Table Of ContentNon-parametric Tests for Censored Data
Non-parametric Tests
for Censored Data
Vilijandas Bagdonavičius
Julius Kruopis
Mikhail S. Nikulin
First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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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
Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Terms and Notation . . . . . . . . . . . . . . . . . . . . . xv
Chapter1. Censored and Truncated Data . . . . . . 1
1.1. Right-censored data . . . . . . . . . . . . . . . . . 2
1.2. Left truncation . . . . . . . . . . . . . . . . . . . . 12
1.3. Left truncation andrightcensoring . . . . . . . 14
1.4. Nelson–AalenandKaplan–Meierestimators . . 15
1.5. Bibliographicnotes . . . . . . . . . . . . . . . . . . 17
Chapter2. Chi-squared Tests . . . . . . . . . . . . . . 19
2.1. Chi-squaredtest for composite hypothesis . . . 19
2.2. Chi-squaredtest for exponential distributions . 31
2.3. Chi-squaredtests for shape-scaledistribution
families . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1. Chi-squaredtest for the Weibulldistribution 39
2.3.2. Chi-squaredtests for the loglogistic
distribution . . . . . . . . . . . . . . . . . . . . 44
2.3.3. Chi-squaredtest for the lognormal
distribution . . . . . . . . . . . . . . . . . . . . 46
2.4. Chi-squaredtests for other families . . . . . . . 51
vi Non-parametricTestsforCensoredData
2.4.1. Chi-squaredtest for theGompertz
distribution . . . . . . . . . . . . . . . . . . . . 53
2.4.2. Chi-squaredtest for distribution with
hyperbolic hazardfunction . . . . . . . . . . . 56
2.4.3. Bibliographicnotes . . . . . . . . . . . . . . . . 59
2.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6. Answers . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter3. HomogeneityTests for Independent
Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
.
3.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2. Weightedlogrank statistics . . . . . . . . . . . . . 64
3.3. Logrank test statistics asweightedsumsof
differences between observed andexpected
numberof failures . . . . . . . . . . . . . . . . . . 66
3.4. Examplesof weights . . . . . . . . . . . . . . . . . 67
3.5. Weightedlogrank statistics asmodifiedscore
statistics . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6. Thefirst twomoments of weighted logrank
statistics . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7. Asymptotic properties of weightedlogrank
statistics . . . . . . . . . . . . . . . . . . . . . . . . 73
3.8. Weightedlogrank tests . . . . . . . . . . . . . . . 80
3.9. Homogeneity testing when alternativesare
crossings of survivalfunctions . . . . . . . . . . 85
3.9.1. Alternatives . . . . . . . . . . . . . . . . . . . . 86
3.9.2. Modified score statistics . . . . . . . . . . . . . 88
3.9.3. Limitdistributionof the modifiedscore
statistics . . . . . . . . . . . . . . . . . . . . . . 91
3.9.4. Homogeneity tests againstcrossing survival
functionsalternatives . . . . . . . . . . . . . . 92
3.9.5. Bibliographicnotes . . . . . . . . . . . . . . . . 97
3.10. Exercises . . . . . . . . . . . . . . . . . . . . . . . . 98
3.11. Answers. . . . . . . . . . . . . . . . . . . . . . . . . 102
TableofContents vii
Chapter4. HomogeneityTests for Related
Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1. Pairedsamples . . . . . . . . . . . . . . . . . . . . 106
4.1.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.1.2. Test statistics . . . . . . . . . . . . . . . . . . . 107
4.1.3. Asymptotic distribution of the test statistic. 107
4.1.4. Thetest . . . . . . . . . . . . . . . . . . . . . . . 116
4.2. Logrank-type tests for homogeneity of related
k > 2 samples . . . . . . . . . . . . . . . . . . . . . 119
4.3. Homogeneity tests for related samplesagainst
crossing marginalsurvivalfunctions
alternatives . . . . . . . . . . . . . . . . . . . . . . 122
4.3.1. Bibliographicnotes . . . . . . . . . . . . . . . . 124
4.4. Exercises . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5. Answers . . . . . . . . . . . . . . . . . . . . . . . . . 126
Chapter5. Goodness-of-fit for Regression Models 127
5.1. Goodness-of-fit for thesemi-parametric Cox
model . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1.1. TheCox model. . . . . . . . . . . . . . . . . . . 127
