Conditioning on a shared outcome of two variables can alter the association between these variables, possibly adding a bias component when estimating effects. In particular, if two causes are marginally independent, they might be dependent in strata of their common effect. Explanations of the phenomenon, however, do not explicitly state when dependence will be created and have been largely informal. We prove that two, marginally independent, causes will be dependent in a particular stratum of their shared outcome if and only if they modify each other's effects, on a probability ratio scale, on that value of the outcome variable. Using our result, we also qualify the claim that such causes will "almost certainly" be dependent in at least one stratum of the outcome: dependence must be created in one stratum of a binary outcome, and independence can be maintained in every stratum of a trinary outcome.12 month embargo; published online: 2017-03-31This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]
