Comment: A Fruitful Resolution to Simpson’s Paradox via Multiresolution Inference

Abstract

Simpson’s Paradox is really a Simple Paradox if one at all. Peeling away the paradox is as easy (or hard) as avoiding a comparison of apples and oranges, a concept requiring no mention of causality. We show how the commonly adopted notation has committed the gross-ery mistake of tagging unlike fruit with alike labels. Hence, the “fruitful” question to ask is not “Do we condition on the third variable?” but rather “Are two fruits, which appear similar, actually similar at their core?.” We introduce the concept of intrinsic similarity to escape this bind. The notion of “core” depends on how deep one looks—the multi resolution inference framework provides a natural way to define intrinsic similarity at the resolution appropriate for the treatment. To harvest the fruits of this insight, we will need to estimate intrinsic similarity, which often results in an indirect conditioning on the “third variable.” A ripening estimation theory shows that the standard treatment comparisons, unconditional or conditional on the third variable, are low hanging fruit but often rotten. We pose assumptions to pluck away higher-resolution (more conditional) comparisons—the multiresolution framework allows us to rigorously assess the price of these assumptions against the resulting yield. One such assessment gives us Simpson’s Warning: less conditioning is most likely to lead to serious bias when Simpson’s Paradox appears.

• Bias-variance tradeoff
• Principal stratification