Mathematical Methods in Survival Analysis, Reliability and Quality of Life

Catherine Huber is an Emeritus professor at Université de Paris René Descartes.  Her research activity concerns nonparametric and semi-parametric theory of statistics and their applications in biology and medicine. She has several publications in particular in the field of survival analysis. She is the co-author and co-editor of several books in the above fields. Nikolaos Limnios is a professor at the University of Technology of Compiègne. His research and teaching activities concern stochastic processes, statistical inference and their applications in particular in reliability and survival analysis. He is the co-author and co-editor of several books in the above fields.Mounir Mesbah is a professor at the Université Pierre et Marie Curie, Paris 6. His research and teaching activities concern statistics and its applications in health science and medicine (biostatistics). He is the co-author of several articles and co-editor of several books in the above fields.Mikhail Nikulin is a professor at the Université Victor Segalen, and a member of the Institute of Mathematics at Bordeaux. His research and teaching activities concern mathematical statistics and its applications in reliability and survival analysis. He is the co-author and co-editor of several books in the above fields.

• Preface 13PART I 15Chapter 1. Model Selection for Additive Regression in the Presence of Right-Censoring 17Elodie BRUNEL and Fabienne COMTE1.1. Introduction 171.2. Assumptions on the model and the collection of approximation spaces 181.2.1. Non-parametric regression model with censored data 181.2.2. Description of the approximation spaces in the univariate case 191.2.3. The particular multivariate setting of additive models 201.3. The estimation method 201.3.1. Transformation of the data 201.3.2. The mean-square contrast 211.4. Main result for the adaptive mean-square estimator 221.5. Practical implementation 231.5.1. The algorithm 231.5.2. Univariate examples 241.5.3. Bivariate examples 271.5.4. A trivariate example 281.6. Bibliography 30Chapter 2. Non-parametric Estimation of Conditional Probabilities, Means and Quantiles under Bias Sampling 33Odile PONS2.1. Introduction 332.2. Non-parametric estimation of p 342.3. Bias depending on the value of Y 352.4. Bias due to truncation on X 372.5. Truncation of a response variable in a non-parametric regression model 372.6. Double censoring of a response variable in a non-parametric model 422.7. Other truncation and censoring of Y in a non-parametric model 442.8. Observation by interval 472.9. Bibliography 48Chapter 3. Inference in Transformation Models for Arbitrarily Censored and Truncated Data 49Filia VONTA and Catherine HUBER3.1. Introduction 493.2. Non-parametric estimation of the survival function S 503.3. Semi-parametric estimation of the survival function S 513.4. Simulations 543.5. Bibliography 59Chapter 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 HEMON4.1. Introduction 614.2. Modeling framework 624.2.1. Aggregated model 624.2.2. Prior distributions 654.3. Simulation framework 654.4. Results 664.4.1. Strong association between relative risk and risk factor, correlated within-area means and variances (mean-dependent case) 674.4.2. Sensitivity to within-area distribution of the risk factor 684.4.3. Application: leukemia and indoor radon exposure 694.5. Discussion 714.6. Bibliography 72Chapter 5. Semi-Markov Processes and Usefulness in Medicine 75Eve MATHIEU-DUPAS, Claudine GRAS-AYGON and Jean-Pierre DAURES5.1. Introduction 755.2. Methods 765.2.1. Model description and notation 765.2.2. Construction of health indicators 795.3. An application to HIV control 825.3.1. Context 825.3.2. Estimation method 825.3.3. Results: new indicators of health state 845.4. An application to breast cancer 865.4.1. Context 865.4.2. Age and stage-specific prevalence 875.4.3. Estimation method 885.4.4. Results: indicators of public health 885.5. Discussion 895.6. Bibliography 89Chapter 6. Bivariate Cox Models 93Michel BRONIATOWSKI, Alexandre DEPIRE and