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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 $t<s$).

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?

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