Table of Contents

Preface

Terms and Notation

Chapter 1. Censored and Truncated Data

1.1. Right-censored data

1.2. Left truncation

1.3. Left truncation and right censoring

1.4. Nelson–Aalen and Kaplan–Meier estimators

1.5. Bibliographic notes

Chapter 2. Chi-squared Tests

2.1. Chi-squared test for composite hypothesis

2.2. Chi-squared test for exponential distributions

2.3. Chi-squared tests for shape-scale distribution families

2.4. Chi-squared tests for other families

2.5. Exercises

2.6. Answers

Chapter 3. Homogeneity Tests for Independent Populations

3.1. Data

3.2. Weighted logrank statistics

3.3. Logrank test statistics as weighted sums of differences between observed and expected number of failures

3.4. Examples of weights

3.5. Weighted logrank statistics as modified score statistics

3.6. The first two moments of weighted logrank statistics

3.7. Asymptotic properties of weighted logrank statistics

3.8. Weighted logrank tests

3.9. Homogeneity testing when alternatives are crossings of survival functions

3.10. Exercises

3.11. Answers

Chapter 4. Homogeneity Tests for Related Populations

4.1. Paired samples

4.2. Logrank-type tests for homogeneity of related k > 2 samples

4.3. Homogeneity tests for related samples against crossing marginal survival functions alternatives

4.4. Exercises

4.5. Answers

Chapter 5. Goodness-of-fit for Regression Models

5.1. Goodness-of-fit for the semi-parametric Cox model

5.2. Chi-squared goodness-of-fit tests for parametric AFT models

5.3. Chi-squared test for the exponential AFT model

5.4. Chi-squared tests for scale-shape AFT models

5.5. Bibliographic notes

5.6. Exercises

5.7. Answers

Appendices

Appendix A. Maximum Likelihood Method for Censored Samples

A.1. ML estimators: right censoring

A.2. ML estimators: left truncation

A.3. ML estimators: left truncation and right censoring

A.4. Consistency and asymptotic normality of the ML estimators

A.5. Parametric ML estimation for survival regression models

Appendix B. Notions from the Theory of Stochastic Processes

B.1. Stochastic process

B.2. Counting process

B.3. Martingale and local martingale

B.4. Stochastic integral

B.5. Predictable process and Doob–Meyer decomposition

B.6. Predictable variation and predictable covariation

B.7. Stochastic integrals with respect to martingales

B.8. Central limit theorem for martingales

Appendix C. Semi-parametric Estimation using the Cox Model

C.1. Partial likelihood

C.2. Asymptotic properties of estimators

Bibliography

Index

First published 2011 in Great Britain and the United States 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 2011

The rights of Vilijandas Bagdonaviçius, Julius Kruopis and Mikhail S. 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

Bagdonavicius, V. (Vilijandas)

Nonparametric tests for censored data / Vilijandas Bagdonavicius, Julius Kruopis, Mikhail Nikulin.

p. cm.

ISBN 978-1-84821-289-3 (hardback)

1. Nonparametric statistics. 2. Statistical hypothesis testing. I. Kruopis, Julius. II. Nikulin, Mikhail (Mikhail S.) III. Title.

QA278.8.B338 2010

519.5--dc22

2010038274

British Library Cataloguing-in-Publication Data

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

ISBN 978-1-84821-289-3

Preface

This book deals with testing hypotheses in non-parametric models. A statistical model is non-parametric if it cannot be written in terms of finite-dimensional parameters. This book is a continuation of our book “Non-parametric Tests for Complete Data” [BAG 10], and it gives generalizations to the case of censored data. The basic notions of hypotheses testing covered in [BAG 10] and many other books are not covered here.

Tests from censored data are mostly considered in books on survival analysis and reliability, such as the monographs by Kalbfleisch and Prentice [KAL 89], Fleming and Harrington [FLE 91], Andersen et al. [AND 93], Lawless [LAW 02], Bagdonaviius and Nikulin [BAG 02], Meeker and Escobar [MEE 98], Klein and Moeschberger [KLE 03], Kleinbaum and Klein [KLE 05], and Martinussen and Scheike [MAR 06].

In the first chapter, the idea of censored and truncated data is explained. In Chapter 2, modified chi-squared goodness-of-fit tests for censored and truncated data are given. The application of modified chi-squared tests to censored data is not well described in the statistical literature, so we have described such test statistics for the most-used families of probability distributions. Chi-squared tests for parametric accelerated failure time regression models, which are widely applied in reliability, accelerated life testing and survival analysis, are given in Chapter 5. These tests may be used not only for censored data but also for complete data. Goodness-of-fit tests for semi-parametric proportional hazards or Cox models are given in Chapter 5.

Homogeneity tests for independent censored samples are given in Chapter 3. We describe classical logrank tests, the original tests directed against alternatives with possible crossings of cumulative distribution functions. Homogeneity tests for dependent censored samples are only touched on very slightly in classical books on survival analysis. In Chapter 4, we give generalizations of logrank tests to the case of dependent samples, and also tests which are powerful against crossing marginal distribution functions alternatives.

