# Mathematical Methods in Survival Analysis, Reliability and Quality of Life

• ID: 2179080
• Book
• 370 Pages
• John Wiley and Sons Ltd
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Reliability and survival analysis are important applications of stochastic mathematics (probability, statistics and stochastic processes) that are usually covered separately in spite of the similarity of the involved mathematical theory. This title aims to redress this situation: it includes 21 chapters divided into four parts: Survival analysis, Reliability, Quality of life, and Related topics. Many of these chapters were presented at the European Seminar on Mathematical Methods for Survival Analysis, Reliability and Quality of Life in 2006.
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Preface 13

PART I 15

Chapter 1. Model Selection for Additive Regression in the Presence of Right–Censoring 17Elodie BRUNEL and Fabienne COMTE

1.1. Introduction 17

1.2. Assumptions on the model and the collection of approximation spaces 18

1.2.1. Non–parametric regression model with censored data 18

1.2.2. Description of the approximation spaces in the univariate case 19

1.2.3. The particular multivariate setting of additive models 20

1.3. The estimation method 20

1.3.1. Transformation of the data 20

1.3.2. The mean–square contrast 21

1.4. Main result for the adaptive mean–square estimator 22

1.5. Practical implementation 23

1.5.1. The algorithm 23

1.5.2. Univariate examples 24

1.5.3. Bivariate examples 27

1.5.4. A trivariate example 28

1.6. Bibliography 30

Chapter 2. Non–parametric Estimation of Conditional Probabilities, Means and Quantiles under Bias Sampling 33Odile PONS

2.1. Introduction 33

2.2. Non–parametric estimation of p 34

2.3. Bias depending on the value of Y 35

2.4. Bias due to truncation on X 37

2.5. Truncation of a response variable in a non–parametric regression model 37

2.6. Double censoring of a response variable in a non–parametric model 42

2.7. Other truncation and censoring of Y in a non–parametric model 44

2.8. Observation by interval 47

2.9. Bibliography 48

Chapter 3. Inference in Transformation Models for Arbitrarily Censored and Truncated Data 49Filia VONTA and Catherine HUBER

3.1. Introduction 49

3.2. Non–parametric estimation of the survival function S 50

3.3. Semi–parametric estimation of the survival function S 51

3.4. Simulations 54

3.5. Bibliography 59

Chapter 4. Introduction of Within–area Risk Factor Distribution in Ecological Poisson Models 61Lea FORTUNATO, Chantal GUIHENNEUC–JOUYAUX, Dominique LAURIER,Margot TIRMARCHE, Jacqueline CLAVEL and Denis HEMON

4.1. Introduction 61

4.2. Modeling framework 62

4.2.1. Aggregated model 62

4.2.2. Prior distributions 65

4.3. Simulation framework 65

4.4. Results 66

4.4.1. Strong association between relative risk and risk factor, correlated within–area means and variances (mean–dependent case) 67

4.4.2. Sensitivity to within–area distribution of the risk factor 68

4.4.3. Application: leukemia and indoor radon exposure 69

4.5. Discussion 71

4.6. Bibliography 72

Chapter 5. Semi–Markov Processes and Usefulness in Medicine 75Eve MATHIEU–DUPAS, Claudine GRAS–AYGON and Jean–Pierre DAURES

5.1. Introduction 75

5.2. Methods 76

5.2.1. Model description and notation 76

5.2.2. Construction of health indicators 79

5.3. An application to HIV control 82

5.3.1. Context 82

5.3.2. Estimation method 82

5.3.3. Results: new indicators of health state 84

5.4. An application to breast cancer 86

5.4.1. Context 86

5.4.2. Age and stage–specific prevalence 87

5.4.3. Estimation method 88

5.4.4. Results: indicators of public health 88

5.5. Discussion 89

5.6. Bibliography 89

Chapter 6. Bivariate Cox Models 93Michel BRONIATOWSKI, Alexandre DEPIRE and Ya acov RITOV

6.1. Introduction 93

6.2. A dependence model for duration data 93

6.3. Some useful facts in bivariate dependence 95

6.4. Coherence 98

6.5. Covariates and estimation 102

6.6. Application: regression of Spearman s rho on covariates 104

6.7. Bibliography 106

Chapter 7. Non–parametric Estimation of a Class of Survival Functionals 109Belkacem ABDOUS

