%0 Journal Article
%T Survival Model Inference Using Functions of Brownian Motion
%A John O¡¯Quigley
%J Applied Mathematics
%P 641-651
%@ 2152-7393
%D 2012
%I Scientific Research Publishing
%R 10.4236/am.2012.36098
%X 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, <i>¦Â</i>(<i>t</i>), can be any real function of time. When <i>¦Â</i>(<i>t</i>) = <i>¦Â</i>, 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; <i>H</i><sub>0</sub>:<i>¦Â</i>(<i>t</i>) = 0 for all <i>t</i> against alternatives such as; <i>H</i><sub>1</sub>:¡Ò<i>¦Â</i>(<i>t</i>)d<i>F</i>(<i>t</i>) ¡Ù 0 or <i>H</i><sub>1</sub>:<i>¦Â</i>(<i>t</i>) ¡Ù 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.
%K Brownian Motion
%K Brownian Bridge
%K Cox Model
%K Integrated Brownian Motion
%K Kaplan-Meier Estimate
%K Non-Proportional Hazards
%K Reflected Brownian Motion
%K Time-Varying Effects
%K Weighted Score Equation
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=20359