# 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