. .   238
         7.4.4 Weibull regression models . .. . .. . .. . .. .   240
   7.5  PVF frailty distributions.....      ...   . ....241
         7.5.1  Weibull and regression models . . . ........   243
         7.5.2 Updating . ..........                             243
   7.6  Lognormal frailty distributions .. . . . . .........  244
   7.7  Other distributions  ...................                 245
   7.8  Comparison of frailty distributions . . . . ........  246
         7.8.1 Theoretical comparison .. . .. . .........        247
         7.8.2 Comparison of fit       .........        ...      247
         7.8.3  Quantification of dependence  ...252
   7.9  Chapter summary ......................                   256



  7.10 Bibliographic comments ...      ...........           257
  7.11 Exercises       ..............          ..   ... .    258

8 Statistical inference for shared frailty models            263
  8.1  Parametric models .....264
  8.2  Simple estimates..                                    265
  8.3  The EM-algorithm           ..                         265
  8.4  The three-stage approach..                            267
  8.5  The full marginal estimate        .268
  8.6  The full conditional estimate . . . . .269
        8.6.1 The gamma model                                270
        8.6.2 The PVF model .           ..272
   8.7  The penalized likelihood approach.. .                 274
   8.8  Other approaches ......... ......275
   8.9  Software  ...............275
   8.10  Goodness-of-fit  ................... .275
   8.11  Asymptotic evaluations .  .......... ...277
        8.11.1 The gamma-Weibull model ... . . ......        278
        8.11.2 The stable-Weibull model ..............       279
        8.11.3 The non-parametric gamma model . ......       282
        8.11.4 The two-stage approach...........             283
   8.12 Applications     ..........................           283
        8.12.1 Tumorigenesis data.................           283
        8.12.2 Exercise data........                         285
        8.12.3 Catheter infections. . . . . . . . .          288
        8.12.4 Amalgam fillings...............               290
        8.12.5 Danish twin data. . . . . . . ...             291
        8.12.6 Other applications         ..                 308
   8.13 Chapter summary ....................                  309
   8.14  Bibliographic comments    ..........309
   8.15 Exercises. .. . .. .. .. . .. ...         . . ..      310

9  Shared frailty models for recurrent events                 312
   9.1  Notation and obsetionion plan    .......314
   9.2  Poisson processes . . . ........316
        9.2.1 Evaluation in SAS . . ........       ....317
   9.3  Constant frailty overdispersion models .....317
        9.3.1  Common observation period .   .....318
        9.3.2  Negative binomial models for period count data . 319
        9.3.3 Negative binomial models .. .. . .. .. . ..    321
        9.3.4  PVF frailty models for period count data . .322
        9.3.5 PVF frailty models          .....328
        9.3.6  Positive stable frailty models ...... ..328
   9.4  Dependence measures ............                      329
   9.5  Regression models ........... .....329



        9.5.1 Period count data.        ...  ... ..........    330
        9.5.2 Interval count data.................             330
        9.5.3 Gap times . . . . .. ......       ..    ...      331
  9.6  Estimation ..........................                   331
        9.6.1 Data of exact times  .................           332
        9.6.2 Period count data ............                   332
        9.6.3  Interval counts for several intervals .. ...    333
  9.7  Asymptotics  . .. . .. ..................               333
  9.8  Applications . .. . .. ..................               334
        9.8.1 Period count data .. .. . .. . .. .. . .. .      335
        9.8.2  Time data for a fixed period   ......339
        9.8.3  Interval count data ..  ........339
        9.8.4 Other applications .    ..........               340
  9.9  Chapter summary .     .    .............                340
  9.10 Bibliographic comments                                  341
  9.11 Exercises           . . .   .. . .. .. . .. . .. .      341

