Probability, Statistics and Modelling in Public Health

Contents......Page 8
1 Introduction......Page 24
2.1 Chronic Disease Modeling......Page 25
2.3 Preliminary Results......Page 26
3 Development of the Chronic Disease Model......Page 27
3.1 Forward Recurrence Time Distribution......Page 28
3.3 Length Biased Sampling and the Survival of Prevalent Cases......Page 29
3.4 Chronological Time Modeling......Page 31
4 Early Detection Disease Model......Page 32
5 Discussion......Page 33
References......Page 34
1 Introduction......Page 35
2 Data......Page 37
3 Three Components of the Individual Aging Process......Page 38
4 The Incorporated Ontogenetic Model of Cancer......Page 39
5 Application of the Ontogenetic Model to Data on Cancer Incidence Rate by Sex......Page 40
6 Conclusion......Page 43
References......Page 44
1 Introduction......Page 46
2 Model......Page 47
3 Decomposition of a counting process associated with Z(T)......Page 48
4.1 The data......Page 50
4.2 Estimation of Λ......Page 51
4.3 Large sample properties of Λ......Page 53
4.4 Estimation of the probability pj(z)......Page 58
References......Page 59
1 Introduction......Page 60
1.1 Data......Page 61
2 Degradation model......Page 63
3 Estimation of the mean degradation......Page 64
4.2 The estimated mean of the disablement process in demented and non-demented subjects......Page 66
4.3 The estimated mean of the disablement process in demented and non-demented men......Page 67
4.4 The estimated mean of the disablement process in demented and non-demented women......Page 68
4.5 The estimated mean of the disablement process in demented men and women......Page 69
4.6 The estimated mean of the disablement process in non-demented men and women......Page 70
4.7 The estimated mean of the disablement process in high and low educated subjects......Page 71
5 Joint model for degradation-failure time data......Page 72
References......Page 73
1 Introduction......Page 76
2.1 The Discrete Time semi-Markov Model......Page 77
2.2 Basic Results on semi-Markov Chains Estimation......Page 81
3 Failure Rates Estimation......Page 82
4 Proofs......Page 86
5 Numerical Example......Page 91
References......Page 93
1 Introduction......Page 96
2 Joint models for degradation and failure time modeling......Page 97
2.1 Failure time as hitting times of stochastic processes......Page 98
2.2 Failure times with degradation-dependent hazard rate......Page 101
2.3 The joint model: a mixed regression model with traumatic censoring......Page 102
3.1 Linear estimation......Page 103
3.2 Nonlinear estimation......Page 105
3.3 Estimation of the reliability functions......Page 107
References......Page 110
1 Introduction......Page 113
2.1 The model......Page 115
2.2 Example......Page 119
2.3 Estimation......Page 122
2.4 Random censoring......Page 127
3 A data example......Page 130
References......Page 140
1.1 Introduction......Page 142
1.2 Some Relations with the Kullback-Leibler Information......Page 144
2.1 A useful Theorem......Page 147
2.2 Appendix......Page 156
References......Page 158
1 Introduction......Page 159
2.1 Preliminaries......Page 160
2.2 The estimators......Page 161
3 Convergence of the estimators......Page 162
References......Page 166
1 Introduction......Page 168
2.1 Proportional hazards......Page 169
2.2 Accelerated life model......Page 171
2.3 Other models......Page 174
2.4 Damage accumulation and plasticity. Period Setting......Page 177
References......Page 180
1 Introduction and Preliminaries......Page 182
2 The Dropout Process......Page 184
3 The Model of Dupuy and Mesbah (2002)......Page 185
4 The Test of Goodness of Fit......Page 187
6 References......Page 189
1 Introduction......Page 192
2 Definitions......Page 193
3.1 Transition Rate Method......Page 194
3.2 A parametric model [CD97]......Page 198
3.3 Counting Method estimates......Page 200
4 Results......Page 201
5 Discussion......Page 203
References......Page 207
1 Introduction......Page 210
Initial definitions......Page 211
Moments of the first exit time distributions......Page 212
Inverse gamma process......Page 213
One-dimensional distribution......Page 214
Example 2......Page 216
Multi-dimensional distribution......Page 219
The direct way of data gathering......Page 220
Approximate maximum likelihood estimates......Page 221
Inverse way of data gathering when dealing with a continuous wear curve......Page 222
References......Page 224
Operating Characteristics of Partial Least Squares in Right-Censored Data Analysis and Its Application in Predicting the Change of HIV-I RNA......Page 225
1 Introduction......Page 226
2 Analysis Methods......Page 227
3 Simulation studies......Page 232
4 A Description of the Data......Page 236
5 The Data Analysis......Page 238
6 Summary and Discussion......Page 247
References......Page 250
1 Introduction......Page 254
2 Framework......Page 255
3 Independent Competing Risks Model......Page 257
4 General Model......Page 258
5 Case of a bounded number of transitions......Page 261
6 A Test of the Hypothesis of Independent Competing Risks......Page 262
7 Proofs......Page 264
References......Page 267
1 Introduction......Page 269
2.1 Random covering......Page 270
2.2 Short-cut covering......Page 271
2.3 The mechanism of truncation and censoring......Page 272
3 The distribution associated with random covering......Page 273
4 The distribution of random vector (L(x), R(x), L(z), R(z))......Page 276
5 The distribution of random vector (L(X), R(X), L(Z), R(Z))......Page 278
6 Maximum likelihood estimators......Page 279
6.1 The bracketing Hellinger ε–entropy......Page 280
6.2 Hellinger and Kullback-Leibler distances......Page 282
