Contents

Preface

PART I

Chapter 1. Model Selection for Additive Regression in the Presence of Right-Censoring

1.1. Introduction

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

1.3. The estimation method

1.4. Main result for the adaptive mean-square estimator

1.5. Practical implementation

1.6. Bibliography

Chapter 2. Non-parametric Estimation of Conditional Probabilities, Means and Quantiles under Bias Sampling

2.1. Introduction

2.2. Non-parametric estimation of p

2.3. Bias depending on the value of Y

2.4. Bias due to truncation on X

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

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

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

2.8. Observation by interval

2.9. Bibliography

Chapter 3. Inference in Transformation Models for Arbitrarily Censored and Truncated Data

3.1. Introduction

3.2. Non-parametric estimation of the survival function S

3.3. Semi-parametric estimation of the survival function S

3.4. Simulations

3.5. Bibliography

Chapter 4. Introduction of Within-area Risk Factor Distribution in Eco-logical Poisson Models

4.1. Introduction

4.2. Modeling framework

4.3. Simulation framework

4.4. Results

4.5. Discussion

4.6. Bibliography

Chapter 5. Semi-Markov Processes and Usefulness in Medicine

5.1. Introduction

5.2. Methods

5.3. An application to HIV control

5.4. An application to breast cancer

5.5. Discussion

5.6. Bibliography

Chapter 6. Bivariate Cox Models

6.1. Introduction

6.2. A dependence model for duration data

6.3. Some useful facts in bivariate dependence

6.4. Coherence

6.5. Covariates and estimation

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

6.7. Bibliography

Chapter 7. Non-parametric Estimation of a Class of Survival Functionals

7.1. Introduction

7.2. Weighted local polynomial estimates

7.3. Consistency of local polynomial fitting estimators

7.4. Automatic selection of the smoothing parameter

7.5. Bibliography

Chapter 8. Approximate Likelihood in Survival Models

8.1. Introduction

8.2. Likelihood in proportional hazard models

8.3. Likelihood in parametric models

8.4. Profile likelihood

8.5. Statistical arguments

8.6. Bibliography

PART II

Chapter 9. Cox Regression with Missing Values of a Covariate having a Non-proportional Effect on Risk of Failure

9.1. Introduction

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

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

9.4. Asymptotic theory

9.5. A simulation study

9.6. Discussion

9.7. Bibliography

Chapter 10. Exact Bayesian Variable Sampling Plans for Exponential Distribution under Type-I Censoring

10.1. Introduction

10.2. Proposed sampling plan and Bayes risk

10.3. Numerical examples and comparison

10.4. Bibliography

Chapter 11. Reliability of Stochastic Dynamical Systems Applied to Fatigue Crack Growth Modeling

11.1. Introduction

11.2. Stochastic dynamical systems with jump Markov process

11.3. Estimation

11.4. Numerical application

11.5. Conclusion

11.6. Bibliography

Chapter 12. Statistical Analysis of a Redundant System with One Standby Unit

12.1. Introduction

12.2. The models

12.3. The tests

12.4. Limit distribution of the test statistics

12.5. Bibliography

Chapter 13. A Modified Chi-squared Goodness-of-fit Test for the Three-parameter Weibull Distribution and its Applications in Reliability

