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Can it be determined somehow whether the correlation coefficient between two random variables would be insufficient for assessing the level and type of co-dependency between them, and that another better dependence measure should additionally be tried as well to check for a non-linear relationship?

For example, if correlation is only $0.3$ (low linear dependence), would this warrant the use of a different dependency measure like Spearman's rho or mutual information more than for a relationship whose correlation level is $0.9$ (high linear dependence)?

(This line of thinking is due to the true but extreme case where a correlation of $0$ (no dependence) can actually be taken to be a sign that an undetected non-linear relationship was highly likely, which warrants a second look using a different measure.)

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    $\begingroup$ Sure; a weak Pearson correlation could mean that a nonlinear relationship exists that deserves a different analysis. What to do instead depends on the details. With a few variables the simple answer is to plot the data and think about what makes sense. With thousands of variables where automation is more surely and sorely needed, there are competing answers on general purpose measures of correlation; some have proved controversial. See e.g. arxiv.org/abs/1401.7645 $\endgroup$ – Nick Cox Aug 27 '20 at 11:39
  • $\begingroup$ so a high correlation, on the other hand, does not need to be questioneed further? $\endgroup$ – develarist Aug 27 '20 at 11:41
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    $\begingroup$ There is nothing here that doesn't need scrutiny apart from correlations that follow from definitions, such as correlation between Celsius and Fahrenheit temperatures. For example, a high correlation could be a side-effect of one or more outliers. Or see jstor.org/stable/2528645 on correlations between powers. $\endgroup$ – Nick Cox Aug 27 '20 at 11:49

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