One of the assumptions of the Rubin causal model is positivity which, for an individual $i$, a set of covariates $X$ and a treatment assignment indicator $Z_i$, is often expressed as $0 < P(Z_i | X) < 1$.

The situation I’m in is as follows:

As newer treatments are approved and adopted over time, older treatments may fall out of use. This may be for economic or administrative reasons, rather than through safety or efficacy concerns. So effectively the old treatments, while still approved, are no longer used, though in theory it might still be possible to prescribe them, hence in theory $0 < P(Z_i | X) < 1$. In practice though I can see in the data that there are no prescriptions after certain dates for some of the older treatments.

(In fact it is more complicated than that because these types of switchover to new treatments are different in different regions due to independent prescribing rules throughout the regions of the country whose data I’m looking at).

So although the study data may cover a number of years, there is in effect limited overlap in terms of time when all treatments are available and actually used, with earlier patients not having had access to the not yet available new treatments. But the later patients all being put on the newer treatments with the old treatments no longer prescribed.

I would have thought that this must be quite a common occurrence and that many published analyses who use causal methods (propensity scoring or inverse probability weighting etc) in observational studies must have faced this issue.

My questions then are:

  1. Can this type of situation be handled within the causal analysis framework or similar type of analysis? For example, try to use causal methods to achieve covariate balancing and continue from there – so it won’t be a true causal analysis as such but as “causal” as possible given the data.
  2. And if there is some literature on this subject, could someone give me some guidance to any references regarding this type of situation?

Any guidance/help appreciated.


This is a great question. The key to understanding this is in knowing what $X$ refers to. Here I presume you have taken it to mean all pretreatment variables, including the calendar time that the treatment was administered. As you identify, if time is part of $X$ and some treatments are not available at certain times, then positivity is not satisfied, and it doesn't make sense to talk about the potential outcomes for treatments that could not have been received.

$X$, however, is not meant to be interpreted as all pretreatment variables, but rather as all variables in a minimally sufficient adjustment set (e.g., all confounders; see my answer here). If time is not a confounder, that is, if calendar time does not causally affect the outcome, then time is not part of a minimally sufficient adjustment set, so $X$ does not include time. So, even though time limits what treatments one can receive, it does not create a positivity violation.

However, maybe time is a confounder because it not only affects what treatment one receives but also the outcomes, perhaps because quality of care has increased over time, in which case the association between the treatment and the outcome could at least partially be explained simply by the improvement in the quality of care over time and not just by the difference in treatment received. By conditioning on the qualities other than treatment that vary across time and that might affect the outcome, you can control for the confounding effect of time so that it is no longer part of the minimally sufficient adjustment set because it is no longer a confounder, and therefore doesn't create the positivity violation. If you are unable to do this, however, you cannot separate the effect of treatment with the effect of time.

  • $\begingroup$ Hi @Noah, thanks for this very informative answer. I was getting hung up on the timing of the treatments rather than seeing the study as a whole. $\endgroup$
    – Ray
    Feb 17 '20 at 10:44

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