Any given test is described in the following way: 1) a hypothesis is formulated; 2) the idea of test construction is given; 3) a statistic on which a test is based is given; 4) the asymptotic distribution of the test statistic is found; 5) a test is formulated; 6) practical examples of application of the tests are given; and 7) at the end of each chapter exercises with answers are given.

The basic facts on probability, stochastic processes and survival analysis used in the book are given in appendices.

Anyone who applies non-parametric methods of mathematical statistics, or who wants to know the ideas behind and mathematical substantiations of the tests, can use this book. If the application of non-parametric tests in reliability and survival analysis is of interest then this book could be the basis of a one-semester course for graduate students.

Knowledge of probability and parametric statistics is needed to follow the mathematical developments. The basic facts on probability and parametric statistics used in the book are also given in appendices.

The book contains five chapters and three appendices. In each chapter, the numbering of theorems, formulas and comments include the chapter number.

This book was written using lecture notes for graduate students in Vilnius and Bordeaux universities.

We thank colleagues and students at Vilnius and Bordeaux universities for their comments on the content of this book, especially Rta Levulien for writing the computer programs needed for the application of tests and the solutions of all exercises.

Vilijandas BAGDONAVIIUS

Julius KRUOPIS

Mikhail NIKULIN

Terms and Notation

– the norm of a matrix A = [aij];

A > B (A ≥ B) – the matrix A − B is positive (non-negative) definite;

a ∨ b (a ∧ b) – the maximum (the minimum) of the numbers a and b;

A(t) – the compensator of a counting process N(t);

AFT – accelerated failure time;

ALT – accelerated life testing;

B(n, p) – binomial distribution with parameters n and p;

Be(γ, η) – beta distribution with parameters γ and η;

cdf – the cumulative distribution function;

CLT – the central limit theorem;

Cov(X, Y) – the covariance of random vectors X and Y;

Cov(X, Y) – the covariance matrix of random vectors X and Y;

EX – the mean of a random variable X;

E(X) – the mean of a random vector X;

Eθ(X), E(X|θ), Varθ(X), Var(X|θ) – the mean or the variance of a random variable X depending on the parameter θ;

FT(x) (fT(x)) – the cdf (the pdf) of the random variable T;

f(x; θ) – the pdf depending on a parameter θ;

F(x;θ) – the cdf depending on a parameter θ;

iid – independent identically distributed;

LS – least-squares (method, estimator);

M(t) – a martingale;

< M > (t) – the predictable variation of a martingale M(t);

< M₁, M₂ > (t) – the predictable covariation of martingales M₁(t) and M₂(t);

ML – maximum likelihood (function, method, estimator);

MPL – maximum partial likelihood;

N(0, 1) – standard normal distribution;

N(μ, σ²) – normal distribution with parameters μ and σ²;

N(t) – the number of observed failures in the interval [0, t];

Nk(μ, ∑) – k-dimensional normal distribution with the mean vector μ and the covariance matrix ∑;

pdf – the probability density function;

P{A} – the probability of an event A;

P{A|B} – the conditional probability of event A;

Pθ{A}, P{A|θ} – the probability depending on a parameter θ;

PH – proportional hazards

– k-dimensional multinomial distribution with parameters n and π = (π₁, …,πk)T, π₁ + … + πk = 1;

S(t; θ) – the survival function depending on a parameter θ;

VarX – the variance of a random variable X;

Var(X) – the covariance matrix of a random vector X;

X, Y, Z,… – random variables;

X, Y, Z,… – random vectors;

XT – the transposed vector, i.e. a vector-line;

– the length of a vector x = (x₁, …,xk)T;

X ~ N(μ, σ²) – random variable X normally distributed with parameters μ, and σ² (analogously in the case of other distributions);

– convergence in probability (n → ∞);

– almost sure convergence or convergence with probability 1 (n → ∞);

– weak convergence or convergence in distribution (n → ∞);

– random variables Xn asymptotically (n → ∞) normally distributed with parameters μ and σ²;

Xn ∼ Yn – random variables Xn and Yn asymptotically (n → ∞) equivalent

Y(t) – the number of objects at risk just prior the moment t;

zα – α critical value of the standard normal distribution;

∑= [σij]k×k – covariance matrix;

X²(n) – chi-squared distribution with n degrees of freedom;

– α critical value of chi-squared distribution with n degrees of freedom;

λ(t; θ) – the hazard function depending on a parameter θ.

Chapter 1

Censored and Truncated Data

Suppose that n objects are observed during an experiment. The failure times of these objects are modeled as iid absolutely continuous positive random variables . The notion of is understood very generally: it may be the time from birth, the beginning of a disease, or the end of an operation to the death of a live organism; the functioning time of a unit from its fabrication or sale to its failure, deterioration or appearance of some defect; the time from registering at a job center to receiving the offer of a specific job, etc.

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