I am interested in finding "real-world" examples of when variables might exhibit marginal independence but are conditionally dependent given some other variable. It seems to me that the converse (marginally dependent but conditionally independent) is more common (e.g. autoregressive processes, confounding variables in general).

Are there any canonical examples of this phenomenon that I am not aware of? My searches turned up many theoretical treatments of this idea, as well as numerical examples of simple distributions where this property holds, but no motivating examples from science/engineering/business/etc.

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    $\begingroup$ This happens when you have two marginally independent variables that are both causes of a third. Conditioning on that third variable makes them dependent. An example would be smoking and some genetic factor that makes you more likely to get cancer. Smoking and the presence of that factor are independent but conditional on whether someone has cancer, they become dependent. $\endgroup$
    – CloseToC
    Jan 12, 2015 at 21:52

1 Answer 1


Elwert F. and Winship C. (2014) Endogenous selection bias: The problem of conditioning on a collider variable Annual Review of Sociology 40:31–53 has a stack of examples - mostly forms of selection bias.


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