Selection Bias and Berkson's Paradox

I am not aware of Berksian's Paradox by name but I do know it by effect.

I came across it in this tweet.

Attacks on the GRE are often misguided:

1) It is less susceptible to class than most other indices
2) It correlates very highly with IQ (which is a great predictor of lots of outcomes)
3) Lack of correlation "among" PhD students is classic selection bias https://t.co/oWGJrRjx6D

— Jay Van Bavel (@jayvanbavel) December 17, 2017

Follow the thread for the discussion. There is dispute about:

Lack of correlation "among" PhD students is classic selection bias

Many people wish to disbelieve the predictive power of IQ and seize on the lack of correlation among PhD students as evidence to support their hypothesis that IQ is non-predicitive.

Jay van Bavel notes, I think correctly, that the non-correlation is due to selection bias.

I have seen this in many cases, where a predictive variable in the population fails to have the same predictive capacity with a subset of the population.

Stuart Buck offer's that perhaps this is not Selection Bias but Berkson's Paradox. Wikipedia's description is of little explanatory use:

Berkson's paradox also known as Berkson's bias or Berkson's fallacy is a result in conditional probability and statistics which is counterintuitive for some people, and hence a veridical paradox. It is a complicating factor arising in statistical tests of proportions. Specifically, it arises when there is an ascertainment bias inherent in a study design. The effect is related to the explaining away phenomenon in Bayesian networks.

The examples are better:

An example presented by Jordan Ellenberg: Suppose Alex will only date a man if his niceness plus his handsomeness exceeds some threshold. Then nicer men do not have to be as handsome to qualify for Alex's dating pool. So, among the men that Alex dates, Alex may observe that the nicer ones are less handsome on average (and vice versa), even if these traits are uncorrelated in the general population.

Other's correct Buck, pointing out that the phenomenon is textbook selection bias. From Wikipedia.

Selection bias is the bias introduced by the selection of individuals, groups or data for analysis in such a way that proper randomization is not achieved, thereby ensuring that the sample obtained is not representative of the population intended to be analyzed.

Berkson's fallacy is a more narrowly defined instance of the class of selection bias.