[R] [Re: Does a survival probability(the probability not, experiencing an event) have to be non-increasing?

[Therneau, Terry M., Ph.D.]

On 03/20/2014 06:00 AM, r-help-request at r-project.org wrote:
> My question is related to a cox model with time-dependent variable.
> When I think about it more, I get a little confused about
> non-increasing assumption for survival probability for an individual.
> For example, for a time-dependent ,say x, assuming increasing x
> increases the risk of event. Assume,time t1 < t2.  If at x at t1<< x
> at t2, obviously, hazard at t1 will less than hazard at t2, assuming
> no other covariaates. But is it possible that s(t2|x at t2) > s(t1|x
> at t1), since at t2, an individual is at greater risk.  This is kind
> of confusing to me.
>
> Thanks for any helpful insights!

Time dependent covariates and survival curves are confusing to a lot of people.
The Cox model is a hazard model
	h(t, x) = h_0(t) exp(x beta)
which means it is a model of the moment-by-moment risk.  A time dependent model
replaces x with x(t) which is the moment-by-moment value of x.

After the model is fit, one can compute the
time dependent cumulative hazard as

       H(t,x) = \integral_0^t  h_0(s) exp(x(s) beta) ds
and the survival is S = exp(-H).

Since everthing inside the integral is positive H(t) has to be an increasing function of 
t, and thus S a decreasing one.  The key thing to note is that H or S depend on the entire 
covariate history for a subject.  If you have a subject whose value of "x" changes from 1 
to 2 at time 10, when computing their survival at time 15 you cannot just use a value of 
"2" all the way from 0 to 15 in the formula.

  Many Cox model programs (e.g.SAS) allow for time dependent covariates when computing the 
Cox fit, but then only allow for fixed covariates when computing a curve.  You can only do 
predictions for people whose covariates never change.  (For some diseases I work with such 
people do not exist, e.g. in PBC your bilirubin WILL rise with time.  So such a curve is 
useless).  This adds to the confusion.

Terry T.