# Predicting survival probability at a future time for proportional hazard model

I have a Weibull proportional hazard model with fitted parameters $$\lambda$$ and $$k$$ and a time dependent covariate $$x(t)$$ with fitted coefficient $$\beta$$, such that the hazard function is given by,

$$h(t) = \frac{k}{\lambda} (\frac{t}{\lambda})^{k-1} e^{\beta x(t)}$$

Now, suppose I have an observation that has survived until time $$s$$. I am interested in predicting the probability of survival until some future time $$t$$ given knowledge of $$x(t)$$ for all future $$t\geq s$$ (but not past $$t).

Normally this would be calculated as

$$Pr(T > t | t > s) = \frac{S(t)}{S(s)}$$

However, I cannot know $$S(t)$$ and $$S(s)$$ because these depend on values of the hazard function before $$s$$, which I don't know because that depends on past values of $$x(t)$$.

But, it seems like I should still be able to get $$Pr(T > t | t > s)$$ without ever having to calculate the survival function, since I know all future values of the hazard function.

Edit: I believe I have a solution:

$$\frac{S(t)}{S(s)} = \frac{e^{-\int_0^t h(\tau)d\tau}}{e^{-\int_0^s h(\tau)d\tau}} = e^{\int_0^s h(\tau)d\tau - \int_0^t h(\tau)d\tau} = e^{-\int_s^t h(\tau)d\tau}$$

Which only depends on values of the hazard function after $$s$$. Does this work?