Why don't you take his model, as described, and see if it really produces this effect?

For the first disagreement, that's a disagreement about his commentary on his second figure. I don't have the data to actually calculate the correlation there, but eyeballing it the groups look like they don't have a positive relationship between education and income anywhere near that of the larger group.

The second disagreement is on interpretation. If you add noise in both dimensions to a multivariate Gaussian model with mean differences between groups, then that impacts any slice of the model (modified by the angle between the mean difference vector and the slice vector). If one subgroup is above and to the right of the other subgroup, that means it's above for every vertical slice and to the right for every horizontal slice. (On northwest-southeast slices, there's no mean difference between the distributions, just population size differences, and the mean difference is maximized on the northeast-southwest slice.)

The particular slicing used in this effect- looking at each vertical slice individually, and each horizontal slice individually- seems reasonable, **except** that in the presence of mean differences it behaves as a filter that preserves the NW-SE noise!

The grandparent was wrong before I edited it, where I speculated that the noise had to be negatively correlated. That's the claim that the *major* axis of the covariance ellipse has to be oriented a particular direction, but that was an overreach, as you see the reverse regression effect if there is *any* noise along the NW-SE axis. Take a look at Yan's first figure- it has noise in both blues and greens, but it's *one-dimensional* noise going NE-SW, and so we don't see reverse regression.

My original thought (when I thought you might need the major axis to be NW-SE, rather than just the NW-SE axis to be nonzero) was that this was just a reversal effect, with the noise providing the reversing factor. That's still true but I'm surprised at how benign the restrictions on the noise are.

That is, I disagree with Yan that this has a different origin than Simpson's Paradox, but I agree with Yan that this is an important example of how pernicious reversal effects are, and that noise generates them *by default*, in some sense. I would demonstrate it with a multivariate Gaussian where the blue mean is [6 6], the green mean is [4 4], and the covariance matrix is [1 .5; .5 1], so that it's obvious that the dominant relationship for each group is a positive relationship between education and income but the NW-SE relationship exists and these slices make it visible.

Thanks, that dicussions's examples were exactly what I was looking for!

I'm glad it was helpful. =)