I am trying to determine mortality rates for untreated patients from an observational dataset where treatment has occurred (thus blocking the possibility of further untreated mortality). You can't just model this with a CoxPH model and consider treatment as a censoring event due to the coxph prohibition on informative censoring. The situation is also complicated by the fact that the covariates that drive mortality also drive probability of treatment. I've had success using inverse probability censor weighting, but now I'd like to try a Bayesian approach. I can successfully model in pymc observed survival and observed rates of treatment using Poisson's regression. I'm struggling to figure out how to tie those models together to figure out the unobserved counterfactual "what would mortality have been if the patients weren't treated". Any suggestions?

  • $\begingroup$ Sounds like you're trying to do causal inference. Have you drawn a DAG for your causal model? $\endgroup$ Commented Apr 9, 2022 at 23:40
  • $\begingroup$ Yes, I'm exactly doing causal inference. The DAG looks like a node of patient covariates pointing to a node of treatments and node of mortality with an additional arrow between treatments and mortality. $\endgroup$
    – Mike
    Commented Apr 10, 2022 at 1:34
  • 2
    $\begingroup$ Can you post the DAG, just so we are on the same page? $\endgroup$ Commented Apr 10, 2022 at 3:09

1 Answer 1


What I ended up doing with (some success, but I'm still fiddling to improve) was use a cure rate model where:

SurvivalObsevered(t) = TreatedPortion(t) + (1-TreatedPortion(t))*SurvivalUntreated(t)

In effect what I'm doing is considering treatment to be a "cure" for untreated mortality. This allows me to model and ultimately solve for the quantity of interest which is untreated survival versus time which is not directly observed.

  • $\begingroup$ (For the record, the approach above may or may not be valid, but what I can say after quite a bit of effort is that I still certainly haven't proven this works. I just wanted to give this update in case others try to follow in my footsteps.) $\endgroup$
    – Mike
    Commented May 5, 2022 at 4:38

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