A family of tests for the presence of regression effect under proportional and non-proportional hazards models is described. The non-proportional hazards model, although not completely general, is very broad and includes a large number of possibilities. In the absence of restrictions, the regression coefficient, β(t), can be any real function of time. When β(t) = β, we recover the proportional hazards model which can then be taken as a special case of a non-proportional hazards model. We study tests of the null hypothesis; H0:β(t) = 0 for all t against alternatives such as; H1:∫β(t)dF(t) ≠ 0 or H1:β(t) ≠ 0 for some t. In contrast to now classical approaches based on partial likelihood and martingale theory, the development here is based on Brownian motion, Donsker’s theorem and theorems from O’Quigley  and Xu and O’Quigley . The usual partial likelihood score test arises as a special case. Large sample theory follows without special arguments, such as the martingale central limit theorem, and is relatively straightforward.
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