Reading up on Simpson's Paradox again made me realize something: women retrieving custody more often than men appears to me to be a perfect example of Simpson's Paradox. Overall, women get custody more often than men, this is true, but if you consider only the cases where men actually asked for or attempted to retrieve custody, this is no longer true[0].
How to interpret that, though? Seems likely to me that who asks for custody would be influenced by their chances of receiving it. How many of those cases are "contested custody"?
Then that article places the claim "Additional evidence, however, indicates that women may be less able to afford the lawyers and experts needed in contested custody cases (see “Family Law Overview”) and that, in contested cases, different and stricter standards are applied to mothers." without providing any data. So it ends up being a propaganda piece, which makes it not very trustworthy.
[0]: http://www.villainouscompany.com/vcblog/archives/2012/04/chi...