7.1. Introduction 109

7.2. Weighted local polynomial estimates 111

7.3. Consistency of local polynomial fitting estimators 114

7.4. Automatic selection of the smoothing parameter 116

7.5. Bibliography 119

Chapter 8. Approximate Likelihood in Survival Models 121Henning LAUTER

8.1. Introduction 121

8.2. Likelihood in proportional hazard models 122

8.3. Likelihood in parametric models 122

8.4. Profile likelihood 123

8.4.1. Smoothness classes 124

8.4.2. Approximate likelihood function 125

8.5. Statistical arguments 127

8.6. Bibliography 129

PART II 131

Chapter 9.Cox Regression with Missing Values of a Covariate having a Non–proportional Effect on Risk of Failure 133Jean–Francois DUPUY and Eve LECONTE

9.1. Introduction 133

9.2. Estimation in the Cox model with missing covariate values: a short review 136

9.3. Estimation procedure in the stratified Cox model with missing stratum indicator values 139

9.4. Asymptotic theory 141

9.5. A simulation study 145

9.6. Discussion 147

9.7. Bibliography 149

Chapter 10.Exact Bayesian Variable Sampling Plans for Exponential Distribution under Type–I Censoring 151Chien–Tai LIN, Yen–Lung HUANG and N. BALAKRISHNAN

10.1. Introduction 151

10.2. Proposed sampling plan and Bayes risk 152

10.3. Numerical examples and comparison 156

10.4. Bibliography 161

Chapter 11. Reliability of Stochastic Dynamical Systems Applied to Fatigue Crack Growth Modeling 163Julien CHIQUET and Nikolaos LIMNIOS

11.1. Introduction 163

11.2. Stochastic dynamical systems with jump Markov process 165

11.3. Estimation 168

11.4. Numerical application 170

11.5. Conclusion 175

11.6. Bibliography 175

Chapter 12. Statistical Analysis of a Redundant System with One Standby Unit 179Vilijandas BAGDONAVIC¡ IUS, Inga MASIULAITYTE and Mikhail NIKULIN

12.1. Introduction 179

12.2. The models 180

12.3. The tests 181

12.4. Limit distribution of the test statistics 182

12.5. Bibliography 187

Chapter 13.A Modified Chi–squared Goodness–of–fit Test for the ThreeparameterWeibull Distribution and its Applications in Reliability 189Vassilly VOINOV, Roza ALLOYAROVA and Natalie PYA

13.1. Introduction 189

13.2. Parameter estimation and modified chi–squared tests 191

13.3. Power estimation 194

13.4. Neyman–Pearson classes 194

13.5. Discussion 197

13.6. Conclusion 198

13.7. Appendix 198

13.8. Bibliography 201

Chapter 14.Accelerated Life Testing when the Hazard Rate Function has Cup Shape 203Vilijandas BAGDONAVIC¡ IUS, Luc CLERJAUD and Mikhail NIKULIN

14.1. Introduction 203

14.2. Estimation in the AFT–GW model 204

14.2.1. AFT model 204

14.2.2. AFT–Weibull, AFT–lognormal and AFT–GW models 205

14.2.3. Plans of ALT experiments 205

14.2.4. Parameter estimation: AFT–GW model 206

14.3. Properties of estimators: simulation results for the AFT–GW model 207

14.4. Some remarks on the second plan of experiments 211

14.5. Conclusion 213

14.6. Appendix 213

14.7. Bibliography 215

Chapter 15. Point Processes in Software Reliability 217James LEDOUX

15.1. Introduction 217

15.2. Basic concepts for repairable systems 219

15.3. Self–exciting point processes and black–box models 221

15.4. White–box models and Markovian arrival processes 225

15.4.1. A Markovian arrival model 226

15.4.2. Parameter estimation 228

15.4.3. Reliability growth 232

15.5. Bibliography 234

PART III 237

Chapter 16. Likelihood Inference for the Latent Markov Rasch Model 239Francesco BARTOLUCCI, Fulvia PENNONI and Monia LUPPARELLI

16.1. Introduction 239

16.2. Latent class Rasch model 240

16.3. Latent Markov Rasch model 241

16.4. Likelihood inference for the latent Markov Rasch model 243

16.4.1. Log–likelihood maximization 244

16.4.2. Likelihood ratio testing of hypotheses on the parameters 245

16.5. An application 246

16.6. Possible extensions 247

16.6.1. Discrete response variables 248

16.6.2. Multivariate longitudinal data 248

16.7. Conclusions 251

16.8. Bibliography 252

Chapter 17. Selection of Items Fitting a Rasch Model 255Jean–Benoit HARDOUIN and Mounir MESBAH

