The following paradox was brought to our attention by Pablo Lardelli from Granada (Spain).
1. Imagine that you design a cohort study to assess the causal effect of X on Y, E[Y|do(X=x)]. Prior knowledge informs you that variable M is a possible confounder of the process X—>Y, which leads you to assume X<---M--->Y.
To adjust for the effect of this confounder, you decide to design a matched cohort study, matching on M non exposed to exposed. You know that matching breaks down the association between X and M in the sample.
The problem arises when you draw the DAG […] and realize that S is a collider on the path X—>S<---M and, since we are conditioning on S (because the study sample is restricted to S=1) we are in fact opening a non causal path between X and Y (through M) in the sample. But this stands in contradiction to everything we are told by our textbooks. Click here for full discussion of matching in DAGs, persistent-unfiathfulness and unit-to-unit interactions.