Can you use event history analysis if your sample was selected on the dependent variable? I have a dataset that includes a random sample of individuals who are currently in a relationship in the United States. I also have data on the date that they met their partner and the date that they considered themselves in a romantic relationship. 
I was planning on doing an event history analysis to compare different types of relationships (IV) and the time it takes for them to transition from meeting --> relationship (DV). In other words, for each year they know each other, what is the risk that they will enter a relationship.
However, someone recently pointed out to me that since I do not have information on all of the people that they have met in their lives, I am in a sense sampling on the dependent variable. Since everyone in my sample is in a relationship, there is no censoring and everyone experiences the event (relationship entry) once. 
So, my question is basically this: Can I use some sort of event history analysis (Cox model, discrete time, etc.) OR am I really only able to do a t-test comparing the outcomes for each group?
 A: The answer to your title question is "No", you can't do an event history analysis. You can do t-tests or other comparisons of times to events.
An event study with romantic relationship as outcome would start the "clock" when a couple first "know each other". 
For any given time interval $h$ following the start, the risk that a romantic relationship first occurred in the interval would be estimated by:
$$ 
r_h = \frac{a_h}{n_h}
$$
wher $a_h$ is the number of events in the interval (enter romantic relationship) and $n_h$ is the number of couples who "know each other" in the interval but were not in a romantic relationship at the interval start.
However, your sample does not permit you to estimate $n_h$, because it excludes people who know potential relationship partners, but are not in a romantic relationship. Thus the observed $n_h$ are too small.
If the survey identified people in romantic relationship by their status at interview, there is also the potential for length-biased sampling, with over-representation of  longer-lasting romantic relationships.
