Cumulative empirical processes for survival models

Following the martingale approach of Khmaladze (1981), we obtain tests for survival models by transforming an appropriate empirical process to one with a known large, sample distribution. In order to carry out this prescription, we appeal to the classic Donsker´s theorem, together with a theorem of O´Quigley (2003). This idea has been recently explored in the context of goodness of fit. The goodness of fit problem can be viewed as a test of the form H₀:β(t)=βˆ and this form can be simply generalized in order to obtain tests of survival effect, such tests then taking the form H₀:β(t)=β₀. We are particularly interested in constructing tests, which relate to Brownian motion. One of these tests coincides exactly with the usual partial likelihood score test. But we have broad generality and other forms of the test, an example being integrated Brownian motion, can achieve very greater power when faced with alternatives of a nonproportional hazards nature, such as declining effects or even crossing hazards. We focus our attention on the univariate case for the purposes of transparency of exposition. Extension to the multivariate case is mostly obvious and straightforward. Some examples are considered.