Time‐varying covariate vs. “matching on time” to handle second event

data analysis
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I’m analyzing outcomes in a cohort of lung transplant recipients (time 0 = lung transplant). Some patients later received a transplant of another organ. I proposed treating the second transplant as a time‐varying covariate to estimate its effect on mortality.

A colleague instead divided patients into two groups — those who ever received a second transplant vs. those who didn’t — and “matched” them on time since the first transplant before comparing survival. He argues this is equivalent.

My concern is that this approach still introduces immortal time bias, since those who received a second transplant must have survived long enough to get it. Am I correct that only a time‐dependent model properly accounts for this? Any citations that explain the difference would be greatly appreciated!

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My initial reaction would be to consider using a multi-state model on the full cohort, since you can have multiple, potentially competing, endpoints, including the need to censor patients at apropos times.

Terry Therneau et al have a great vignette on this approach, using R via their survival package, on CRAN. The main package link is:

and the specific vignette of interest is here:

I am presuming that perhaps your available cohort does not include patients that have been diagnosed and perhaps are on a waiting list for lung transplantation, but have not yet received the transplant? Starting with time 0 at transplant can be another source of immortal time bias to consider, since patients would have to survive long enough after a relevant diagnosis and then being placed on the waiting to list, to get their initial lung transplant.

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I think Marc’s approach is probably the best. A couple of random observations:

  • Time-dependent covariates can work well (see the class re-analysis of the Stanford heart transplant data) but you need to have the patient re-characterized (updated baseline variables) just before the moment of transplant.
  • Landmark analysis can sometimes be useful where you define a strict qualification time and start the follow-up clock over. That seems less possible in your setting.
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thank you both for your expertise and input! Marc’s idea is actually a very exciting one given the nature of the data (just will require a large amount of data wrangling). We do have their listing date both for lung and kidney.

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Genuinely curious what the underlying scientific question might be. Maybe a failure of imagination on my part, but this sounds like a narrowly technical question dealing with statistical abstractions — statistical ‘navel-gazing’, as it were.

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Lol - A fair question! Lots of folks who receive a lung transplant have trouble with kidneys due to the immunosuppressant drugs, and some later require a kidney transplant. We are trying to estimate the effect that having a kidney transplant has on their survival. However…the estimates that the analysis came up with are (in my opinion) completely unrealistic (a 5 year survival rate of 97% - whereas the median survival time for a lung transplant these days is only about 60%) which made me wonder if we’re looking at a statistical artifact.

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I had wondered about this when I first saw the post. If it’s the case that lung transplant has a causal effect on need for kidney transplant then I think you have some additional issues related to collider stratification so I think a vanilla MSM or time-varying approach would still be biased.

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Interesting - that’s a new term to me. Do you have any suggested methods/references I should look into?

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Chapters 8 and 20 here would be the most relevent for what might be happening. You’ll have a better sense of the overall design and DGP than I would from my quick skim so it may or may not actually be a problem.

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Mainly tacrolimus? Confounding by indication will be rife here. There’s confounding by the need for the kidney transplant (inadequate monitoring of tac levels? low adherence to immunosuppressant regimen? other suboptimalities in care?) and also by conditioning of the 2nd transplant on the patient’s clinical status and clinically assessed propensity to benefit. It does seem likely you would end up with an artifactual/spurious inference that wouldn’t be useful (might even be harmful!) to clinical decision-making.

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Very true - given the rarity, we’re likely going to be limited to UNOS data to get any sort of adequate sample size to even examine the question which will severely limit the type of data that are available. Will be important to highlight the (many) limitations of the study if we get to that point. (I’m not a clinician so I won’t speak to the causes, but yes - my guess is it’s the tacro driving the process.)

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If you are going to be using UNOS STAR datasets, you would also have access to data for patients that are on waiting lists for lung (or combined heart/lung) transplants and not yet transplanted, as opposed to just those who have had a lung transplant as a starting point.

That would ostensibly give you more flexibility in how you structure your data, how you define Time 0, and the various states that a patient might be in over time.

It would also likely place more emphasis on your inclusion/exclusion criteria for your analyses, as I located some more complex scenarios during a shallow dive today, including this one at Frank’s institution in 2023:

VUMC performs novel reoperative lung and kidney transplant

and which also references simultaneous lung/kidney transplants, which are less common, but appear to be increasing in prevalence, for example, from 2022:

Simultaneous Lung-Kidney Transplantation in the United States

I also found this one on combined transplants from 2015, which indicates being the first such presentation:

Combined Lung-Kidney Transplantation: An Analysis of the UNOS/OPTN Database

More recently (August 2025) was this paper, which references a 2023 change in UNOS allocation criteria for “rescue” kidney transplants for lung transplant recipients:

Rescue kidneys in lung transplantation: A retrospective analysis of recipients who might have benefitted from a kidney safety net

which therefore might suggest a potential impact on your analyses with respect to time periods before and after that change in protocol.

Thus, prior history, including prior organ transplants, pre-existing chronic kidney disease, the need for dialysis pre-transplant, and other risk factors would be relevant to the propensity for later kidney transplant, as opposed to just more narrowly the lung transplant itself and peri-transplant medical/immunosuppressive protocols.

As you reference, sample size is going to be important here in terms of the potential complexity of the cohort that you may be able to reasonably model in a stable manner.

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I am not sure you are accurately relating what your colleague did - you cannot match on time since first transplant since the never receivers of a second transplant have no time to match on. What he must have done is what is called PTDM (prescription time distribution matching) which means randomly allocate immortal time in the ever receivers to the never receivers (with replacement) and origin will be lung transplant but entry to risk set is at this allocated time. This will remove immortal time bias, unless a majority of the time allocation happens after exit of the participant from the study.

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Apologies for the earlier lack of clarity and thank you for following up! The cohort includes all lung transplant (LTx) recipients who eventually required dialysis. Some of these patients later received a kidney transplant (KTx). Those who received a KTx were matched to those who did not, based on time from LTx to dialysis start, so both groups had a similar exposure period before dialysis.

My concern is that this design may introduce bias by conditioning on survival until dialysis—excluding patients who died earlier—and that the groups may therefore not be truly comparable at the time of matching.

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If they all had dialysis then you have already conditioned on dialysis and matching on time to dialysis is not relevant here. Some got KTx and others did not but KTx is not the exposure of interest (LTx is the exposure of interest) so all that needs to be done is adjust for KTx in a Cox regression model assuming Ktx is prognostic for mortality and irrespective of if it is a confounder or not - nothing else is of concern here. Immortal time will only be an issue if KTx was the exposure of interest,