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Suppose we have a simple competing risks model and have a calculated $CIF_K$ for two events $\{k_1, k_2\} \in K$. By definition $CIF_{k_1}(t_i)$ gives us the probability that event $k_1$ has occurred on or before time $t_i$. Is it possible for us to recalculate $CIF_{K}$ given a known minimum survival time?

That is, if we consider a patient that has already lived to time $t_d$ without experiencing either event, is it possible to derive an updated probability $CIF'_{K}(t_i - t_d)$ given the additional information that no event $K$ occurred before time $t_d$ where $t_0$ < $t_d$ $\le$ $t_i$?

We would expect $CIF'_K(t_d)$ = $h_K(t_d)$, the instantaneous hazard rate at time $t_d$. But how does this lead us to $CIF'_K(t_d + 1)$?

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This is done using the rules of conditional probability. Letting $T$ denote the time until the event $K$ you can write the conditional distribution function of interest as:

$$\begin{align} CIF_K(t|t_d) &\equiv \mathbb{P}(T \leqslant t | T > t_d) \\[12pt] &= \frac{\mathbb{P}(t_d < T \leqslant t)}{\mathbb{P}(T > t_d)} \\[6pt] &= \frac{CIF_K(t) - CIF_K(t_d)}{1 - CIF_K(t_d)} \cdot \mathbb{I}(t > t_d \geqslant 0). \\[6pt] \end{align}$$

The resulting conditional density and conditional hazard functions are then given by:

$$\begin{align} f_K(t|t_d) &= \frac{d}{dt} CIF_K(t|t_d) \\[6pt] &= \frac{d}{dt} \frac{CIF_K(t) - CIF_K(t_d)}{1 - CIF_K(t_d)} \cdot \mathbb{I}(t > t_d \geqslant 0) \\[6pt] &= \frac{f_K(t)}{1 - CIF_K(t_d)} \cdot \mathbb{I}(t > t_d \geqslant 0), \\[12pt] h_K(t|t_d) &= \frac{f_K(t|t_d)}{1 - CIF_K(t|t_d)} \cdot \mathbb{I}(t > t_d \geqslant 0) \\[6pt] &= \bigg[ \frac{f_K(t)}{1 - CIF_K(t_d)} \bigg/ \frac{1 - CIF_K(t)}{1 - CIF_K(t_d)} \bigg] \cdot \mathbb{I}(t > t_d \geqslant 0) \\[6pt] &= \frac{f_K(t)}{1 - CIF_K(t)} \cdot \mathbb{I}(t > t_d \geqslant 0) \\[6pt] &= \frac{f_K(t)}{1 - CIF_K(t)} \cdot \mathbb{I}(t > t_d \geqslant 0) \\[12pt] &= h_K(t) \cdot \mathbb{I}(t > t_d \geqslant 0). \\[6pt] \end{align}$$

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