# Survival analysis when the event might never occur

Suppose I select a bunch of individuals and send a message for each one. The individual may or may not reply. If they reply, I measure the time between my message and the reply. So I would like to do some survival analysis here.

The problem I have is that I'm confused by the fact that for some (actually the majority) of the messages sent, no reply will ever be received. The time-to-reply for those guys are actually undefined. I'm not sure if this is similar to right-censoring. In right-censoring, the event will eventually happen sometime, but my measurements were made before that time. In my case, no amount of waiting will lead to the event ever happening.

What I was intending to do was to derive some a model for:

$$P(\text{I will eventually receive a reply} | \text{time passed since message})$$

based on some model for $$P(\text{time-to-reply} | \text{there was a reply})$$.

The problem I have is that if I just naively use Bayes' Theorem here I'll eventually write something like:

$$P(reply | time) = \frac{P(time | reply) P(reply)} {P(time | reply) P(reply) + P(time | no\,reply) P(no\,reply)}$$

Where:

But $$P(time | no\,reply)$$ doesn't make any sense because the variable $$time$$ is not defined in the case of no reply.

Do someone know any literature on this that can help me sort this mess?

• AFAIK there is no restriction on the time-to-event being finite in survival analysis. That being said, this makes expected value calculations wonky, but that's why we often consider the median time. – Cam.Davidson.Pilon Nov 28 '18 at 18:38

## 1 Answer

You may want to look at cure models. The background there are studies on when a cancer returns after treatment, and fortunately for some it never returns (they are cured). Estimating the proportion of cured individuals and distinguishing them from censored cases is of obvious interest. The structure of your problem seems similar enough, for this set of models to be of use. A review can be found here.