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**Survival Model Inference Using Functions of Brownian Motion** ()

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;*H*_{0}:*β*(*t*) = 0 for all*t*against alternatives such as;*H*_{1}:∫*β*(*t*)d*F*(*t*) ≠ 0 or*H*_{1}:*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 [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.Share and Cite:

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.Conflicts of Interest

The authors declare no conflicts of interest.

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