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If I know the Pearson correlation between A and B and also know the Pearson correlation between A and C, can I infer anything about the correlation between B and C?

Assume that I no longer have access to the raw values for A, B, and C.

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    $\begingroup$ This is becoming a FAQ: some version of this question is asked every few months. If you're really good at searching (I found a near duplicate at stats.stackexchange.com/questions/72790 by including "correlation" and "three" in the search terms, but that didn't quite work out and I had to look much harder) you ought to be able to find quite a few good and interesting answers. $\endgroup$ – whuber Aug 10 '15 at 19:31
  • $\begingroup$ Thanks for the link. I wasn't quite sure how to formulate a good search on this topic. $\endgroup$ – Mike A. Aug 10 '15 at 20:03
  • $\begingroup$ @MikeA. When you're wondering whether such a general statement is true a good strategy is often to look for a single counterexample. Here it's not difficult. Take $A$, $B$ and $C$ to have common pairwise correlation $\rho$. Now take $A^\prime$, $B^\prime$ and $C^\prime$ to be such that $A^\prime$ and $B^\prime$ have correlation $\rho$, but $A^\prime = C^\prime$ so that $B^\prime$ and $C^\prime$ also have correlation $\rho$. You clearly can't infer the correlation between the first and third based on the other two correlations. $\endgroup$ – dsaxton Aug 10 '15 at 20:44
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On the whole, the answer to your question is "no". Structurally, it's easy to generate data so that A->B is significant and B->C is significant but A is not related to C.

There are two circumstances where this question crops up:

  1. B may be considered a "mediator" in that A causes the B which causes the C. So if you are interested in A->B->C there are specific tests for that (like in SEM) BUT it's rare I believe such results (when significant) if we don't at least find A->C in the sample. It's important to be sure that B is not a confounder in this circumstance. Take smoking (A), dietary intake (B) and cardiovascular risk. A->C (positive) and B->C (positive) but A->B negative. if you looked just at the directional effects here, you would conclude smoking is beneficial for heart disease since it reduces appetite.

  2. C was not actually measured. From a strictly practical perspective, in general scientists will greatly shoot down any attempt to make inference of that nature. Suppose I am interested in treating lowstage breast cancer with a mild chemotherapeutic agent. Tumors have to shrink before they disappear, so tumor shrinkage might be my B whereas survival is my C. Chemotherapy is very toxic, so I wouldn't be surprised to find I achieve B but not having measured C I cannot conclude that the A->C effects are "negligible".

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