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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, βt, can be any real function of time. When βt = 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:βt = 0 for all t against alternatives such as; H1:∫βtdFt ≠ 0 or H1:βt ≠ 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 1 and Xu and O’Quigley 2. 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.


Brownian Motion; Brownian Bridge; Cox Model; Integrated Brownian Motion; Kaplan-Meier Estimate; Non-Proportional Hazards; Reflected Brownian Motion; Time-Varying Effects; Weighted Score Equation

Cite this paper

J. O’Quigley -Survival Model Inference Using Functions of Brownian Motion,- Applied Mathematics, Vol. 3 No. 6, 2012, pp. 641-651. doi: 10.4236-am.2012.36098.

Author: John O’Quigley



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