I have an experiment that perturbs variable x and causes a change in variable z. There is a concurrent change in variable y. How can I determine whether variable y is on the causal path between x and z or if it is an irrelevant epiphenomenon?
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1$\begingroup$ while it is difficult/impossible to "prove" causation, look into the subject of statistical mediation, which seems to be exactly what you're describing. This has been studied in many social sciences with the seminal paper by Baron and Kenny in the 1980s with many incremental advances since then by many statisticians in the field of causal inference. $\endgroup$– MacroCommented Jul 1, 2011 at 23:46
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4 Answers
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Experimental design rather than statistics alone will give you a solid answer. You may well have already thought about this and ruled it out as a possibility, but can you design an experiment that tests whether Z changes as a function of Y when X is held constant? Further, perhaps you could build in trials that test the extent to which Z changes as a function of X when Y is held constant. Isolating predictors in this way is your best bet of capturing the causal relationships.
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$\begingroup$ But if Z changes as a function of Y by only a tiny amount this might not account fully for the effect of X. $\endgroup$ Commented Jul 3, 2011 at 20:52
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$\begingroup$ Thanks for the answers but I need a quantitative solution. $\endgroup$ Commented Jul 5, 2011 at 17:12
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I think you're looking for an instrumental variable test, which will allow you to assess whether $y$ mediates the correlation between $x$ and $z$. The section on "interpretation as two-stage least squares" provides a nice intuition for this procedure.
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$\begingroup$ Thank you - I will look into the instrumental variable test. And thanks everyone for very helpful comments. $\endgroup$ Commented Jul 12, 2011 at 21:41
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You might try comparing...
-the corr. of x with z while controlling for y
-the corr. of y with z while controlling for x
It won't give you a definitive answer, since as @Macro said statistics are limited in that way, but it could give you clues.
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The front-door adjustment seems applicable assuming your causal diagram can be drawn in the following manner:
If the causal effect of X on Y is confounded by some unknown distribution U and mediated by another variable M, then it can be estimated from observational data. Note, from inspection of the DAG above, we need to assume the confounding variable U has no or limited (to the point where the interaction term can be ignored) influence on the mediating variable M.
If the true causal relationships between the variables is indeed as shown, conditioning on M will prevent the effect of X on influencing Y. Using correlation analysis in light of this might lead to a false conclusion.
For a derivation of the front-door criterion, please see the following post by Carlos Cinelli. His derivation uses the rules of do-calculas which I explicitly previously defined here.
