I understand the basics of Simpson's paradox, but I'm not yet confident that I'd always be able to "avoid" it, or to spot cases where others have failed to do so. To be more precise, I'm not sure I would always be able to spot situations in which some fallacious argument of the form
$$\forall i, P(A|C_i) > P(B|C_i) \Leftrightarrow P(A) > P(B),$$
or an equally fallacious variant thereof, is being invoked (most likely tacitly).
How often do such errors occur in the peer-reviewed literature? How often are authors called on them? Are there common but hard-to-detect forms of such arguments?
For example, I routinely come across "averages of averages" being treated as equivalent to "global averages". Is this a potential backdoor to Simpson's-paradox-like fallacies?