[caddemon]

I think Berkson's paradox is more specific than just correlations arising from non-random sample selection. Correlations that are not representative of the general population could still be useful, if it's a meaningful correlation within some subgroup of interest. The problem is when the features you are correlating relate too closely to the features that were used for sample selection - then you can end up with a trivial result.

I've always learned of Simpson's paradox as relating more to different sample sizes when partitioning data, which can happen entirely arbitrarily - for example a baseball player getting injured part way through the season.

The fact that one player's at bats get partitioned differently than another's is not caused by the on field performance, so there's no "double dipping" going on like I would imagine with Berkson's. Conversely I'm having trouble fitting a Berkson's example into the framework of Simpson's paradox, since there's no reason the poorly-selected subpopulation can't theoretically be exactly half of the general population. And if all of the samples are of equal size Simpson's paradox doesn't exist anymore (because with equal bin sizes the mean of means is equivalent to the overall mean).

[kgwgk]

> I've always learned of Simpson's paradox as relating more to different sample sizes when partitioning data

When you look at proportions based on binary outcomes it may be related to imbalanced groups but it's more general than that.

In the context discussed here of correlations between continous variables the groups can be of similar size.

See for example the chart here: https://towardsdatascience.com/simpsons-paradox-d2f4d8f08d42

[caddemon]

Interesting, I only ever heard of Simpson's paradox in the context of comparing overall averages versus subgroup averages.

I guess this paradox could then be thought of as a special case of Simpson's paradox? Since the out group will exclude people with both traits there should also be a negative correlation there, which disappears in the overall population. But in Berkson's case it seems they're implying the subgroup correlation is spurious whereas with Simpson's it could go either way.

[kgwgk]

> Since the out group will exclude people with both traits there should also be a negative correlation there

Not necessarily. Imagine the traits are distributed uniformly and independently in [-1 1]. There is no correlation:

    ******
    ******
    ******
    ******
    ******
    ******

If you select people with at least one positive trait you will find negative correlation in the group + but the correlation will still be zero in the group -.

    ++++++
    ++++++
    ++++++
    ---+++
    ---+++
    ---+++

[caddemon]

Makes sense, I was picturing more of a diagonal boundary but you're right the paradox doesn't specify the shape of the boundary. Thanks!