If A is correlated with B, and B is correlated with C, does it follow that A and C are correlated?
Not necessarily and usually not.
Here’s an example:
A: Being rich is positively correlated with living in a rich(er) state
B: Being from a rich state is positively correlated with voting for Democrats
C: Is being rich positively correlated with voting for Democrats?
No. Being rich is associated with voting for Republicans. How can that be?
It’s because correlations are not necessarily transitive. Definition of transitive: the property that if the relation holds between a first element and a second and between the second element and a third, it holds between the first and third elements.
More non-transitive examples:
Example from medicine:
A: Having high HDL (good) cholesterol is correlated with lower risk of heart disease
B: Taking niacin is positively correlated with higher HDL.
C: Is taking niacin then correlated with lower risk of heart disease?
No. Large scale clinical trials have concluded that those people who take niacin have as many heart attacks as everyone else even though they have higher HDL. Maybe some other quality that makes HDL be higher is responsible to not having heart attacks – it’s just not niacin. This example is similar to occurrences all over health, medicine and nutrition. Be wary!
Example from a study looking at baseball at-bats:
A: The number of triples hit by a player correlates positively with the overall number of base hits
B: The number of base hits correlates with the number of home runs a player hits.
C: Yet the number of triples a player hits correlates negatively with the number of home runs he hits
How? Players who hit a lot of triples were usually lithe and fast, traits that do not lend themselves to home run hitting, and players who homered a lot were generally big and slow, traits that do not lend themselves to hitting a lot of triples.
A: Good exam scores on a test are correlated with a deep understanding of the material
B: Memorizing material from flash cards are correlated with high exam scores
C: Does it follow that a deep understanding of the material is correlated with memorizing flash cards? Of course not.
Very similar to the notion that correlations are not necessarily transitive that there is no transitive property in college football. Here is an amazing 2016 college football season “transitive wheel” that makes this point (note the beginning and ending team) – truly awesome: College Football Transitive Wheel
Unlike correlations, causation IS transitive meaning that if A causes B and B causes C then A causes C. But, as we know correlation does not mean causation.
Source for some of these examples is from the fantastic book “How Not to be Wrong: The Power of Mathematical Thinking” by Jordan Ellenberg.