Goodness-of-fit testing in survival models
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The statistical analysis of lifetime data is an important topic in many areas, such as the biomedical and engineering sciences. The main objective in many of these studies is to examine the relationship between lifetimes and their associated covariates by means of a survival model. In these models it is sometimes appropriate to make specific distributional assumptions about the shape of the survival function. Therefore, one of the main objectives of this thesis is to evaluate a variety of tests for testing the goodness-of-fit (GOF) of parametric survival models (in particular the proportional hazards and accelerated failure time models) and compare their performance to one another. The approach adopted in this thesis, to conduct GOF, involves exploiting the property that, if the survival model is correctly specified, the errors follow a unit exponential distribution. Aspects of the models under consideration include the shape of the survival function and time-invariance of the covariates. In addition, interest lies in testing GOF of the proposed model against a more general model. These aspects are investigated by first evaluating the GOF of survival models in the complete sample case by considering various tests for exponentiality. The results of an extensive Monte Carlo study show that tests based on the characteristic function and those based on the Laplace transform have the best overall power performance. However, in lifetime studies, it is not always possible to obtain complete information on lifetimes for all the subjects or units under study. This kind of data is called censored and a special case of this is Type-II right censored data which commonly arises in the testing of, for example, the lifetimes of equipment. The GOF testing procedures considered for Type-II right censored data can be divided into two categories: (i) an approach involving transforming the data to a complete sample, and then, on the newly constructed complete sample, employ any of the numerous tests for exponentiality designed for complete samples, and (ii) an approach where the test statistic is modified to accommodate Type-II right censoring. It was found, after performing a thorough Monte Carlo study, that both of these two approaches are useful for conducting GOF testing for exponentiality if the censoring proportion in a data set is lower than 30%. However, for higher censoring proportions, we would recommend using the approach that first transforms the sample to a complete sample. Additioally, it was found that tests based on the characteristic function or the Laplace transform generally outperformed the other tests for many of the settings considered. Lastly, we present a new way of modifying three well-known tests for exponentiality based on either the characteristic function or the Laplace transform to accommodate Type-II right censored data. In a limited Monte Carlo study we found that these newly modified test statistics outperformed both the existing modified tests as well as the tests based on transformed data.