Nonparametric Methods for the Exploration and Analysis of Survival Data

Traditional survival analysis methods are primarily those of Kaplan-Meier curves, the log-rank test and Cox's Proportional Hazards model. Only the first of these techniques is routinely used to provide a graphical representation of the data. The idea of a regression curve is used to describe the relationship between survival time and a continuous covariate is rarely considered. This is presumably due to the complexity of estimating a mean when there are censored observations. Median survival times are often quoted for a set of analysed data and extending this to a median curve across a continuous covariate would provide an intuitive description of the effect of this covariate on survival time. In this thesis, a combination of two nonparametric procedures using kernel estimates provides a doubly-smooth quantile estimator for the pth (0 ≤ p ≤ 1) quantile of survival time given a covariate. Similar percentile curves can be derived for both Cox's model and a smooth proportional hazards models. While these allow a more explicit form of the curve to be written down, the doubly-smooth estimator has no assumptions about the baseline hazard rate or the shape of the covariate effect and is therefore more flexible. Assessing and comparing the fits of each of these approaches can be achieved by the calculation of a form of likelihood statistic. Due to the complexity of the mathematical properties of the nonparametric method, testing procedures are carried out using resampling techniques such as bootstrapping and permutation tests. One extension of this methodology is to consider the additional effect of a binary covariate on survival time. This is analogous to an analysis of covariance in a Normal regression model and interest lies in how to characterise the behaviour of the curves from each of the two levels. As before, percentile curves can be obtained and appropriate testing procedures applied. An algorithm based on serum creatinine curves was developed to detect graft deterioration in kidney transplant patients. These diagnoses had previously only been made by the subjective, experienced opinions of physicians, whereas the algorithm provided an explicit rule for detecting these cases. Survival times were also obtained and these data were analysed using standard techniques. Percentile curves were used to provide more information where the interpretation of a co- variate effect was difficult. In the absence of censored data, a different form of nonparametric smoothing was considered to assess the development of children suffering from cerebral palsy. Percentile curves were obtained using cubic splines to describe the growth of children with this condition and to compare them with those of normal children. This not only vindicated the belief that children suffering from cerebral palsy tend to be smaller and lighter than normal children of similar age but also provided standard curves useful in monitoring the development of these children