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One has a simple dataset of 3 independent variables, e.g., x, y, z.

Now:

  • y and z are logically connected (this is known a priori) and indeed a nice & tight correlation (small scatter) between them is established;

  • two correlations are measured between x and y and between x and z, with some scatter in both of them. Characterising those correlations would be great because they're not expected a priori, and would add something new to a theory.

I wonder: is x correlated to y (or z) only because of the underlying correlation between y and z? In other words, what is the "more fundamental" correlation? x-y or x-z? (with the other being only a byproduct of the underlying y-z relation)

What is the best statistical tool to face this problem?

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  • $\begingroup$ PS: I have been told that PCA could help, but I am not sure how. Any hints? $\endgroup$ – jordan Jul 24 at 12:52
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The best is to do conditional test of independence.

For that you can use an adapted chi squared test of independence but considering conditional probability. You can find for example here https://newonlinecourses.science.psu.edu/stat504/node/112/ (not the best reference but it gives the formula).

You can also look to bayesian network to go further into those kind of conditional independence, it gives a theoretical framework to begin with.

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