10 Multivariate frailty models                                 345
   10.1 Comparison of dependence ...       ..........348
        10.1.1 Estimation ...................349
        10.1.2 Application to twin data........                350
   10.2  Differential effect of a shared frailty ..... . .. .   352
   10.3  Shared frailty models for bivariate marginals ......   353
        10.3.1 Applications................... .               353
   10.4 The multiplicative stable model ......     .......      354
        10.4.1 The trivariate model...............             355
        10.4.2 Combined model for monozygotic and dizygotic
               twins ........       .............              357
         10.4.3 Treatment by center interaction  ... . . . . .  359
         10.4.4 Goodness-of-fit..................               359
         10.4.5 Estimation...................            ..359
         10.4.6 Applications................           ....     360
   10.5 Additive models .....      ..........        .. ...     362
         10.5.1 Bivariate additive gamma models ........        364
         10.5.2 Compound symmetry model .. . .....        .     369
         10.5.3 Combined model for monozygotic and dizygotic
               twins .. .. . .. . .....369
         10.5.4 Treatment by center interaction  .......  ..    370
         10.5.5 Father-mother-child model............           370
         10.5.6 General models  .............         ....      371
         10.5.7 Estimation........        .   .......... ... 373
         10.5.8 Asymptotic evaluation...............            373
         10.5.9 Applications................... ..              374
   10.6 Multivariate lognormal frailty. .   .. . .. .. .  .     379
   10.7  Negative dependence models ..............380



  10.8 Other approaches ... . . . .. .. . .. .. . ... . ..   380
  10.9 Chapter summary ...........................           381
  10.10 Bibliographic comments ..................            382
  10.11 Exercises................... ........                 383

11 Instantaneous and short-term frailty models                385
  11.1 Independent increments frailty model . ..........     387
        11.1.1 Bivariate parallel data ...............        387
        11.1.2 Recurrent events.................              388
        11.1.3 Estimation for recurrent events ..........     390
        11.1.4 Estimation for parallel data ............      391
        11.1.5 Asymptotics ...................           ..    391
        11.1.6 Application   ................            .    392
   11.2  Polya and independent increments additive model . ...  392
        11.2.1 Application  ..................... .394
   11.3 Piecewise gamma model ..................              395
        11.3.1 Bivariate parallel data..... .. .. .. .   .    397
        11.3.2 Recurrent events..................             398
        11.3.3Applications...................           ...   398
   11.4 Moving average model ...................              400
   11.5 Hidden cause of death model.........       .....      401
   11.6 The Woodbury-Manton model ..............              402
   11.7 Chapter summary ......................                403
   11.8 Bibliographic comments ..................             403
   11.9 Exercises...................           .........404

12 Competing risks models                                     406
   12.1  Effects of changing cause-specific hazards ..... . ...  408
   12.2  The multivariate parallel approach  ...... . . ....  409
   12.3  The multi-state approach .................. . 410
   12.4  The frailty approach................... .             411
   12.5  What can be estimated? ............. .......          412
   12.6  Competing risks for twin data ...............        412
   12.7  Applications .............................            414
   12.8  Chapter summary        ......................         417
   12.9  Bibliographic comments .     ..  .............   .    417
   12.10 Exercises ........................... .......        417

13 Marginal and copula modeling                               419
   13.1 The coordinate-wise approach. ..............          421
        13.1.1 Recurrent events..................             422
        13.1.2 Computational aspects ................         423
        13.1.3 Asymptotic results.................            425
        13.1.4 Applications.....................              425
   13.2 The independence working model ...... . . . ....      427



        13.2.1 Asymptotic evaluation....... .. ..       ...    428
        13.2.2 Software............ ....... ...               429
        13.2.3 Applications     ......     ..    ......  .    429
   13.3 Discussion of marginal modeling........      ... ..    431
   13.4 Recurrent events.    ............... ..        ....    433
   13.5 Copulas   ......           ...    ..      ....435
   13.6 The combined approach     ..   ............ .          436
        13.6.1Estimation ......................               436
        13.6.2 Asymptotics. . .. .......           .   ..     437
        13.6.3 Applications.....................               437
   13.7 Chapter summary .     ..........       .... .. ..      439
   13.8 Bibliographic comments  ... .   ............           439
   13.9 Exercises .................... .. . . ..           .   440

