| Many of the popular nonparametric two-sample test statistics for censored survival data, such as the logrank, Gehan and Peto-Peto test statistics, have been shown to be special cases of a general statistic, differing only in the choice of weight function. Gill has formulated this more general statistic in terms of counting processes and martingales for which very powerful theoretical results apply. In this dissertation an urn-model motivated nonparametric trend statistic is shown to unify existing and generate new test statistics for the two-sample, k-sample trend and single continuous covariate problems. The already mentioned two-sample statistics, the Cox regression score test of (beta) = 0, the modified Kendall rank correlation, the O'Brien logit rank test, the Jonckheere k-sample test and some new statistics which are robust to outliers in the covariable space are all special cases of the general trend statistic. The general trend statistic can also be formulated in terms of counting processes and martingales. In this framework desirable statistical properties can be checked. It is shown under regularity conditions that the power of this statistic approaches one as the sample size increases when the alternative hypothesis of ordered hazards is true. Under regularity conditions this general statistic converges in distribution to that of a normal random process under a sequence of contiguous hazard alternatives that approaches the null hypothesis. Using this last result the Pitman asymptotic relative efficiencies between various versions of the trend statistic are made for relative and excess risk models. |
