# Would you choose a joint frailty model or a multi-state model for survival analysis with recurrent events and competing risks?

I am going to analyse data with multiple recurring events and additional terminal event. The recurrent events are of the same kind, no hierarchy in them (like in the Prentice-Williams-Petersen). The terminal event, the "elimination from experiment" is single and precludes all over events, it is an absorbing state.

I found two methods of survival analysis able to handle it: joint frailty models and multi-state model. The first is parametric, the second seems a non-parametric one. I am trying to figure out which one may be better and when.

It's like Cox and AFT. The first is semi-parametric, doesn't have any distributional assumptions, and deals with proportional hazards, while the second is parametric, doesn't care of the proportionality in hazards and deals with time-to-event. They answer different questions, however, under special conditions (Weibull distribution), the two can return the same results differently expressed (scaled), so PH agrees with the TTE.

My question about frailty and multi-state models is of this nature - do they answer different questions? Can they be comparable under certain conditions? If they do the same - which one seems to be better and when?

If you had to choose, which one would you use and why in general? Let us assume both methods are available in your favourite statistical tool.

Balan TA, Putter H. A tutorial on frailty models. Statistical Methods in Medical Research. 2020;29(11):3424-3454. doi:10.1177/0962280220921889, https://journals.sagepub.com/doi/full/10.1177/0962280220921889

Rondeau V, Mathoulin-Pelissier S, Jacqmin-Gadda H, Brouste V, Soubeyran P. Joint frailty models for recurring events and death using maximum penalized likelihood estimation: application on cancer events. Biostatistics. 2007 Oct;8(4):708-21. doi: 10.1093/biostatistics/kxl043. Epub 2007 Jan 30. PMID: 17267392., https://academic.oup.com/biostatistics/article/8/4/708/299068

Rondeau, V., Marzroui, Y., & Gonzalez, J. (2012). frailtypack: An R Package for the Analysis of Correlated Survival Data with Frailty Models Using Penalized Likelihood Estimation or Parametrical Estimation. Journal of Statistical Software, 47(4), 1 - 28. doi:http://dx.doi.org/10.18637/jss.v047.i04

Charles‐Nelson, A, Katsahian, S, Schramm, C. How to analyze and interpret recurrent events data in the presence of a terminal event: An application on readmission after colorectal cancer surgery. Statistics in Medicine. 2019; 38: 3476– 3502. https://doi.org/10.1002/sim.8168

## 1 Answer

Briefly, the frailty and multi-state models answer different questions, but you have to answer both types of questions in your situation.

Even if you didn't have the terminal absorbing event, you would need to use a frailty or similar approach to deal with within-individual correlations of the repeated event (beyond what's accounted for by the covariates of your model). Standard modeling of events assumes independence among observations, which clearly isn't the case with the same individual being able to have the repeated event multiple times.

In the R survival package, a frailty term is one way to account for within-individual correlations, similar to a random-effects approach.* (I understand that it's now preferred to use the coxme package for that type of modeling.) Another way to account for within-individual correlations is a cluster term, similar to a generalized estimating equations (GEE) approach. That essentially fits the model as if events are independent, but then adjusts the coefficient covariance estimates to take the within-individual clustering into account.

As you have a terminal absorbing event in addition to the baseline state and the possible repeated event, you should also think about this as a "multi-state" situation. You have to allow for repeated transitions between the baseline state and the repeated event, in addition to the final transition to the absorbing state.

Technically, a full multi-state model might treat the repeated-event state as a state having an actual lifetime, modeling the return from that state to the baseline state (and potentially from that state directly to the terminal absorbing state). If the repeated-event state doesn't have an appreciable lifetime in your case, I think that there are ways to code data in a way that gets around that complication. With only a single transition into the terminal absorbing state, I think you have a choice whether or not to include the "frailty" estimated from the repeated event into your modeling of that terminal transition. That level of detail, however, is getting beyond my expertise.

*A random-effects/frailty term can be used both with a semi-parametric Cox model and a fully parametric AFT model. Cox models are parametric in terms of covariates; AFT models further parameterize the baseline hazard. Both can model times-to-events, albeit in different ways. In both Cox and AFT models, a random/frailty term just adds an additional parameter to estimate.

• Thank you very much for such a detailed answer. Regarding the tools, currently the most complete and mature is the frailtypack - jstatsoft.org/article/view/v047i04. It can fit all types including the joint frailty, which allows me to include both recurrent and terminal events (each constitutes own frailty, that's why they are here joint). The authors made it to handle exactly situations like mine. I have no experience with multi-state, there is a number of implementations in R. I don't fully get how to obtain survival curves from it. I hoped someone has some experience with them. Apr 28, 2021 at 17:41
• By the way, you perfectly explained the difference between cluster and frailty! I had hard times with getting that in survival. Now I see - frailty is for the conditional model, cluster - marginal, exactly like in mixed models (estimates conditional to random-effects) and GEE (estimates marginal, population-average). Thank you so much! Apr 28, 2021 at 17:45
• @MissDoberty a true multi-state approach would model transitions both to and from the intermediate state that can be entered multiple times. If you don't need to model the return from the intermediate state to the baseline state, then a multi-state model would seem to be overkill if frailtypack can do what you need. I"m struck, however, that frailtypack uses a cluster term to specify grouping in its formulas. I suspect that the cluster is interpreted internally as a frailty, but checking out that suspicion is a bit beyond my expertise.
– EdM
Apr 28, 2021 at 18:12