# Does non-zero Spearman rank correlation imply dependence of original variables?

### Introduction

Let say we have random variables $$X$$ and $$Y$$, and we take their rank transforms to be $$g(X)$$ and $$g(Y)$$. The Spearman rank correlation coeffiicient can be considered to be $$R[g(X),g(Y)] = \frac{\mathbb{E}\left[ (g(X) - \mathbb{E}[g(X)])(g(Y) - \mathbb{E}[g(Y)]) \right]}{\sqrt{\mathbb{E}[(g(X) - \mathbb{E}[g(X)])]}\sqrt{\mathbb{E}[(g(Y) - \mathbb{E}[g(Y)])]}}$$

where $$R$$ is the Pearson product-moment correlation coefficient (PPMCC). It is also known from the algebra of random variables that $$X \perp \!\!\! \perp Y \implies R[X,Y] = 0$$. In other words, if two random variables are independent, then their PPMCC will be zero.

From the above, we can understand that a non-zero Spearman rank correlation coefficient implies that the rank-transformed variables $$g(X)$$ and $$g(Y)$$ are mutually dependent.

### Question

Can we further deduce that $$X$$ and $$Y$$ are mutually dependent if their Spearman rank correlation is non-zero? I.e. that $$R[g(X),g(Y)] \neq 0 \implies \lnot (X \perp \!\!\! \perp Y)$$?

### Discussion

Perhaps the rank transform being a monotonic transform could be useful if combined with the notion copulas. Specifically, the copula that maps the marginal cumulative distribution functions to the joint cumulative distribution function is invariant to monotonic transformations of the random variables. Thus ranking should not change the copula. If the copula was not the independence copula, then it is a copula that describes some sort of dependence. I see a potential argument here, but I have not closed the gap.

• Commented Feb 18, 2022 at 2:43
• A nonzero correlation explicitly exhibits a dependence among the variables. It's hard to know what would need to be added to that. I believe you might be asking the contrapositive of the question at stats.stackexchange.com/questions/94872.
– whuber
Commented Feb 18, 2022 at 21:36

Let $$(\Omega, \mathcal{F}, P)$$ be a probability space and let $$\{ X_j \}_{j=1}^n$$, $$2 \leq n < \infty$$, be a collection of random variables on $$(\Omega, \mathcal{F}, P)$$. Then $$\{ X_j \}_{j=1}^n$$ are independent if-and-only-if $$\mathbb{E} \left[ \prod_{j=1}^n f_j(X_j) \right] = \prod_{j=1}^n \mathbb{E} \left[ f_j(X_j) \right]$$ for all bounded Borel-measurable functions $$f_j: \mathbb{R} \mapsto \mathbb{R}, \forall j \in \{ 1, \cdots, n \}$$.