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Suppose that X, Y, and Z are random variables. X and Y are positively correlated and Y and Z are likewise positively correlated. Does it follow that X and Z must be positively correlated?

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  • $\begingroup$ Your question is covered in numerous answers on site. See, for example, here, or here, or a number of others $\endgroup$
    – Glen_b
    Nov 12, 2015 at 6:51
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    $\begingroup$ But I didn't found any example to explain this answer. $\endgroup$ Nov 12, 2015 at 6:57
  • $\begingroup$ The first link directly shows that the bounds on the correlation between X and Z can be negative (in the notation of that question, it shows that $\rho_{BC}$ is (attainably) bounded by $\rho_{AB}\rho_{AC}\pm {\sqrt{1-\rho_{AB}^2} \sqrt{1-\rho_{AC}^2}}$). e.g. take $\rho_{AB}=\rho_{AC}=0.5$ and the lower bound is negative. Aside from the need to substitute $A$ for $Y$ and so forth, how does that not directly answer your question? $\endgroup$
    – Glen_b
    Nov 12, 2015 at 7:05
  • $\begingroup$ It was not understandable to me $\endgroup$ Nov 12, 2015 at 7:07
  • $\begingroup$ Then explain what you don't understand about it. Do you not understand what $\rho_{AB}$ means? (Or do you not understand what $\pm$ is? Or is it the $\sqrt{}$ symbol that's the problem?) I'll be happy to help clarify any part of the formula. $\endgroup$
    – Glen_b
    Nov 12, 2015 at 7:09

2 Answers 2

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We may prove that if the correlations are sufficiently close to 1, then X and Z must be positively correlated.

Let’s assume C(x,y) is the correlation coefficient between x and y. Like wise we have C(x,z) and C(y,z). Here is an equation which comes from solving correlation equation mathematically :

C(x,y) = C(y,z) * C(z,x) - Square Root ( (1 - C(y,z)^2 ) * (1 - C(z,x)^2 ) )

Now if we want C(x,y) to be more than zero , we basically want the RHS of above equation to be positive. Hence, you need to solve for :

C(y,z) * C(z,x) > Square Root ( (1 - C(y,z)^2 ) * (1 - C(z,x)^2 ) )

We can actually solve the above equation for both C(y,z) > 0 and C(y,z) < 0 together by squaring both sides. This will finally give the result as C(x,y) is a non zero number if following equation holds true:

C(y,z) ^ 2 + C(z,x) ^ 2 > 1

Wow, this is an equation for a circle. Hence the following plot will explain everything :

correlation circle

If the two known correlation are in the A zone, the third correlation will be positive. If they lie in the B zone, the third correlation will be negative. Inside the circle, we cannot say anything about the relationship. A very interesting insight here is that even if C(y,z) and C(z,x) are 0.5, C(x,y) can actually also be negative.

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Here is a great post by Terence Tao on the topic. Words from the man himself:

I came across the (important) point that correlation is not necessarily transitive: if $X$ correlates with $Y$, and $Y$ correlates with $Z$, then this does not imply that $X$ correlates with $Z$.

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