Statistical Models and Methods for Reliability and Survival Analysis

Vincent Couallier, Léo Gerville-Réache, Catherine Huber-Carol, Nikolaos Limnios, Mounir Mesbah, Vincent Couallier, Léo Gerville-Réache, Catherine Huber-Carol, Nikolaos Limnios, Mounir Mesbah

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781848216198
• eBook ISBN: 9781118826997
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Statistical Models and Methods for Reliability and Survival Analysis brings together contributions by specialists in statistical theory as they discuss their applications providing up-to-date developments in methods used in survival analysis, statistical goodness of fit, stochastic processes for system reliability, amongst others. Many of these are related to the work of Professor M. Nikulin in statistics over the past 30 years. The authors gather together various contributions with a broad array of techniques and results, divided into three parts - Statistical Models and Methods, Statistical Models and Methods in Survival Analysis, and Reliability and Maintenance.

The book is intended for researchers interested in statistical methodology and models useful in survival analysis, system reliability and statistical testing for censored and non-censored data.

PART 1

Statistical Models and Methods

Chapter 1

Unidimensionality, Agreement and Concordance Probability

The evaluation and comparison of various methods often arise in medical research. For example, the evaluation of reproducibility of a new measurement technique often needs a comparison with the established technique, and image interpretation is often read by two or more observers. In this chapter, we provide a review of the measures of agreement and association, describe the statistical models underlying the Cronbach’s alpha coefficient (CAC) and the backward reliability curve (BRC), the kappa coefficient, and present a general approach based on the concept of concordance probability. In particular, we illustrate the relationship between the concordance probability and various existing measures of agreement and association, namely Kendall’s τ , Somer’s D, area under receiver operating characteristic (ROC) curve and Harrell’s c-index. In addition, we review the estimation of concordance probability and present its large sample properties. Recent developments in the analysis of right censored data are also presented.

1.1. Introduction

The evaluation and comparison of various methods often arise in medical research. For example, the evaluation of reproducibility of a measurement technique often needs a comparison with the established technique, and the interpretation of a computerized tomography (CT) or magnetic resonance imaging (MRI) scan is often read by two or more observers. There is considerable literature on the measure of agreement (see [CHO 04], [BAR 07], [WAT 10], [SHO 04] and [LIN 10]). The methods vary with different types of measurement, i.e. continuous or categorical measurements. When the response variable is continuous, there are several intuitive approaches, namely comparison of means, Cronbach’s coefficient alpha (CAC), various correlation coefficients and the test of slope being 1 in a simple linear regression, as well as alternative methods, the limits of agreement [BLA 86, BLA 99], the concordance correlation coefficient [LIN 89], mean squared deviation and total deviation index [LIN 00], and coverage probability approach [LIN 02]. When the response variable is categorical, kappa statistic, Somer’s D-statistic and logistic regression are commonly used. When one measure is binary and the other measure is continuous, the methods of the ROC curve and logistic regression approach are often applied. These methods are related to typical concordance correlation between repeated measurements through an underlying linear or nonlinear parametric model. Recently developed concordance probability is a non-parametric approach. The concordance probability is commonly used as a measure of discriminatory power and predictive accuracy of statistical models. We show that the concordance probability also provides a unified measure of agreement for different types of measurement.

In this chapter, we present a review of the statistical models underlying the CAC and the BRC in section 1.2, and the kappa coefficient in section 1.3. In section 1.4, we introduce the concordance probability and describe its relationship with Kendall’s τ , Somer’s D and area of ROC curve of sensitivity and 1–specificity for different cutoffs. In section 1.5, we review the estimation of concordance probability and present its large sample properties. In section 1.6, we present recent developments on how to use the concordance probability to assess the agreement among different measures. We present the extension of the approach to the right censored data in section 1.7 and conclude with some discussion in section 1.8.

1.2. From reliability to unidimensionality: CAC and curve

1.2.1. Classical unidimensional models for measurement

Latent variable models involve a set of observable variables A = {X₁, X₂, …, X_k} and a latent (unobservable) variable θ of dimension d ≤ k. In such models, the dimensionality of A is captured by the dimension of θ, the value of d. When d = 1, the dimensionality of set A is called unidimensional.

In a health-related quality of life (HrQoL) study, me...

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Preface
• Biography of Mikhail Stepanovitch Nikouline
• Part 1. Statistical Models and Methods
• Part 2. Statistical Models and Methods in Survival Analysis
• Part 3. Reliability and Maintenance
• List of Authors
• Index