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