I have heard that the Cox model can be extended to handle recurrent events and competing risks. But is this possible to analyse data, where both occur? For example two kinds of fractures that can occur multiple times - I want to model the time-to-fracture in general and want to use only the Cox model, without any modern additions, like frailty models. Could you maybe recommend any books or examples on that topic?
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1$\begingroup$ As for modeling recurring events, I recommend Modeling Survival Data: Extending the Cox Model by Terry M. Therneau and Patricia M. Grambsch and this book for competing risks, which is not so rigorous but there are many insightful examples in R. $\endgroup$– treskovCommented Sep 21, 2020 at 15:10
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To start, the R survival package vignettes contain a good amount of information on how to deal with competing risks and recurrent events until you get more detailed references by Therneau and Grambsch or Pintile suggested in a comment. The package includes help with this type of analysis both in the main vignette and in one specifically on Multi-state models and competing risks.
The basic approach (in R) is to use the Surv(startTime, stopTime, status) syntax for the outcome with a multi-category status indicator, one for each event type and one for censoring, instead of the simple censored/event status dichotomy for single event types. You break up the data into separate rows for each individual, identified by that individual, so that each event has an associated start time and stop time. In principle any set of multiple states with recurrences can be accommodated, although with more complexity there are more chances to mis-specify some aspects of the model.
Your wish to avoid "modern additions, like frailty models" cannot be granted, however. If you have recurrent events, then events occurring within the same individual are necessarily correlated and the assumption of independence among events no longer holds. At a minimum, you need to adjust the model accordingly with frailty or cluster terms, or to use a more extended mixed-effects approach as provided in the coxme package. You also might need to consider whether having an event of a certain type makes an individual more or less likely to have the same or another event type later, even with the same covariate values.