Ya’acov RITOV6.1. Introduction 936.2. A dependence model for duration data 936.3. Some useful facts in bivariate dependence 956.4. Coherence 986.5. Covariates and estimation 1026.6. Application: regression of Spearman’s rho on covariates 1046.7. Bibliography 106Chapter 7. Non-parametric Estimation of a Class of Survival Functionals 109Belkacem ABDOUS7.1. Introduction 1097.2. Weighted local polynomial estimates 1117.3. Consistency of local polynomial fitting estimators 1147.4. Automatic selection of the smoothing parameter 1167.5. Bibliography 119Chapter 8. Approximate Likelihood in Survival Models 121Henning LAUTER8.1. Introduction 1218.2. Likelihood in proportional hazard models 1228.3. Likelihood in parametric models 1228.4. Profile likelihood 1238.4.1. Smoothness classes 1248.4.2. Approximate likelihood function 1258.5. Statistical arguments 1278.6. Bibliography 129PART II 131Chapter 9.Cox Regression with Missing Values of a Covariate having a Non-proportional Effect on Risk of Failure 133Jean-Francois DUPUY and Eve LECONTE9.1. Introduction 1339.2. Estimation in the Cox model with missing covariate values: a short review 1369.3. Estimation procedure in the stratified Cox model with missing stratum indicator values 1399.4. Asymptotic theory 1419.5. A simulation study 1459.6. Discussion 1479.7. Bibliography 149Chapter 10.Exact Bayesian Variable Sampling Plans for Exponential Distribution under Type-I Censoring 151Chien-Tai LIN, Yen-Lung HUANG and N. BALAKRISHNAN10.1. Introduction 15110.2. Proposed sampling plan and Bayes risk 15210.3. Numerical examples and comparison 15610.4. Bibliography 161Chapter 11. Reliability of Stochastic Dynamical Systems Applied to Fatigue Crack Growth Modeling 163Julien CHIQUET and Nikolaos LIMNIOS11.1. Introduction 16311.2. Stochastic dynamical systems with jump Markov process 16511.3. Estimation 16811.4. Numerical application 17011.5. Conclusion 17511.6. Bibliography 175Chapter 12. Statistical Analysis of a Redundant System with One Standby Unit 179Vilijandas BAGDONAVIC¡ IUS, Inga MASIULAITYTE and Mikhail NIKULIN12.1. Introduction 17912.2. The models 18012.3. The tests 18112.4. Limit distribution of the test statistics 18212.5. Bibliography 187Chapter 13.A Modified Chi-squared Goodness-of-fit Test for the ThreeparameterWeibull Distribution and its Applications in Reliability 189Vassilly VOINOV, Roza ALLOYAROVA and Natalie PYA13.1. Introduction 18913.2. Parameter estimation and modified chi-squared tests 19113.3. Power estimation 19413.4. Neyman-Pearson classes 19413.5. Discussion 19713.6. Conclusion 19813.7. Appendix 19813.8. Bibliography 201Chapter 14.Accelerated Life Testing when the Hazard Rate Function has Cup Shape 203Vilijandas BAGDONAVIC¡ IUS, Luc CLERJAUD and Mikhail NIKULIN14.1. Introduction 20314.2. Estimation in the AFT-GW model 20414.2.1. AFT model 20414.2.2. AFT-Weibull, AFT-lognormal and AFT-GW models 20514.2.3. Plans of ALT experiments 20514.2.4. Parameter estimation: AFT-GW model 20614.3. Properties of estimators: simulation results for the AFT-GW model 20714.4. Some remarks on the second plan of experiments 21114.5. Conclusion 21314.6. Appendix 21314.7. Bibliography 215Chapter 15. Point Processes in Software Reliability 217James LEDOUX15.1. Introduction 21715.2. Basic concepts for repairable systems 21915.3. Self-exciting point processes and black-box models 22115.4. White-box models and Markovian arrival processes 22515.4.1. A Markovian arrival model 22615.4.2. Parameter estimation 22815.4.3. Reliability growth 23215.5. Bibliography 234PART III 237Chapter 16. Likelihood Inference for the Latent Markov Rasch Model 239Francesco BARTOLUCCI, Fulvia PENNONI and Monia LUPPARELLI16.1. Introduction 23916.2. Latent class Rasch model 24016.3. Latent Markov Rasch model 24116.4. Likelihood inference for the latent Markov Rasch model 24316.4.1. Log-likelihood maximization 24416.4.2. Likelihood ratio testing of hypotheses on the parameters 24516.5. An application 24616.6. Possible extensions 24716.6.1. Discrete response variables 24816.6.2. Multivariate longitudinal data 24816.7. Conclusions 25116.8. Bibliography 