Suppose we have a population $P$ that we are interested in. We want to track the onset of particular conditions (i.e. $A$, $B$ and $C$). Suppose that as blood pressure drops, these events occur. So $C$ is associated with a higher level drop of blood pressure. How would we use survival analysis to analyze this situation?
Solved – Survival Analysis with Multiple Events
competing-riskssurvival
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You've come across an issue that can occur with Cox-PH models (actually, just about all survival regression models). That is, if no events occur in one group, then the estimated effect of that group will be $-\infty$. This is very similar to the issue in general linear models with, say, the binomial family, when you have one group with all 0's or all 1's.
If you are just interested in comparing groups (without adjusting for other covariates), this can be still be done with log-rank statistics: see the function survdiff in R's survival package.
My approach would be as follows: use the Cox-PH model on all the groups that observed at least one event. Then, for the group that had no events, use the log-rank statistic to compare with some baseline group of interest. Make note in the report that the log-rank statistic was used because the Cox-PH model resulted in degenerate estimate in the group with no events.
This might be handled by standard recurrent-event analysis allowing for time-dependent covariates (in this case, in particular whether the drug was being provided during each time interval). The general principles and illustrations for analysis in SAS, Stata and R are nicely presented in a recent paper by Amorim and Cai.
In R, the simplest approach would be to format the data with columns for (patientID, startTime, stopTime, event, drugStatus, otherCovariateValues...) with a separate row starting with each change of drug (or of other time-dependent covariates) or ending with each event. See this vignette for ways to format data this way. Including an ID for each patient allows correction for the correlations of results within individuals, e.g. by including a term of cluster(patientID) as a predictor in the model. This approach would be appropriate if the occurrence of an event is simply a function of the covariates and drug status at the time of the event, and if subsequent recurrent events were independent of prior events (given the present values of drug/covariates).
There isn't a problem here with events happening absent the drug. Presumably your interest is in whether the drug affects the probability/timing of an event occurring, and this analysis would address this issue directly provided that the above assumptions hold.
It is possible to model more complicated situations in which prior events themselves influence the probability of subsequent events, cumulative exposure to drug rather than instantaneous exposure is related to events, and so on. The references linked above are useful starting points for exploring these possibilities.
Best Answer
I was going to suggest competing risk models. There has been a recent revival of this subject because of its importance in medicine. Here is a very detailed article from Statistics in Medicne 2005 that is a nice tutorial with many references and software tools in R and SAS. I went to a conference where Jason Fine presented his work on competing risk and the Fine-Gray model. I think I mentioned it and Crowder's book in an answer to a previous question on CV. You can find Fine's two major papers and much more in the tutorial style article I gave in the link above.
It appears that Julien's link is a similar one to mine, also a tutorial covering competing risk but his has more references because it also includes multistate models but does not include all the references and software in my link. Oddly they are both tutorials for Statistics in Medicine.
Note: the link to Statistics in Medicine paper is not working any more, but the paper got cached on archive.org and can be found there. The paper is referenced as: Gichangi, A. & Vach, W. (2005). The analysis of competing risks data: A guided tour. Statistics in Medicine, but it does not seem to have appeared in Statistics in Medicine journal yet.