The PHSELECT Procedure

The Cox proportional hazards regression model is a semiparametric model that assumes a parametric form for the effects of the explanatory variables, but it allows an unspecified form for the underlying survivor function. The survival time of each member of a population is assumed to follow its own hazard function, , expressed as

where is an arbitrary and unspecified baseline hazard function, is the vector of explanatory variables for the jth individual, and is the vector of unknown regression parameters that is associated with the explanatory variables.

The population under study can consist of a number of subpopulations, each of which has its own baseline hazard function. A stratified analysis is needed to adjust for such subpopulation differences. Under the stratified model, the hazard function for the jth individual in the ith stratum is expressed as

where is the baseline hazard function for the ith stratum and is the vector of explanatory variables for the individual. The regression coefficients are assumed to be the same for all individuals across all strata.

To estimate , Cox (1972, 1975) introduced the partial likelihood function, which eliminates the unknown baseline hazard functions and accounts for censored survival times.