Judea Pearl Writes:
For a long time I could not figure out why SEM researchers find it hard to embrace the “causal inference approach” to mediation, which is based on counterfactuals. My recent conversations with David Kenny and Bengt Muthen have opened my eyes, and I am now pretty sure that I have found both the obstacle and the key to making causal mediation an organic part of SEM research.
Here is the key:
Why are we tempted to “control for” the mediator M when we wish to estimate the direct effect of X on Y? The reason is that, if we succeed in preventing M from changing then whatever changes we measure in Y are attributable solely to variations in X and we are justified then in proclaiming the effect observed as “direct effect of X on Y”. Unfortunately , the language of probability theory does not possess the notation to express the idea of “preventing M from changing” or “physically holding M constant”. The only operation probability allows us to use is “conditioning” which is what we do when we “control for M” in the conventional way (i.e., let M vary, but ignore all samples except those that match a specified value of M). This habit is just plain wrong, and is the mother of many confusions in the practice of SEM.
To find out why, you are invited to visit: http://ftp.cs.ucla.edu/pub/stat_ser/r421.pdf, paragraph starting with “In the remaining of this note, …”, on page 2.