Andrew Gelman in one of his recent blog posts says:
I do not think counterfactuals or potential outcomes are necessary for Simpson’s paradox. I say this because one can set up Simpson’s paradox with variables that cannot be manipulated, or for which manipulations are not directly of interest.
Simpson’s paradox is part of a more general issue that regression coefs change if you add more predictors, the flipping of sign is not really necessary.
Here’s an example that I use in my teaching that illustrates both points:
I can run a regression predicting income from sex and height. I find that the coef of sex is \$10,000 (i.e., comparing a man and woman of the same height, on average the man will make \$10,000 more) and the coefficient of height is \$500 (i.e., comparing two men or two women of different heights, on average the taller person will make \$500 more per inch of height).
How can I interpret these coefs? I feel that the coef of height is easy to interpret (it’s easy to imagine comparing two people of the same sex with different heights), indeed it would seem somehow “wrong” to regress on height without controlling for sex, as much of the raw difference between short and tall people can be “explained” by being differences between men and women. But the coef of sex in the above model seems very difficult to interpret: why compare a man and a woman who are both 66 inches tall, for example? That would be a comparison of a short man with a tall woman. All this reasoning seems vaguely causal but I don’t think it makes sense to think about it using potential outcomes.
I pondered over it (and even commented on the post) and think there's something that begs to be understood with greater clarity here.
Until the part on interpretation of gender it is so okay. But I do not see what's the problem behind comparing a short man and a tall woman. Here's my point: In fact it makes even greater sense (given the assumption that men are taller on average). You cannot compare a 'short man' and a 'short' woman for exactly the same reason, that the difference in income is explained in some part by the difference in heights. Same goes for tall men and tall women and even more so for short women and tall men (which is further out of the question, so to speak). So basically the effect of height is eliminated only in the case when short men and tall women are compared (and this helps in interpreting the coefficient on gender). Doesn't it ring a bell on similar underlying concepts behind the popular matching models?
The idea behind Simpson's paradox is that the population effect might be different from the sub-group wise effect(s). This is in some sense related to his point 2 and the fact that he acknowledges that height should not be controlled for alone (what we say omitted variable bias). But I could not relate this to the controversy on the coefficient on gender.
Maybe you might be able to express it more clearly? Or comment on my understanding?