Mathematical Methods in Survival Analysis, Reliability and Quality of Life

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.

Autorentext

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.

Inhalt

Preface 13

PART I 15

Chapter 1. Model Selection for Additive Regression in the Presence of Right-Censoring 17
Elodie 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 33
Odile 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 49
Filia 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 61
Lea 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 75
Eve 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 93
Michel 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 109<br /&g...