During the covid crisis, we have seen explanations of why vaccinated people end up less hospitalized than unvaccinated people under the paradox that there were more vaccinated people hospitalized than unvaccinated people (at least amongst populations where vaccination was widely deployed).

The following figure helps understand the phenomenon by showing the proportion of vaccinated and unvaccinated people ending up at the hospital.

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I was wondering if we could represent this phenomenon with a causal DAG (Directed Acyclic Graph) showing a bias when conditioning on hospitalization (I was first thinking of a collider bias). I guess the DAG would have 3 nodes: Vaccine, Covid and Hospitalization, but I have yet to find a DAG that shows a bias when conditioning on hospitalization.

  • $\begingroup$ The issue here with a DAG is that there is a feedback loop present. Fear of COVID causes some people to vaccinate, so that there is an arrow from COVID to Vaccine. But then if the vaccine has an effect on the virus (not transmission, but severity), then there is an arrow from vaccine to COVID. DAGs are useful, but loops are not terribly easy to deal with. Surely the vaccine does not have a direct effect on hospitalizations, so that your DAG would be $V\leftrightarrow C\to H.$ If this is correct, then you are right in thinking there is no collider bias. $\endgroup$ Commented Jan 5, 2022 at 14:07
  • $\begingroup$ Thank you for your answer. If we suppose people are randomly vaccinated but in a higher proportion (so that there is no arrow from Covid to Vaccine, and for illustration suppose that 90% of the population is vaccinated), I understand the DAG would be $V \rightarrow C \rightarrow H$. Now, comparing hospitalized patients withtout taking into account the high vaccination rate in the population could lead one to conclude that Vaccine has a negative effect on Hospitalization (see the bubble conditioning on Hospitalized). How can we show that this conclusion is false with such a DAG? $\endgroup$
    – Tanguy
    Commented Jan 5, 2022 at 14:46
  • $\begingroup$ I guess I would take a step back and reiterate what the question is: you are interested in the causal effect of $V$ on $H.$ That causal effect is mediated through the virus $C.$ It is unheard-of to condition on the effect in any causal analysis! In a linear regression setting, that would be the equivalent of including the LH variable $Y$ in the RHS! All this means is that the LH picture you have in your question is simply the wrong perspective if you are investigating the causal effect of vaccination on hospitalizations. $\endgroup$ Commented Jan 5, 2022 at 15:08
  • $\begingroup$ Getting to brass tacks: it is inappropriate to compare the proportion of unvaccinated to vaccinated in hospitals. It is appropriate to compare the proportion of vaccinated who are hospitalized to unvaccinated who are hospitalized. Incidentally, with the PCR tests being essentially declared invalid, I would say that we essentially have no valid data on COVID at all: the case numbers are totally off, likely the deaths and hospitalizations as well. I think we basically know almost nothing about COVID. $\endgroup$ Commented Jan 5, 2022 at 15:10
  • $\begingroup$ The so-called "paradox" to which you refer sounds like the standard base-rate consideration in any such setting: when most people are vaccinated, they will predominate in hospitals for that reason alone. It's unclear what your proposed analysis could add to that. $\endgroup$
    – whuber
    Commented Jan 5, 2022 at 18:39