Statistics

Correlation is not transitive, in general at least: A simulation approach

Let $$\rho_{XY}$$ be the correlation between the stochastic variables $$X$$ and $$Y$$ and similarly for $$\rho_{XZ}$$ and $$\rho_{YZ}$$. If we know two of these, can we say anything about the third? In a recent blog post I dealt with the problem mathematically and I used the concept of a partial correlation coefficient. Here I will take a simulation approach. First z is simulated. Then x and y is simulated based on z in a regression context with a slope between $$-1$$ and $$1$$.

Correlation is not transitive, in general at least

Update Aug 10, 2019: I wrote a new blog post about the same as below but using a simulation approach. Let $$\rho_{XY}$$ be the correlation between the stochastic variables $$X$$ and $$Y$$ and similarly for $$\rho_{XZ}$$ and $$\rho_{YZ}$$. If we know two of these, can we say anything about the third? Yes, sometimes, but not always. Say we have $$\rho_{XZ}$$ and $$\rho_{YZ}$$ and they are both positive. Intuition would then make us believe that $$\rho_{XY}$$ is probably also positive then.