# Comparing Cox Proportional Hazards Models (variable selection)

I am using a cox proportional hazards model to run a survival analysis in r on a number of non-nested, distinct covariates such as Age, Blood Type, Cancer, etc:

 A, B, C, D, E


When I run the model on the omnibus null hypothesis:

surv ~ A + B + C + D


The effects of all of the covariates are insignificant because the number of subjects that have measurements for every covariate is relatively small. However, when I isolate single or other combinations of covariates in different cox models:

surv ~ A
surv ~ A + C
surv ~ B + D


I'm showing significant effects because the sample set is larger (i.e. the number of observations discarded by the model shrinks).

What I'm having difficulty understanding is how to do the following:

• Comparing the different cox models for the best fit, i.e. is surv ~ A + B + D a better model than surv ~ A + C ? Should I be comparing the likelihood, wald or logrank scores?
• Is it possible to run every possible combination of covariates to determine the best model? I have about 15 covariates.