14 Multivariate non-parametric estimates                       442
   14.1  Distribution concepts and observations ..... . . ...    446
   14.2 The Hanley and Parnes estimate ........ . . ...        447
        14.2.1 Homogeneous censoring. .. . .. .. . ..    .    448
        14.2.2 Heterogeneous censoring.. . .. .. . ..     .    451
   14.3 The Dabrowska estimate ......       .... . ...         452
   14.4 The Pruitt estimate.  .......      ........ ..         453
   14.5 Comparison   .................... ........             454
   14.6 Other estimates and extensions....... . . . ....       456
   14.7 Asymptotics ..........................                 457
   14.8 Graphical illustrations...................             457
        14.8.1Cuts  ..............         .....   ......     457
        14.8.2 Conditional on coordinate survival ........    458
        14.8.3 Conditional on coordinate interval survival . .  458
        14.8.4 Three-dimensional plot...............          459
        14.8.5 Contour curves of the survivor function ......   459
        14.8.6 Contour curves after uniform transformation . . .  459
        14.8.7 Contour curves of the bivariate density . . .    459
        14.8.8 Bivariate density after uniform transformation . . 459
        14.8.9 Markov assumption check . .. . .. . .. .   .   460
   14.9  Quantitative evaluation of dependence ..... . . ...     461
         14.9.1 Degree of dependence.. .. . .. . .. .. .   .   461
         14.9.2 Early/late dependence. .. . .. .. . .. .   .   461
         14.9.3 Short-term dependence................          462
   14.10 Model checking ................... ..                 462
   14.11 Applications.......................... ...            463
         14.11.1 Danishtwins.. .. .. . .. . .. .. . .. . .. .  463
         14.11.2Kidney catheter data...............         .  477
         14.11.3 Exercise data...................              478
         14.11.4Heart transplant data.................         478
   14.12 Chapter summary ................. ...                 479




   14.13 Bibliographic comments ...............                 480
   14.14 Exercises.........       .......        ........ .     480

15 Summary                                                     483
   15.1 Summary of the theory ...................               483
   15.2 Course of analysis..   .   .......               ...    486
   15.3 Summary of the applications ................            491
        15.3.1 Twins .........                   .....491
        15.3.2 Length of leukemia remission.. ......           492
        15.3.3 Exercise data................ .                 493
        15.3.4 Kidney catheter infections.........             493
        15.3.5 Epileptic seizures ............     ..493
        15.3.6 Amalgam data ............ .                     494
        15.3.7 Albuminuria ......... ..494
        15.3.8: Malignant melanoma ................            494
   15.4 Future development .....................                494
   15.5 Exercises .................             .......    .    495

A  Mathematical results                                         497
   A.1  Laplace transforms ...              .    .. ... .497
   A.2  Exponential families.   .                 ...498
   A.3  Distribution theory ................500
         A.3.1 Gamma distributions ....       .....             500
         A.3.2 Inverse Gaussian distributions .....501
         A.3.3 Positive stable distributions        . .         502
         A.3.4  Power variance function distributions ...504
         A.3.5 Lognormal distributions ........ ..506
         A.3.6 Generalized inverse Gaussian distributions .508
         A.3.7 Non-central gamma distributions . .. . .. . .    508
   A.4  Mathematical functions ...............                  509
         A.4.1  The gamma function and related functions ..509
         A.4.2 Bessel functions..............                   510
         A.4.3 Hypergeometric functions  .......                511
   A.5  Bibliography . .. . .....511
   A.6  Exercises . .. .. . .. .. . .. . .. .. . .. . .. .. .   512

B  Iterative solutions                                          513
   B.1  Newton-Raphson iteration .................              514
   B.2  Extensions and modifications..............              514
   B.3  Standard error evaluation ..............           .    516

References                                                      517

Index                                                           531