The all other things being equal assumption
I get the impression that a lot of people around the web confound absolutely everything when talking about correlations on the topic of social sciences. They supply a biased, incomplete, or truncated reality of what actually really is. A good example of this is when it is said that in Country #1, there is a negative association between corruption and GDP. The typical counter-argument is that Country #2 has more corruption but also higher GDP. Variants of that argument are simply innumerable.
The obvious mistake is the serious neglect of confounding factors. What differ between Country #1 and #2 are not only GDP and corruption but thousands other parameters which have not been taken into account in the bivariate correlation. And even in a multiple regression.
Researchers do not generally fall into this trap, and although they do not express it explicitly, base the conclusion that variable #1 correlate with #2 given the all-else-equal assumption. That is, when everything else is constant, there is a connection between these two variables presently tested.
Of course, it is not possible to include all possible confounders because humans are not omniscient, are not Gods. They include only a set of possible mediators for which they suspect a possible candidate for mediation in the two principal variables of interest. The problem emerges when a study provides results for individual differences in a given country, it is countered by defectuous arguments stating that other countries have more of “this” or more of “that”, which is taken as contradicting the relationship established in the study.
This over-simplistic view totally ignores the fact that within-group variation is not necessary equivalent to between-group variation, for reasons detailed above. The meaning of one parameter is not necessarily equivalent across (i.e., racial) groups. For example, what makes population #1 happy does not necessarily makes population #2 happy. And even when this is the case, there still might be a large difference in the correlations when comparing diverse groups. Also, it is possible that a “score” on a certain variable X is attained through different causal pathways. In this case, score on variable X is not comparable, generalizeable to other groups, unless we detect and correct the causes behind this “anomaly”. The problem becomes clearer when this group-biased variable X is used for correlation with other variables. Inferences can’t be valid, undeniably.
In fact, there is no such “all-else-equal” state in the reality because everything moves together, changes together, over time, perpetually. A correlation at any given time point could not be valid when studied in another time point. In other words, when a relation-causality is established, because elements composing the real world continue to change, some previously established evidence could not be true at a later date.
The all-else-equal assumption is nothing more (and nothing less) than a mental tool. One simply assumes, on some theoretical basis, that one factor causes another one under certain restrictions. So, when a critique of a certain theory neglects this assumption, it completely misses the target.