Can someone explain precisely and unambiguously what "correlation does not equal
correlation" means? After years of reading/hearing this, I still don't understand it. I'm less concerned about lay usage of this phrase and more interested in what someone might mean by it in a serious discussion of research methodology, statistics, or epistemology.
To help explain my confusion, here's my view of why "correlation does not imply
causation" aptly expresses caution against inferring direct/proximal causation from the existence of a correlation (as I've told my stats students): The fallacy in question commonly arises when someone observes that two events (or variables, etc.) are correlated (i.e., occur together over subjects, times, places, etc.) and concludes from this that one event causes the other. In other words, he reasons that if
the events are correlated then
one causes the other; that is, he thinks the events' correlation implies
their causal connection.
To correct this fallacy, it seems appropriate to assert that two events' correlation does not imply
their causal connection -- that is, correlation does not imply causation. Is there a similar explanation of what "correlation equals
correlation" or "correlation does not equal
To describe this a bit more succinctly, let's define two events, A and B, as statements about some relationship between random variables X
directly causes or is directly caused by Y
(To be more rigorous, I should define "correlated" and "directly causes" more precisely.) It's easy to pick a counterexample to show that the implication A ==> B is false; for instance, suppose a third variable, say Z
, causes both X
and induces a correlation between them. Hence, A doesn't imply B. Here again, I don't know what statements such as "A equals B" or "A doesn't equal B" mean, because it seems unusual to make statements about the (in)equality of events A and B. Can someone explain this to me? Is "equals" just short for "is equivalent to"?
BTW, Philip Stark's online stats textbook (SticiGui) -- especially Ch. 2, 12, and 13 -- is a rare example of logic and logical fallacies integrated into intro stats material: www.stat.berkeley.edu/~stark/SticiGui/index.htm