5.1.2. Alternativesto theCox model basedon
expanded models . . . . . . . . . . . . . . . . . 128
5.1.3. Thedata andthemodified score statistics . 129
5.1.4. Asymptotic distribution of the modified
score statistic . . . . . . . . . . . . . . . . . . . 133
5.1.5. Tests . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2. Chi-squaredgoodness-of-fit tests for
parametric AFTmodels . . . . . . . . . . . . . . . 142
5.2.1. Accelerated failuretime model . . . . . . . . 142
5.2.2. ParametricAFTmodel . . . . . . . . . . . . . 144
5.2.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.2.4. Idea of test construction . . . . . . . . . . . . . 145
5.2.5. Asymptotic distribution of Hn andZ . . . . . 146
5.2.6. Test statistics . . . . . . . . . . . . . . . . . . . 151
5.3. Chi-squaredtest for the exponentialAFT model. 153
viii Non-parametricTestsforCensoredData
5.4. Chi-squaredtests for scale-shapeAFT models. 159
5.4.1. Chi-squaredtest for the WeibullAFT model 163
5.4.2. Chi-squaredtest for the lognormalAFT
model . . . . . . . . . . . . . . . . . . . . . . . . 166
5.4.3. Chi-squaredtest for the loglogisticAFT
model . . . . . . . . . . . . . . . . . . . . . . . . 169
5.5. Bibliographicnotes . . . . . . . . . . . . . . . . . . 172
5.6. Exercises . . . . . . . . . . . . . . . . . . . . . . . . 173
5.7. Answers . . . . . . . . . . . . . . . . . . . . . . . . . 174
APPENDICES. . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Appendix A. Maximum Likelihood Method for
Censored Samples. . . . . . . . . . . . . . . . . . . . . 179
A.1.MLestimators: rightcensoring . . . . . . . . . 179
A.2. ML estimators: left truncation. . . . . . . . . . 181
A.3.ML estimators: left truncation andright
censoring . . . . . . . . . . . . . . . . . . . . . . . 182
A.4.Consistency andasymptoticnormality of the
ML estimators . . . . . . . . . . . . . . . . . . . . 186
A.5.ParametricML estimation for survival
regression models . . . . . . . . . . . . . . . . . . 187
Appendix B. Notions from the Theory of
Stochastic Processes . . . . . . . . . . . . . . . . . . 191
B.1. Stochastic process. . . . . . . . . . . . . . . . . . 191
B.2. Countingprocess . . . . . . . . . . . . . . . . . . 193
B.3. Martingaleandlocal martingale. . . . . . . . . 194
B.4. Stochastic integral . . . . . . . . . . . . . . . . . 195
B.5. Predictable process andDoob–Meyer
decomposition . . . . . . . . . . . . . . . . . . . . 197
B.6. Predictable variationandpredictable
covariation . . . . . . . . . . . . . . . . . . . . . . 198
B.7. Stochastic integralswithrespect to
martingales. . . . . . . . . . . . . . . . . . . . . . 204
B.8. Central limittheorem for martingales . . . . . 207
TableofContents ix
Appendix C. Semi-parametric Estimation
using the Cox Model . . . . . . . . . . . . . . . . . 211
C.1. Partiallikelihood . . . . . . . . . . . . . . . . . . 211
C.2. Asymptotic properties of estimators . . . . . . 213
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Preface
Thisbook 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.
Testsfromcensored dataaremostlyconsidered inbookson
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],
Bagdonavicˇius and Nikulin [BAG 02], Meeker and Escobar
[MEE 98],Klein andMoeschberger [KLE 03],Kleinbaumand
Klein[KLE 05],andMartinussenandScheike [MAR 06].
Inthefirstchapter,theideaofcensoredandtruncateddata
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
xii Non-parametricTestsforCensoredData
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 aregiven inChapter 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
testsfordependentcensoredsamplesareonlytouchedonvery
slightly in classical books on survival analysis. In Chapter
4, we give generalizations of logrank tests to the case of
dependent samples,andalsotests whicharepowerful against
crossing marginaldistributionfunctionsalternatives.
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
isformulated; 6) practical examples of application of the tests
are given; and 7) at the end of each chapter exercises with
answersare given.
The basic facts on probability, stochastic processes and
survivalanalysisusedin thebook are givenin 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.
