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I have two variables X and Y. X is continuous, while Y is semicontinuous. Specifically, Y continuous on strictly positive values and has positive mass at 0.

What would be the optimal way to measure dependence between these two variables?

I am thinking of maybe separately measuring the dependence on the binary part of Y (correlation of X with Y_positive/Y_zero) and then, on positive Y values, measure the spearman correlation with X. But I do not know if this is a sensible approach.

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    $\begingroup$ You're going to need to be more specific about what your definition of correlation is, and/or what it is you're optimizing there. $\endgroup$
    – Glen_b
    Commented Jul 18 at 10:24
  • $\begingroup$ I want a measure of dependence or association between the two variables. I would say a statistic that measures this quantitatively, analogous to the spearman correlation for continuous variables. $\endgroup$
    – D1X
    Commented Jul 18 at 10:37
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    $\begingroup$ Dependence/association is much more general than what people normally call correlation, while Spearman correlation measures degree of a monotonic relationship. For example, this is functional dependence; this shows dependence (association) but no monotonic correlation. If you do just mean monotonic relationship, what's wrong with one of the usual measures? (I can think of about 5 right now but there are many more; they work on non-continuous variables, though some - such as the Kendall correlation - do adjust for ties) $\endgroup$
    – Glen_b
    Commented Jul 18 at 23:57

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There's no problem with measuring a Pearson correlation (or Spearman, Kendall, etc.) when there are values at zero.

But if you are not restricted to using a correlation coefficient, you could also use a zero-inflated model or hurdle model to describe variation in $Y$. There are some subtle differences between the two described at the linked thread, but both allow you to model the probability of a value being equal to zero and variation in the non-zero values. This is akin to your proposed solution and possibly no different than fitting two separate models to the zero and non-zero data.

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