Comment: Understanding Simpson’s Paradox

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Notes

This contrasts the historical account of Hernán, Clayton, and Keiding (2011) according to which “Such discrepancy [between marginal and conditional associations in the presence of confounding] had been already noted, formally described and explained in causal terms half a century before the publication of Simpson’s article...” Simpson and his predecessor did not have the vocabulary to articulate, let alone formally describe and explain causal phenomena.

Lindley later regretted that choice (Pearl 2009, p. 384), and indeed, his treatment of exchangeability was guided exclusively by causal considerations (Meek and Glymour 1994).

Statistics teachers would enjoy the challenge of explaining how the sentence “treatment does not change gender” can be expressed mathematically. Lindley and Novick tried, unsuccessfully of course, to use conditional probabilities.

Interestingly, Simpson’s examples also had different causal structure; in the former, the third variable (gender) was a common cause of the other two, whereas in the latter, the third variable (paint on card) was a common effect of the other two (Hernán, Clayton, and Keiding 2011). Yet, although this difference changed Simpson’s intuition of what is “more sensible,” it did not stimulate his curiosity as a fundamental difference, worthy of scientific exploration.

In Simpson’s paradox, we witness the simultaneous orderings: (a1 + b1)/(c1 + d1) > (a2 + b2)/(c2 + d2), (a1/c1) < (a2/c2), and (b1/d1) < (b2/d2).

The no-change provision is probabilistic; it permits the action to change the classification of individual units so long as the relative sizes of the subpopulations remain unaltered.

When such determination cannot be made from the given graph, as is the case in Figure 2(b), the do-calculus alerts us to this fact.

By “structure” we mean the list of variables that need be consulted in computing each variable V_i in the simulation.

Expressions such as “should be carefully examined” were used by statisticians in the precausal era to convey helplessness in handling causal questions.