Cover image for Nonparametric tests for censored data
Title:
Nonparametric tests for censored data
Publication Information:
London : ISTE ; Hoboken, NJ : Wiley, 2010
Physical Description:
xviii, 230 p. ; 24 cm.
ISBN:
9781848212893

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30000010283086 QA278.8 B338 2010 Open Access Book Book
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Summary

Summary

This book concerns testing hypotheses in non-parametric models.Generalizations of many non-parametric tests to the case ofcensored and truncated data are considered. Most of the testresults are proved and real applications are illustrated usingexamples. Theories and exercises are provided. The incorrect use ofmany tests applying most statistical software is highlighted anddiscussed.


Author Notes

Vilijandas Bagdonavicius is Professor of Mathematics at the University of Vilnius in Lithuania. His main research areas are statistics, reliability and survival analysis.
Julius Kruopis is Associate Professor of Mathematics at the University of Vilnius in Lithuania. His main research areas are statistics and quality control.


Table of Contents

Prefacep. xi
Terms and Notationp. xv
Chapter 1 Censored and Truncated Datap. 1
1.1 Right-censored datap. 2
1.2 Left truncationp. 12
1.3 Left truncation and right censoringp. 14
1.4 Nelson-Aalen and Kaplan-Meier estimatorsp. 15
1.5 Bibliographic notesp. 17
Chapter 2 Chi-squared Testsp. 19
2.1 Chi-squared test for composite hypothesisp. 19
2.2 Chi-squared test for exponential distributionsp. 31
2.3 Chi-squared tests for shape-scale distribution familiesp. 36
2.3.1 Chi-squared test for the Weibull distributionp. 39
2.3.2 Chi-squared tests for the loglogistic distributionp. 44
2.3.3 Chi-squared test for the lognormal distributionp. 46
2.4 Chi-squared tests for other familiesp. 51
2.4.1 Chi-squared test for the Gompertz distributionp. 53
2.4.2 Chi-squared test for distribution with hyperbolic hazard functionp. 56
2.4.3 Bibliographic notesp. 59
2.5 Exercisesp. 59
2.6 Answersp. 60
Chapter 3 Homogeneity Tests for Independent Populationsp. 63
3.1 Datap. 64
3.2 Weighted logrank statisticsp. 64
3.3 Logrank test statistics as weighted sums of differences between observed and expected number of failuresp. 66
3.4 Examples of weightsp. 67
3.5 Weighted logrank statistics as modified score statisticsp. 69
3.6 The first two moments of weighted logrank statisticsp. 71
3.7 Asymptotic properties of weighted logrank statisticsp. 73
3.8 Weighted logrank testsp. 80
3.9 Homogeneity testing when alternatives are crossings of survival functionsp. 85
3.9.1 Alternativesp. 86
3.9.2 Modified score statisticsp. 88
3.9.3 Limit distribution of the modified score statisticsp. 91
3.9.4 Homogeneity tests against crossing survival functions alternativesp. 92
3.9.5 Bibliographic notesp. 97
3.10 Exercisesp. 98
3.11 Answersp. 102
Chapter 4 Homogeneity Tests for Related Populationsp. 105
4.1 Paired samplesp. 106
4.1.1 Datap. 106
4.1.2 Test statisticsp. 107
4.1.3 Asymptotic distribution of the test statisticp. 107
4.1.4 The testp. 116
4.2 Logrank-type tests for homogeneity of related k > 2 samplesp. 119
4.3 Homogeneity tests for related samples against crossing marginal survival functions alternativesp. 122
4.3.1 Bibliographic notesp. 124
4.4 Exercisesp. 125
4.5 Answersp. 126
Chapter 5 Goodness-of-fit for Regression Modelsp. 127
5.1 Goodness-of-fit for the semi-parametric Cox modelp. 127
5.1.1 The Cox modelp. 127
5.1.2 Alternatives to the Cox model based on expanded modelsp. 128
5.1.3 The data and the modified score statisticsp. 129
5.1.4 Asymptotic distribution of the modified score statisticp. 133
5.1.5 Testsp. 137
5.2 Chi-squared goodness-of-fit tests for parametric AFT modelsp. 142
5.2.1 Accelerated failure time modelp. 142
5.2.2 Parametric AFT modelp. 144
5.2.3 Datap. 144
5.2.4 Idea of test constructionp. 145
5.2.5 Asymptotic distribution of H n and Zp. 146
5.2.6 Test statisticsp. 151
5.3 Chi-squared test for the exponential AFT modelp. 153
5.4 Chi-squared tests for scale-shape APT modelsp. 159
5.4.1 Chi-squared test for the Weibull AFT modelp. 163
5.4.2 Chi-squared test for the lognormal AFT modelp. 166
5.4.3 Chi-squared test for the loglogistic AFT modelp. 169
5.5 Bibliographic notesp. 172
5.6 Exercisesp. 173
5.7 Answersp. 174
Appendicesp. 177
Appendix A Maximum Likelihood Method for Censored Samplesp. 179
A.1 ML estimators: right censoringp. 179
A.2 ML estimators: left truncationp. 181
A.3 ML estimators: left truncation and right censoringp. 182
A.4 Consistency and asymptotic normality of the ML estimatorsp. 186
A.5 Parametric ML estimation for survival regression modelsp. 187
Appendix B Notions from the Theory of Stochastic Processesp. 191
B.1 Stochastic processp. 191
B.2 Counting processp. 193
B.3 Martingale and local martingalep. 194
B.4 Stochastic integralp. 195
B.5 Predictable process and Doob-Meyer decompositionp. 197
B.6 Predictable variation and predictable covariationp. 198
B.7 Stochastic integrals with respect to martingalesp. 204
B.8 Central limit theorem for martingalesp. 207
Appendix C Semi-parametric Estimation using the Cox Modelp. 211
C.1 Partial likelihoodp. 211
C.2 Asymptotic properties of estimatorsp. 213
Bibliographyp. 225
Indexp. 231