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Nonparametric Tests for Censored Data Book

Nonparametric Tests for Censored Data
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Nonparametric Tests for Censored Data, 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 example, Nonparametric Tests for Censored Data
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  • Nonparametric Tests for Censored Data
  • Written by author Vilijandas Bagdonavicus
  • Published by Wiley, John & Sons, Incorporated, 2/7/2013
  • 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 example
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Preface xi

Terms and Notation xv

Chapter 1 Censored and Truncated Data 1

1.1 Right-censored data 2

1.2 Left truncation 12

1.3 Left truncation and right censoring 14

1.4 Nelson-Aalen and Kaplan-Meier estimators 15

1.5 Bibliographic notes 17

Chapter 2 Chi-squared Tests 19

2.1 Chi-squared test for composite hypothesis 19

2.2 Chi-squared test for exponential distributions 31

2.3 Chi-squared tests for shape-scale distribution families 36

2.3.1 Chi-squared test for the Weibull distribution 39

2.3.2 Chi-squared tests for the loglogistic distribution 44

2.3.3 Chi-squared test for the lognormal distribution 46

2.4 Chi-squared tests for other families 51

2.4.1 Chi-squared test for the Gompertz distribution 53

2.4.2 Chi-squared test for distribution with hyperbolic hazard function 56

2.4.3 Bibliographic notes 59

2.5 Exercises 59

2.6 Answers 60

Chapter 3 Homogeneity Tests for Independent Populations 63

3.1 Data 64

3.2 Weighted logrank statistics 64

3.3 Logrank test statistics as weighted sums of differences between observed and expected number of failures 66

3.4 Examples of weights 67

3.5 Weighted logrank statistics as modified score statistics 69

3.6 The first two moments of weighted logrank statistics 71

3.7 Asymptotic properties of weighted logrank statistics 73

3.8 Weighted logrank tests 80

3.9 Homogeneity testing when alternatives are crossings of survival functions 85

3.9.1 Alternatives 86

3.9.2 Modified score statistics 88

3.9.3 Limit distribution of the modified score statistics 91

3.9.4 Homogeneity tests against crossing survival functions alternatives 92

3.9.5 Bibliographic notes 97

3.10 Exercises 98

3.11 Answers 102

Chapter 4 Homogeneity Tests for Related Populations 105

4.1 Paired samples 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 The test 116

4.2 Logrank-type tests for homogeneity of related k > 2 samples 119

4.3 Homogeneity tests for related samples against crossing marginal survival functions alternatives 122

4.3.1 Bibliographic notes 124

4.4 Exercises 125

4.5 Answers 126

Chapter 5 Goodness-of-fit for Regression Models 127

5.1 Goodness-of-fit for the semi-parametric Cox model 127

5.1.1 The Cox model 127

5.1.2 Alternatives to the Cox model based on expanded models 128

5.1.3 The data and the modified score statistics 129

5.1.4 Asymptotic distribution of the modified score statistic 133

5.1.5 Tests 137

5.2 Chi-squared goodness-of-fit tests for parametric AFT models 142

5.2.1 Accelerated failure time model 142

5.2.2 Parametric AFT model 144

5.2.3 Data 144

5.2.4 Idea of test construction 145

5.2.5 Asymptotic distribution of Hn and Z 146

5.2.6 Test statistics 151

5.3 Chi-squared test for the exponential AFT model 153

5.4 Chi-squared tests for scale-shape APT models 159

5.4.1 Chi-squared test for the Weibull AFT model 163

5.4.2 Chi-squared test for the lognormal AFT model 166

5.4.3 Chi-squared test for the loglogistic AFT model 169

5.5 Bibliographic notes 172

5.6 Exercises 173

5.7 Answers 174

Appendices 177

Appendix A Maximum Likelihood Method for Censored Samples 179

A.1 ML estimators: right censoring 179

A.2 ML estimators: left truncation 181

A.3 ML estimators: left truncation and right censoring 182

A.4 Consistency and asymptotic normality of the ML estimators 186

A.5 Parametric ML estimation for survival regression models 187

Appendix B Notions from the Theory of Stochastic Processes 191

B.1 Stochastic process 191

B.2 Counting process 193

B.3 Martingale and local martingale 194

B.4 Stochastic integral 195

B.5 Predictable process and Doob-Meyer decomposition 197

B.6 Predictable variation and predictable covariation 198

B.7 Stochastic integrals with respect to martingales 204

B.8 Central limit theorem for martingales 207

Appendix C Semi-parametric Estimation using the Cox Model 211

C.1 Partial likelihood 211

C.2 Asymptotic properties of estimators 213

Bibliography 225

Index 231


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Nonparametric Tests for Censored Data, 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 example, Nonparametric Tests for Censored Data

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Nonparametric Tests for Censored Data, 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 example, Nonparametric Tests for Censored Data

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Nonparametric Tests for Censored Data, 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 example, Nonparametric Tests for Censored Data

Nonparametric Tests for Censored Data

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