Survival Model Inference Using Functions of Brownian Motion ()

John O’Quigley

Universite Pierre et Marie Curie, Paris, France.

**DOI: **10.4236/am.2012.36098
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Universite Pierre et Marie Curie, Paris, France.

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.

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O’Quigley, J. (2012) Survival Model Inference Using Functions of Brownian Motion. *Applied Mathematics*, **3**, 641-651. doi: 10.4236/am.2012.36098.

Conflicts of Interest

The authors declare no conflicts of interest.

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