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.

中文翻译：

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冲突偏置核心的一个定理。

以两个变量的共同结果为条件可能会更改这些变量之间的关联，并可能在估计效果时增加偏差成分。尤其是，如果两个原因在边缘上是独立的，则它们可能在其共同作用的层次上是相互依赖的。但是，该现象的解释并没有明确说明何时会产生依赖性，并且这种解释在很大程度上是非正式的。我们证明，当且仅当两个因果在概率比率量表上根据结果变量的值修改彼此的影响时，才有两个边际独立的因果将依赖于其共同结果的特定层次。使用我们的结果，我们还可以证明这种原因“几乎可以肯定”将取决于结果的至少一个层次：必须在二元结果的一个层次中创建依赖性，