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?
