how does phreg estimate cumulative hazard?

[pau13rown]

i'd like to see some explanation and equations for this but the sas manual isn't clear. I know lifetest uses the Nelson-Aalen estimate, but i'm unsure about phreg 

Regarding phreg and estimating S(t) the manual says: "This method exponentiates the negative empirical cumulative hazard function to compute the survival function estimates. ....(Allison 2010). The product limit method is the default".

thus it's an empirical estimate? but how are the covariates accounted for? I'd like to see some equations or a reference to persuade myself. The Allison reference is a sas book and doesn't give the detail 

there was a useful comment on sas communities where someone said: "You can only get the cumulated hazard function from phreg..... In the Cox-model the maximum-likelihood estimate of the cumulated hazard function is a step function ..." But without an estimate of the baseline hazard (which cox is not concerned with), how contrive the cumulative hazard for a set of covariates? thanks

[edit: just to be clear: i know how to obtain cumulative hazard using phreg, i use a baseline statement etc, i just dont know exactly what it's doing]

[Rick_SAS]

Just to be clear, the "manual" that you link to are course notes, not the documentation. Have you looked at the "Details" chapter of the PHREG documentation? The section "Survivor Function Estimates" gives the formulas for the various estimators, including the Breslow estimator (which uses the empirical cumulative hazard function) and the product-limit estimator.

[FreelanceReinh · Accepted solution]

Hello @pau13rown,

@Rick_SAS was faster ...

I confirm:

What you refer to as "the manual" are "Course Notes" which I have never seen before. The first place where I would search for those mathematical details is the "Details" section of the procedure in question, in this case the subsection Survivor Function Estimators of the PROC PHREG Details. (This is for the latest software version, SAS/STAT 15.1. You may want to use one of the older versions depending on your SAS/STAT version.)

This is much more detailed and heavy on formulas than the Course Notes, whose formula for S(t, X) can be found among others in subsection "Product-Limit Estimator." The sentence you quoted seems to refer to the "Fleming-Harrington Estimator" (the preceding subsection in the documentation). The default, however, is the "Breslow Estimator" (first subsection) -- contrary to what is stated in the Course Notes (saying that "[t]he product limit method is the default.")! Anyway, all flavors of cumulative hazard function estimators can be found here (see definitions of Lambda "hat" terms) as far as I see. I think last year I answered a question in this forum where I needed to manually calculate something applying some of the formulas on this page to input data -- and I was able to confirm the PROC PHREG output.

Edit: Here's last year's post which I mentioned: Re: Calculation of estimates using contrast statement, and the estimated survival for proc phreg 

[pau13rown]

excellent, thanks both. I originally looked at that documentation but checked the "overview" section for details. I must have been rushed and didn't see "details". cheers