# Survival Function vs Conditional Survival Function?

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:

• 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 $\frac{a}{b}.$ Feb 4, 2023 at 8:12
• Thank you for pointing this out! I will keep this in mind! Feb 4, 2023 at 17:06

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.