252Chapter 17. Selection of Items Fitting a Rasch Model 255Jean-Benoit HARDOUIN and Mounir MESBAH17.1. Introduction 25517.2. Notations and assumptions 25617.2.1. Notations 25617.2.2. Fundamental assumptions of the Item Response Theory (IRT) 25617.3. The Rasch model and the multidimensional marginally sufficient Rasch model 25617.3.1. The Rasch model 25617.3.2. The multidimensional marginally sufficient Rasch model 25717.4. The Raschfit procedure 25817.5. A fast version of Raschfit 25917.5.1. Estimation of the parameters under the fixed effects Rasch model 25917.5.2. Principle of Raschfit-fast 26017.5.3. A model where the new item is explained by the same latent trait as the kernel 26017.5.4. A model where the new item is not explained by the same latent trait as the kernel 26017.5.5. Selection of the new item in the scale 26117.6. A small set of simulations to compare Raschfit and Raschfit-fast 26117.6.1. Parameters of the simulation study 26117.6.2. Results and computing time 26417.7. A large set of simulations to compare Raschfit-fast, MSP and HCA/CCPROX 26917.7.1. Parameters of the simulations 26917.7.2. Discussion 27017.8. The Stata module “Raschfit” 27017.9. Conclusion 27117.10.Bibliography 273Chapter 18. Analysis of Longitudinal HrQoL using Latent Regression in the Context of Rasch Modeling 275Silvia BACCI18.1. Introduction 27518.2. Global models for longitudinal data analysis 27618.3. A latent regression Rasch model for longitudinal data analysis 27818.3.1. Model structure 27818.3.2. Correlation structure 28018.3.3. Estimation 28118.3.4. Implementation with SAS 28118.4. Case study: longitudinal HrQoL of terminal cancer patients 28318.5. Concluding remarks 28718.6. Bibliography 289Chapter 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 MOLLET19.1. Introduction 29119.2. Material and methods 29219.2.1. Health care providers and patients 29219.2.2. Psychometric validation of the DHP 29319.2.3. Psychometric methods 29319.2.4. Comparative analysis of quality of life by treatment group 29419.3. Results 29519.3.1. Internal validation of the DHP 29519.3.2. Comparative analysis of quality of life by treatment group 30319.4. Discussion 30419.5. Conclusion 30519.6. Bibliography 30619.7. Appendices 309PART IV 315Chapter 20. Deterministic Modeling of the Size of the HIV/AIDS Epidemic in Cuba 317Rachid LOUNES, Hector DE ARAZOZA, Y.H. HSIEH and Jose JOANES20.1. Introduction 31720.2. The models 31920.2.1. The k2X model 32220.2.2. The k2Y model 32220.2.3. The k2XY model 32320.2.4. The k2 XYX+Y model 32420.3. The underreporting rate 32420.4. Fitting the models to Cuban data 32520.5. Discussion and concluding remarks 32620.6. Bibliography 330Chapter 21.Some Probabilistic Models Useful in Sport Sciences 333Leo GERVILLE-REACHE, Mikhail NIKULIN, Sebastien ORAZIO, Nicolas PARIS and Virginie ROSA21.1. Introduction 33321.2. Sport jury analysis: the Gauss-Markov approach 33421.2.1. Gauss-Markov model 33421.2.2. Test for non-objectivity of a variable 33421.2.3. Test of difference between skaters 33521.2.4. Test for the less precise judge 33621.3. Sport performance analysis: the fatigue and fitness approach 33721.3.1. Model characteristics 33721.3.2. Monte Carlo simulation 33821.3.3. Results 33921.4. Sport equipment analysis: the fuzzy subset approach 33921.4.1. Statistical model used 34021.4.2. Sensorial analysis step 34121.4.3. Results 34221.5. Sport duel issue analysis: the logistic simulation approach 34321.5.1. Modeling by logistic regression 34421.5.2. Numerical simulations 34521.5.3. Results 34521.6. Sport epidemiology analysis: the accelerated degradation approach 34721.6.1. Principle of degradation in reliability analysis 34721.6.2. Accelerated degradation model 34821.7. Conclusion 35021.8. Bibliography 350Appendices 353A. European Seminar: Some Figures 353A.1. Former international speakers invited to the European Seminar 353A.2. Former meetings supported by the European Seminar 353A.3. Books edited by the organizers of the European Seminar 354A.4. Institutions supporting the European Seminar (names of colleagues) 355B. Contributors 357Index 367