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Consider the Survival Function:

$$S(t) = P(T_1 > t) = 1 - F(t)$$

Where:

  • $S(t)$ is the survival function, which gives the probability of surviving past time $t$.

  • $T_1$ is the random variable representing the time of the first event (such as death or failure).

  • $P(T_1 > t)$ is the probability that the event has not occurred by time $t$.

  • $F(t)$ is the cumulative distribution function (CDF) of the time to event, which gives the probability that the event will occur by time $t$.

As I understand, the Survival Function describes "the probability that the individual is still alive past time t".

Initially, I had thought that the Survival Function was by definition "conditional". For example - to be alive at time "t" is implicitly conditional that I have survived all times between time = 0 and time = t -1.

However, then I found out there is something separate called the Conditional Survival Function. Apparently, the Conditional Survival Function is described as follows:

$$CS(t|s) = P(T > t + s | T > s) = \frac{S(t+s)}{S(s)}$$

Where:

  • $CS(t|s)$ is the conditional survival function, which gives the probability of surviving past time $t + s$ given that the event has not occurred by time $s$.

  • $T$ is the random variable representing the time of the event.

  • $S(t)$ is the survival function, which gives the probability of surviving past time $t$.

  • $s$ and $t$ are positive time intervals.

In other words, the conditional survival function provides the survival probability after time $t + s$, given that the individual has survived to time $s$.

This brings me to my question:

  • Why is so much attention given to the Conditional Survival Function? The Conditional Survival Function is created by dividing two (Ordinary) Survival Functions. Could I not just divide both of them and thereby estimate the Conditional Survival Function?
  • And in what kinds of situations/applications would we be interested in estimating the Conditional Survival Function vs the (Ordinary) Survival Function?

References:

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    $\begingroup$ Just a passing by comment: if $\hat a$ is a "good" estimator of $a$ and $\hat b$ is a "good" estimator of $b, $ it doesn't necessarily mean $\frac{\hat a}{\hat b}$ is a "good" estimator of $\frac{a}{b}.$ $\endgroup$ Feb 4, 2023 at 8:12
  • $\begingroup$ Thank you for pointing this out! I will keep this in mind! $\endgroup$
    – stats_noob
    Feb 4, 2023 at 17:06

1 Answer 1

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There are a couple of different types of "conditional survival functions" that might be getting confused here. One is conditional upon covariate values. That's what's discussed in this link that you provided. That's different from the future survival conditional upon survival to some time, as discussed in your other links.

With respect to your first question, even if your formula is correct for defining future survival conditional on survival to time $s$, that doesn't directly provide an estimate of the error in the conditional survival. Point estimates without error estimates aren't very useful. That requires additional work, as in the linked references.

With respect to your second question, people of my age are typically more interested in the time we have left than in our survival functions from birth. Feel free to generalize to similar situations, like a company that wants to know how much longer customers will stay active after 1 year of activity.

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