Testing adequacy of fit in the random right censoring framework

Abstract

Statistical inference of observed lifetimes in survival analysis, reliability theory, medical studies and nu- merous engineering fields are frequently of interest for researchers. In order to perform inference, it is important to test the hypothesis that observed times are realised from a specified class of distributions. Testing this hypothesis is complicated by the fact that random right censoring often occurs in the men- tioned fields of study; i.e., not all of the lifetimes of interest are observed. Goodness-of-fit testing has been studied in depth in the statistical literature; however, tests in the presence of random right censoring are relatively scarce, especially for more exible distributions such as the Weibull and gamma distributions. Therefore, the main goal of this thesis is to modify existing goodness-of-fit tests to accommodate ran- dom right censoring, specifically for three classes of distributions; the exponential, Weibull and gamma distributions. We compare the finite sample performance of these tests to existing tests, such as the classical Kolmogorov-Smirnov and Cramér-von Mises tests which have been modified to allow for random censoring. The majority of the newly modified tests are based on either the characteristic function or the Laplace transform. Some of the tests developed for the Weibull distribution for use with censored data are also new in the full sample case. We compare the finite sample performance of the tests mentioned above against a broad range of alterna- tives, in both the complete sample case as well as in the presence random right censoring. These classes of tests are based on Stein's method for the approximation of integrals. Furthermore, we modify existing tests based on Stein's method and the Laplace transform for the gamma distribution to allow for the presence of random right censoring. We compare the performance of these tests against a wide range of alternatives as well as three censoring distributions. In addition, we explore the computational assumptions relating to the Kaplan-Meier estimator which is commonly used in statistical literature. The effect of these assumptions are investigated in an extensive Monte Carlo simulation study along with a practical example. The practical example under consideration is used throughout the study; the data set contains 66 observations of which 14 are censored. A further contribution is found in new bootstrap algorithms for testing the adequacy of fit of survival models. The thesis concludes with a model based application of goodness-of-fit testing for the Cox proportional hazards model.