17.1. Introduction 255

17.2. Notations and assumptions 256

17.2.1. Notations 256

17.2.2. Fundamental assumptions of the Item Response Theory (IRT) 256

17.3. The Rasch model and the multidimensional marginally sufficient Rasch model 256

17.3.1. The Rasch model 256

17.3.2. The multidimensional marginally sufficient Rasch model 257

17.4. The Raschfit procedure 258

17.5. A fast version of Raschfit 259

17.5.1. Estimation of the parameters under the fixed effects Rasch model 259

17.5.2. Principle of Raschfit–fast 260

17.5.3. A model where the new item is explained by the same latent trait as the kernel 260

17.5.4. A model where the new item is not explained by the same latent trait as the kernel 260

17.5.5. Selection of the new item in the scale 261

17.6. A small set of simulations to compare Raschfit and Raschfit–fast 261

17.6.1. Parameters of the simulation study 261

17.6.2. Results and computing time 264

17.7. A large set of simulations to compare Raschfit–fast, MSP and HCA/CCPROX 269

17.7.1. Parameters of the simulations 269

17.7.2. Discussion 270

17.8. The Stata module Raschfit 270

17.9. Conclusion 271

17.10.Bibliography 273

Chapter 18. Analysis of Longitudinal HrQoL using Latent Regression in the Context of Rasch Modeling 275Silvia BACCI

18.1. Introduction 275

18.2. Global models for longitudinal data analysis 276

18.3. A latent regression Rasch model for longitudinal data analysis 278

18.3.1. Model structure 278

18.3.2. Correlation structure 280

18.3.3. Estimation 281

18.3.4. Implementation with SAS 281

18.4. Case study: longitudinal HrQoL of terminal cancer patients 283

18.5. Concluding remarks 287

18.6. Bibliography 289

Chapter 19. Empirical Internal Validation and Analysis of a Quality of Life Instrument in French Diabetic Patients during an Educational Intervention 291Judith CHWALOW, Keith MEADOWS, Mounir MESBAH, Vincent COLICHE and Etienne MOLLET

19.1. Introduction 291

19.2. Material and methods 292

19.2.1. Health care providers and patients 292

19.2.2. Psychometric validation of the DHP 293

19.2.3. Psychometric methods 293

19.2.4. Comparative analysis of quality of life by treatment group 294

19.3. Results 295

19.3.1. Internal validation of the DHP 295

19.3.2. Comparative analysis of quality of life by treatment group 303

19.4. Discussion 304

19.5. Conclusion 305

19.6. Bibliography 306

19.7. Appendices 309

PART IV 315

Chapter 20. Deterministic Modeling of the Size of the HIV/AIDS Epidemic in Cuba 317Rachid LOUNES, Hector DE ARAZOZA, Y.H. HSIEH and Jose JOANES

20.1. Introduction 317

20.2. The models 319

20.2.1. The k2X model 322

20.2.2. The k2Y model 322

20.2.3. The k2XY model 323

20.2.4. The k2 XYX+Y model 324

20.3. The underreporting rate 324

20.4. Fitting the models to Cuban data 325

20.5. Discussion and concluding remarks 326

20.6. Bibliography 330

Chapter 21.Some Probabilistic Models Useful in Sport Sciences 333Leo GERVILLE–REACHE, Mikhail NIKULIN, Sebastien ORAZIO, Nicolas PARIS and Virginie ROSA

21.1. Introduction 333

21.2. Sport jury analysis: the Gauss–Markov approach 334

21.2.1. Gauss–Markov model 334

21.2.2. Test for non–objectivity of a variable 334

21.2.3. Test of difference between skaters 335

21.2.4. Test for the less precise judge 336

21.3. Sport performance analysis: the fatigue and fitness approach 337

21.3.1. Model characteristics 337

21.3.2. Monte Carlo simulation 338

21.3.3. Results 339

21.4. Sport equipment analysis: the fuzzy subset approach 339

21.4.1. Statistical model used 340

21.4.2. Sensorial analysis step 341

21.4.3. Results 342

21.5. Sport duel issue analysis: the logistic simulation approach 343

21.5.1. Modeling by logistic regression 344

21.5.2. Numerical simulations 345

21.5.3. Results 345

21.6. Sport epidemiology analysis: the accelerated degradation approach 347

21.6.1. Principle of degradation in reliability analysis 347

21.7. Conclusion 350

21.8. Bibliography 350

Appendices 353

A. European Seminar: Some Figures 353

A.1. Former international speakers invited to the European Seminar 353

A.2. Former meetings supported by the European Seminar 353

A.3. Books edited by the organizers of the European Seminar 354

A.4. Institutions supporting the European Seminar (names of colleagues) 355

B. Contributors 357

Index 367

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