Design: People get drug A and time to an event is measured. Not everyone has the event. Later, most of those people get drug B (those censored and uncensored) and again, time to event is measured. Again, not everyone has the event.
Question: Doctor wants to see if the time to the first event can predict the time to the second event. I'm fairly weak in my survival analysis training, and definitely never learned how one should go about answering this question.
I think one has to decide if time to reaction to drug A is censored or if a reaction to drug A just never happens for some subjects. If the latter is the case, you could use a linear predictor as the one James suggests with some functional form of the time to reaction for those who have a reaction and then a term for those who don't have a time. You can use this linear predictor in some parametric survival model, e.g. a Cox regression.
If all the subjects actually have a reaction to drug A but we just only observe a lower limit of this time for some patient, we have censored covariate. This article introduces a survival model with a censored covariate:
http://www.ncbi.nlm.nih.gov/pubmed/24319625
It might be useful to you.
To answer the question of whether or not the time to reaction on drug A can be used as a predictor in time to reaction to drug B, you can fit your model with and without the terms corresponding to reaction time on drug A and use e.g. cross validation to estimate expected prediction error for both models.
Here is a simple idea. You can say that you have one continuous predictor $X_1$ (time to the first event) and one categorical binary predictor $X_2$ that is equal to 1/0 depending on whether $X_1$ was censored or not. Then the model will be
$Y = X_1 + X_2 + X_1 * X_2$
Now, dealing with the response is not that straightforward, but you could bin it into a few ordered categories where the top category will be ${Y > t_2}$ where $t_2$ is the censoring threshold for the second event. The you can fit the model using ordinal regression.