6.3 Estimation in the presence of a nuisance parameter......Page 285
References......Page 288
1 Introduction......Page 289
2 A Degradation Model......Page 290
2.1 The distribution of (T[sub(n)])......Page 291
2.2 Marking the sequence (T[sub(n)])......Page 293
3 Maximum Likelihood Estimates......Page 294
4 Moment Estimates......Page 297
5 Comparison of Maximum Likelihood and Moment Estimates......Page 299
6 Conclusion......Page 300
References......Page 301
2 Methods of statistical modelling......Page 303
3 Results......Page 304
References......Page 307
1 Introduction......Page 309
2 Degradation-Threshold-Shock-Models......Page 311
2.1 Degradation-Threshold-Models......Page 315
2.2 Degradation-Shock-Models......Page 316
3 Maximum Likelihood Estimation......Page 317
4 Concluding remarks......Page 319
References......Page 320
1 Introduction......Page 322
2 The WLW Method and Definitions of Test Statistics......Page 324
3 Asymptotic Properties of the Test Statistics under Contiguous Alternatives......Page 326
4.1 Equal μ[sub(1)], μ[sub(2)], … , μ[sub(K)]......Page 327
4.3 Special Correlation Structures......Page 329
5 Determining Sample Size and K......Page 330
6 Example: Recurring Opportunistic Infections in HIV/AIDS......Page 333
7 Discussion......Page 334
References......Page 337
1 Stating the Problem......Page 342
2.1 Model with censoring......Page 345
2.2 The Nelson-Aalen estimator for the cumulative hazard function......Page 346
2.3 A kernel estimator for the hazard function......Page 347
3 Testing the Hazard Rate......Page 348
3.1 An asymptotic α-test......Page 349
3.2 Application to the example......Page 350
4 Some further remarks......Page 351
5 About the Extension to the Model with Covariates......Page 352
References......Page 14
1 Introduction......Page 355
2.2 The expected log-likelihood......Page 357
2.3 Case of right-censored data......Page 358
3.1 Likelihood cross-validation: LCV......Page 359
3.2 Direct bootstrap method for estimating ELL (ELL[sub(boot)] and ELL[sub(iboot)])......Page 360
4 Simulation......Page 361
4.1 Kernel estimator......Page 362
4.2 Penalized likelihood estimator......Page 365
5.1 Method......Page 366
5.2 Example......Page 368
6 Conclusion......Page 369
References......Page 370
1 Introduction......Page 373
2 Efficient influence functions......Page 375
3 Efficient estimators......Page 380
References......Page 385
1 Introduction......Page 387
3 Inspection Policies......Page 389
4.1 System Renewal......Page 390
5 Optimal Policies......Page 391
5.2 Total Cost Criterion......Page 392
6 Lévy Processes as Degradation Models......Page 393
7.1 Maximum Process......Page 394
7.3 The Absolute Value......Page 395
7.4 Bessel Processes......Page 396
7.5 Models for Imperfect Inspection......Page 397
References......Page 398
1 Introduction and notations......Page 401
2 Asymptotic distribution of (LR[sub(1)], LR[sub(2)])′ under H[sub(0)] in a copula model......Page 403
2.1 Preliminary results for the martingales under H[sub(0)]......Page 404
2.2 Asymptotic distribution of (LR[sub(1)], LR[sub(2)])′ under H[sub(0)]......Page 407
2.3 What if the joint censoring distributions or the joint survival functions differ in groups A and B under H[sub(0)]?......Page 409
3 Simulations study......Page 411
4 Application......Page 412
5 Discussion......Page 413
References......Page 414
1 Introduction......Page 415
2 Measures of explained variation......Page 416
2.1 Definition of the explained variation......Page 417
2.2 Estimation of the explained variation......Page 418
3 Misspecification and definition of the predictive accuracy......Page 420
4 The failure time model......Page 422
5 Which estimation method to choose - model based or not?......Page 424
7 Appendix......Page 425
References......Page 426
1 Introduction......Page 428
2.1 Natural History of Breast Cancer......Page 430
2.2 Survival Distributions and Mortality......Page 433
2.3 Sensitivities of Mammography and Clinical Breast Examinations......Page 434
2.4 Costs of Screening Programs......Page 435
3 Optimization of Screening Strategies and Sensitivity Analyses......Page 436
4 Discussion......Page 438
References......Page 439
1 Introduction......Page 444
2.1 IRT models......Page 446
2.3 Estimation of the parameters......Page 447
2.4 Sequential Analysis......Page 448
2.6 Study Framework......Page 451
3 Results......Page 453
4 Discussion......Page 457
5 Conclusion......Page 458
6 References......Page 459
7.3 3. Fisher\'s information: V(X) statistic under H[sub(0)](μ = μ[sub(0)] = 0)......Page 461
8.1 Stopping boundaries for the one-sided SPRT and TT......Page 462
1 Patients and method......Page 463
2 Results......Page 464
References......Page 468
1 Introduction......Page 469
2 Fuzzy data......Page 470
3 Empirical reliability functions for fuzzy life times......Page 471
4 Generalized classical statistical inference for fuzzy data......Page 472
5 Generalized Bayesian inference in case of fuzzy information......Page 473
References......Page 474
1 Introduction......Page 475
2 Two-Sample Models......Page 476
2.1 Two-sample location-scale model......Page 477
2.2 Two-sample transformation model......Page 478
3 Hazards Regression......Page 479
4 Non-proportional Hazards Model......Page 483
5 Extensions and Brief Discussion......Page 485
References......Page 486
1 Introduction......Page 489
2 The General Compound Renewal Damage Process, and The Associated Failure Distribution......Page 490
3 Compound Poisson With Exponential Damage......Page 492
4 Compound Poisson With Erlang Damage......Page 496
References......Page 497
E......Page 499
R......Page 500
W......Page 501