13.1. Introduction

13.2. Parameter estimation and modified chi-squared tests

13.3. Power estimation

13.4. Neyman-Pearson classes

13.5. Discussion

13.6. Conclusion

13.7. Appendix

13.8. Bibliography

Chapter 14. Accelerated Life Testing when the Hazard Rate Function has Cup Shape

14.1. Introduction

14.2. Estimation in the AFT-GW model

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

14.4. Some remarks on the second plan of experiments

14.5. Conclusion

14.6. Appendix

14.7. Bibliography

Chapter 15. Point Processes in Software Reliability

15.1. Introduction

15.2. Basic concepts for repairable systems

15.3. Self-exciting point processes and black-box models

15.4. White-box models and Markovian arrival processes

15.5. Bibliography

PART III

Chapter 16. Likelihood Inference for the Latent Markov Rasch Model

16.1. Introduction

16.2. Latent class Rasch model

16.3. Latent Markov Rasch model

16.4. Likelihood inference for the latent Markov Rasch model

16.5. An application

16.6. Possible extensions

16.7. Conclusions

16.8. Bibliography

Chapter 17. Selection of Items Fitting a Rasch Model

17.1. Introduction

17.2. Notations and assumptions

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

17.4. The Raschfit procedure

17.5. A fast version of Raschfit

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

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

17.8. The Stata module “Raschfit”

17.9. Conclusion

17.10. Bibliography

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

18.1. Introduction

18.2. Global models for longitudinal data analysis

18.3. A latent regression Rasch model for longitudinal data analysis

18.4. Case study: longitudinal HrQoL of terminal cancer patients

18.5. Concluding remarks

18.6. Bibliography

Chapter 19. Empirical Internal Validation and Analysis of a Quality of Life Instrument in French Diabetic Patients during an Educational Inter-vention

19.1. Introduction

19.2. Material and methods

19.3. Results

19.4. Discussion

19.5. Conclusion

19.6. Bibliography

19.7. Appendices

PART IV

Chapter 20. Deterministic Modeling of the Size of the HIV/AIDS Epidemic in Cuba

20.1. Introduction

20.2. The models

20.3. The underreporting rate

20.4. Fitting the models to Cuban data

20.5. Discussion and concluding remarks

20.6. Bibliography

Chapter 21. Some Probabilistic Models Useful in Sport Sciences

21.1. Introduction

21.2. Sport jury analysis: the Gauss-Markov approach

21.3. Sport performance analysis: the fatigue and fitness approach

21.4. Sport equipment analysis: the fuzzy subset approach

21.5. Sport duel issue analysis: the logistic simulation approach

21.6. Sport epidemiology analysis: the accelerated degradation approach

21.7. Conclusion

21.8. Bibliography

Appendices

A. European Seminar: Some Figures

B. Contributors

Index

First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd, 2008

The rights of Catherine Huber, Nikolaos Limnios, Mounir Mesbah and Mikhail Nikulin to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Mathematical methods in survival analysis, reliability and quality of life / edited by Catherine Huber … [et al.].

p. cm.

Includes bibliographical references and index.

ISBN: 978-1-84821-010-3

1. Failure time data analysis. 2. Survival analysis (Biometry) I. Huber, Catherine.

QA276.M342 2008

519.5′46--dc22

2007046232

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-010-3

Preface

The European Seminar on Mathematical Methods for Survival Analysis, Reliability and Quality of Life was created in 1997 by C. HUBER, N. LIMNIOS, M. NIKULIN and M. MESBAH, and, thanks to our respective laboratories and also to the supporting universities (see list below), it ran regularly during the last 10 years. 2007 is a special year as our European Seminar celebrates its 10th birthday. This seminar aims to give a review of recent research in the field of survival analysis, reliability, quality of life, and related topics, from both statistical and probabilistic points of view. Three or four sessions take place every year at the participating universities.

Besides these regular annual sessions, the European seminar supported many in-ternational conferences and workshops in France and abroad: for instance, in 2000, GOF2000 (Goodness Of Fit) in Paris and MMR2000 (Mathematical Methods in Reliability) in Bordeaux, in 2004, an international workshop on “Semi-parametric Models” organized at Mont Saint Michel and, more recently, Biostat2006 in Cyprus. More than 14 international workshops, 26 seminar sessions and about 150 talks were organized during the last ten years (see Appendix A).

Reliability and survival analysis are important applications of stochastic mathematics (probability, statistics and stochastic processes) that were usually treated separately in spite of the similarity of the involved mathematical theory. Reliability is oriented towards technical systems studies (without excluding the human factor), while survival analysis and quality of life are oriented towards biological and medical studies (without excluding the technical factor). The lifetime T of a technical system, of a human patient or of a bacteria is a non-negative random variable, and the same function of the time t, Prob(T > t), is stated as the reliability function, denoted R(t) in reliability theory, and as the survival function, denoted S(t), in medical applications. Nevertheless, even if the function to investigate is the same, the objectives are not always identical. In the field of reliability, most of the time, systems are ergodic and large, which is not the case in survival analysis. Thus, techniques developed in order to evaluate or to estimate the reliability/survival function are not always based on the same fundamental results. However they also include several common techniques: Cox models, degradation models, multi-state approaches (i.e., Markov and semi-Markov models), point processes, etc.

