Don't really understand your equations but.

If X is the (random variable of) time of event, then the survival

 S(x) = 1 -  Pr( X < x )

has to be non increasing since:

If t1 < t2. then

S(t1) - S(t2) = Pr(X < t2) - Pr( X < t1 ) = Pr( t1 < X < t2) > 0

which means that

S(t1) > S(t2).

But you must not confuse this with the conditional probability of survival
given the age.


On Wed, Mar 19, 2014 at 10:41 AM, Zhiyuan Sun <sam.d.sun@gmail.com> 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!
>
> Zhiyuan
>
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