Computational Details Overview
Cox's proportional hazards model is a distribution free model in which predictors are related to lifetime multiplicatively.
The form of the Cox proportional hazards model is as follows:
where
is the baseline hazard and
is the vector of regression coefficients. This model does not impose any distributional assumption on the baseline hazard. It is referred to as proportional because the ratio of hazard rates of two individuals is constant and not dependent on time.
For estimation of the regression coefficients, Cox devised the following likelihood. Suppose that the data consists of D unique failure times arranged in the order
and let Ri represent the risk set at time ti, that is, the individuals still at risk just prior to time ti. The conditional probability of that the ith individual in Ri dies is:
The likelihood that Cox proposed is based on the product of these quantities.