While it is recognized that quality of life is ultimately as important as quantity of life (survival time), efforts to implement quality of life measurements often fail. Statistical methods to analyze time of events are nowadays well established; opinions are largely agreed on between statisticians, clinicians and industrial professionals. Unfortunately, in the quality of life field, there is no standard instrument to measure it, no standard methodology to validate measurement instruments (questionnaires) and no standard statistical methodology to analyze obtained measurements. Specific development and application of modern psychometrical measurement models (latent variable models including the Rasch model) and connection with utility theory are important issues. A more recent issue is the joint analysis of the latent quality of life and external variables such as treatment, evolution (longitudinal analysis) and/or survival time.

The present book includes 21 chapters divided into four parts:

I. Survival analysis

II. Reliability

III. Quality of life

IV. Related topics

We would like to especially thank Julien Chiquet for his very efficient technical support on this book, and ISTE Ltd for their ongoing support.

Catherine HUBER,

Nikolaos LIMNIOS,

Mounir MESBAH,

Mikhail NIKULIN.

PART I

Survival Analysis

Chapter 1

Model Selection for Additive Regression in the Presence of Right-Censoring 1

1.1. Introduction

Statistical tools for handling regression problems when the response is censored have been developed in the last two decades. The response was often assumed to be a linear function of the covariates, but non-parametric regression models provide a very flexible method when a general relationship between covariates and response is first to be explored. In recent years, a vast literature has been devoted to non-parametric regression estimators for completely observed data. However, few methods exist under random censoring. First, Buckley and James [BUC 79] and Koul, Susarla and Van Ryzin [KOU 81] among others introduced the original idea of transforming the data to take the censoring into account, for linear regression curves. Then, Zheng [ZHE 88] proposed various classes of unbiased transformations. Dabrowska [DAB 87] and Zheng [ZHE 88] applied non-parametric methods for estimating the univariate regression curve. Later, Fan and Gijbels [FAN 94] considered a local linear approximation for the data transformed in the same way by using a variable bandwidth adaptive to the sparsity of the design points. Györfi et al. [GYÖ 02] also studied the consistency of generalized Stone’s regression estimators in the censored case. Heuchenne and Van Keilegom [HEU 05] considered a nonlinear semi-parametric regression model with censored data. Park [PAR 04] extended a procedure suggested in Gross and Lai [GRO 96] to a general non-parametric model in the presence of left-truncation and right-censoring, by using B-spline developments. Recently, Kohler et al. [KOH 03] proposed an adaptive mean-square estimator built with polynomial splines.

However, for modeling the relationship between a response and a multivariate re-gressor, new methodologies have to be found to solve the problem of practical implementation in higher dimension. The main objective of the article is to propose a multivariate method of model selection for an additive regression function of a low-dimensional covariate vector. In fact, the particular case of additive models seems to be more realistic in practice and may constitute a way to make the dimension of the covariate greater than 1. Suppose that images is a d-dimensional covariate in a compact set, without loss of generality we assume that images is a [0, 1]d-valued vector. Let images, images be independent identically distributed random variables. Let T > 0 be a fixed time for collecting the data. Therefore, the response variables before censoring are denoted by Yi, T = Yi ∧ T, where a ∧ b denotes the infimum of a and b. Then, the model is defined for i = 1, …, n, by:

(1.1) images

with images. For identifiability we suppose that images for j = 2, …, d. A comment on the model is required. The setting is analogous to Kohler et al. [KOH 03] and the fixed time T is due to the fact that functionals of the survival function under censoring cannot be estimated on the complete support, as mentioned by Gross and Lai [GRO 96]. Note that in the empirical setting, the procedure does work without any truncation. This fixed bound is introduced for technical and theoretical purposes. Of course, it is often mentioned that the function of interest would be r in the regression model images, instead of its biased version rT.

The method consists of building projection estimators of the d components rT, 1, …, rT, d on different projection spaces. The strategy is based on a standard mean-square contrast as in Baraud [BAR 00] together with an optimized version of the data transformation proposed by Fan and Gijbels [FAN 94]. The algorithm is explained and performed through several empirical trials and real data, and in particular improved in the bivariate setting. The model and the assumptions are presented in section 1.2, the method is described in section 1.3 and the main theoretical result is given in section 1.4. Finally, practical implementation is to be found in section 1.5 together with several examples of various dimensions.

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

1.2.1. Non-parametric regression model with censored data

We consider the following censoring mechanism. Let C1, C2, …, Cn be n censoring times independent of images and consequently independent of images as well. The and the couples (, ) are observed where

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