Nonparametric Tests for Censored Data

Statistical analysis of data sets usually involves construction of a statistical model of the distribution of data within the available sample – and by extension the distribution of all data of the same category in the world. Statistical models are either parametric or non-parametric – this distinction is based on whether or not the model can be described in terms of a finite-dimensional parameter – and the models must be tested to ascertain whether or not they conform to the data, or are accurate.

This book addresses the testing of hypotheses in non-parametric models in the specific case of censored or truncated data samples. In particular, the applicability of standard tests to incomplete data sets is considered – for example the use of the chi-squared test for parametric accelerated failure time regression models, which are widely used in reliability, accelerated life testing, and survival analysis, is detailed.

Classical non-parametric tests (goodness-of-fit, homogeneity, randomness, independence) of censored data are considered, and explained. Tests featured include the chi-squared and modified chi-squared tests, rank and homogeneity tests, and most of the test results are proved, with real applications illustrated using examples. The incorrect use of many tests, and their application using commonly deployed statistical software is highlighted and discussed.

Theories and exercises are provided, making this book suitable for use in a one semester course in non-parametric statistics and tests.

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

2.4. Chi-squared tests for other families 51

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

3.10. Exercises 98

3.11. Answers 102

Chapter 4. Homogeneity Tests for Related Populations 105

4.1. Paired samples 106

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

4.4. 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

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

5.3. Chi-squared test for the exponential AFT model 153

5.4. Chi-squared tests for scale-shape AFT models 159

Bibliographic notes 172

5.6. Exercises 173

Answers 174

APPENDICES 177

Appendix A. 179

Appendix B. 191

Appendix C. 211

Bibliography 225

Index 231

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.

Mikhail S. Nikulin is a member of the Institute of Mathematics in Bordeaux, France.