Sampling From The Risk Set In A Cox Survival Analysis: A Large Sample Study (martingales, Estimator, Counting Processes).

The survival times of n individuals (tau)(,1), ...,(tau)(,n) are assumed to be independent with unknown distributions. Associated with each is a time dependent covariate z(,t), which is observed on time t, t (ELEM) 0,(tau)(,i) . For modeling the stochastic relationship between survival time and the covariate, Cox suggested the proportional hazards model, (lamda)(t;z(,t)) = (lamda)(,o)(t)exp((beta)(,o)z(,t)), t (GREATERTHEQ) 0, where (lamda)(,o)(t) and (beta)(,o) are parameters. This dissertation investigates a method for estimating (beta)(,o) based on right censored data, (tau)(,1), ...,(tau)(,n), which are either survival times or lower bounds for the survival times. Associated with them are vari- ables (delta)(,1), ...,(delta)(,n),(delta)(,i) = 0 if (tau)(,i) is a censor time ((tau)(,i) < (tau)(,i)), and (delta)(,i) = 1 if (tau)(,i) is a survival time ((tau)(,i) = (tau)(,i)). A method for estimating (beta)(,o) based on censored data was sugges- ted by Cox. At a survival time (tau)(,i) ((delta)(,i) = 1), the observed event is the death of individual i from among those at risk, R((tau)(,i)) = j: (tau)(,j) (GREATERTHEQ) (tau)(,i) . Using the Cox hazard, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) is the probability of such an event. The product of such factors over all survival times gives the Cox likelihood, and the value (beta)(,C) which maximizes it is the Cox regression estimate. In contrast, Thomas suggested that inference on (beta)(,o) be based on the function (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where K((tau)(,i)) is a random sample of fixed size from the set R((tau)(,i))/(i). In comparison to the Cox likelihood, the denominator of the terms of L(,T) contains the case i and a random sample of the risk set at (tau)(,i), instead of the entire risk set. The argument for L(,T) to be considered as a likelihood is similar to that given by Cox. The value (beta)(,T) which maximizes L(,T) estimates (beta)(,o). In this thesis, an investigation is made of the large sample proper- ties of (beta)(,T). It is assumed that followup takes place during some fixed time interval and that the number of deaths increases, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) It is not assumed that the size of the sampled risk set increases. Con- ditions are given which assure the weak consistency and asymptotic normality of (beta)(,T). The efficiency of (beta)(,T) relative to (beta)(,C) is investigated at the null hypothesis value (beta)(,o) = 0. The approach we take is an exten- sion of that taken by Naes and Andersen & Gill, namely modeling the survival experience of individual i as a counting process with intensity based on (lamda)(t;z(,t)('i)), t (GREATERTHEQ) 0. This approach relies heavily on basic martingale theory and U-statistic theory.Ph.D.Biological SciencesBiostatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/127904/2/8621273.pd