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I am running a regression test for which I overall have two variables (x and y) and I would like to test if any of them is dependent on the other.

y = ax + intercept

No matter which one of the variables is the dependent one, I get the same R-squared and p-values (either with Matlab or with Rstudio). Intercept and predictor coefficients do change according to which variable is the independent one, but R-squared and p-value don't. Why is this happening? Can anyone explain and/or add any relevant paper?

Thanks in advance!

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    $\begingroup$ In a simple linear regression (one predictor) like you have, $R^2 = cor(x, y)^2$. Since $cor(x, y) = cor(y, x)$, the $R^2$ does not change, either. $\endgroup$
    – Dave
    Commented Jun 8, 2021 at 16:09
  • $\begingroup$ Thanks for your answer! I understand the maths but still this doesn't make sense to me. If y is dependent on x, then why is it equally possible that x is dependent on y? Isn't the null hypothesis in linear regression saying that y doesn't depend on x (the direction does matter)? While null hypothesis in correlation (e.g Pearson's) assumes that there isn't any kind of depency, no matter where it comes from... $\endgroup$
    – Katerina Kornilia
    Commented Jun 9, 2021 at 8:42

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I will try to explain the intuition. In simple linear regression, you model association between x and y. Thus it is irrelevant if x explain y or y explain x. Either way the relation between the two is the same. Therefore you would expect to get the same R^2 and p-value no-matter the order.

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  • $\begingroup$ Thanks for the explanation! $\endgroup$
    – Katerina Kornilia
    Commented Jun 9, 2021 at 15:10

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