Nonparametric Tests for Censored Data

, by Bagdonavicius, Vilijandas; Kruopis, Julius; Nikulin, Mikhail S.

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ISBN: 9781848212893 | 1848212895
Cover: Hardcover
Copyright: 1/18/2011

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This book concerns testing hypotheses in non-parametric models. Generalizations of many non-parametric tests to the case of censored and truncated data are considered. Most of the test results are proved and real applications are illustrated using examples. Theories and exercises are provided. The incorrect use of many tests applying most statistical software is highlighted and discussed.

Preface	p. xi
Terms and Notation	p. xv
Censored and Truncated Data	p. 1
Right-censored data	p. 2
Left truncation	p. 12
Left truncation and right censoring	p. 14
Nelson-Aalen and Kaplan-Meier estimators	p. 15
Bibliographic notes	p. 17
Chi-squared Tests	p. 19
Chi-squared test for composite hypothesis	p. 19
Chi-squared test for exponential distributions	p. 31
Chi-squared tests for shape-scale distribution families	p. 36
Chi-squared test for the Weibull distribution	p. 39
Chi-squared tests for the loglogistic distribution	p. 44
Chi-squared test for the lognormal distribution	p. 46
Chi-squared tests for other families	p. 51
Chi-squared test for the Gompertz distribution	p. 53
Chi-squared test for distribution with hyperbolic hazard function	p. 56
Bibliographic notes	p. 59
Exercises	p. 59
Answers	p. 60
Homogeneity Tests for Independent Populations	p. 63
Data	p. 64
Weighted logrank statistics	p. 64
Logrank test statistics as weighted sums of differences between observed and expected number of failures	p. 66
Examples of weights	p. 67
Weighted logrank statistics as modified score statistics	p. 69
The first two moments of weighted logrank statistics	p. 71
Asymptotic properties of weighted logrank statistics	p. 73
Weighted logrank tests	p. 80
Homogeneity testing when alternatives are crossings of survival functions	p. 85
Alternatives	p. 86
Modified score statistics	p. 88
Limit distribution of the modified score statistics	p. 91
Homogeneity tests against crossing survival functions alternatives	p. 92
Bibliographic notes	p. 97
Exercises	p. 98
Answers	p. 102
Homogeneity Tests for Related Populations	p. 105
Paired samples	p. 106
Data	p. 106
Test statistics	p. 107
Asymptotic distribution of the test statistic	p. 107
The test	p. 116
Logrank-type tests for homogeneity of related k > 2 samples	p. 119
Homogeneity tests for related samples against crossing marginal survival functions alternatives	p. 122
Bibliographic notes	p. 124
Exercises	p. 125
Answers	p. 126
Goodness-of-fit for Regression Models	p. 127
Goodness-of-fit for the semi-parametric Cox model	p. 127
The Cox model	p. 127
Alternatives to the Cox model based on expanded models	p. 128
The data and the modified score statistics	p. 129
Asymptotic distribution of the modified score statistic	p. 133
Tests	p. 137
Chi-squared goodness-of-fit tests for parametric AFT models	p. 142
Accelerated failure time model	p. 142
Parametric AFT model	p. 144
Data	p. 144
Idea of test construction	p. 145
Asymptotic distribution of H_n and Z	p. 146
Test statistics	p. 151
Chi-squared test for the exponential AFT model	p. 153
Chi-squared tests for scale-shape APT models	p. 159
Chi-squared test for the Weibull AFT model	p. 163
Chi-squared test for the lognormal AFT model	p. 166
Chi-squared test for the loglogistic AFT model	p. 169
Bibliographic notes	p. 172
Exercises	p. 173
Answers	p. 174
Appendices	p. 177
Maximum Likelihood Method for Censored Samples	p. 179
ML estimators: right censoring	p. 179
ML estimators: left truncation	p. 181
ML estimators: left truncation and right censoring	p. 182
Consistency and asymptotic normality of the ML estimators	p. 186
Parametric ML estimation for survival regression models	p. 187
Notions from the Theory of Stochastic Processes	p. 191
Stochastic process	p. 191
Counting process	p. 193
Martingale and local martingale	p. 194
Stochastic integral	p. 195
Predictable process and Doob-Meyer decomposition	p. 197
Predictable variation and predictable covariation	p. 198
Stochastic integrals with respect to martingales	p. 204
Central limit theorem for martingales	p. 207
Semi-parametric Estimation using the Cox Model	p. 211
Partial likelihood	p. 211
Asymptotic properties of estimators	p. 213
Bibliography	p. 225
Index	p. 231
Table of Contents provided by Ingram. All Rights Reserved.

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Nonparametric Tests for Censored Data

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