  Studying the multivariate extension of copula correlation yields a dimension
reduction principle, which turns out to be strongly related with the `simple
measure of conditional dependence' $T$ recently introduced by Azadkia &
Chatterjee (2021). In the present paper, we identify and investigate the
dependence structure underlying this dimension-reduction principle, provide a
strongly consistent estimator for it, and demonstrate its broad applicability.
For that purpose, we define a bivariate copula capturing the scale-invariant
extent of dependence of an endogenous random variable $Y$ on a set of $d \geq
1$ exogenous random variables ${\bf X} = (X_1, \dots, X_d)$, and containing the
information whether $Y$ is completely dependent on ${\bf X}$, and whether $Y$
and ${\bf X}$ are independent. The dimension reduction principle becomes
apparent insofar as the introduced bivariate copula can be viewed as the
distribution function of two random variables $Y$ and $Y^\prime$ sharing the
same conditional distribution and being conditionally independent given ${\bf
X}$. Evaluating this copula uniformly along the diagonal, i.e. calculating
Spearman's footrule, leads to Azadkia and Chatterjee's `simple measure of
conditional dependence' $T$. On the other hand, evaluating this copula
uniformly over the unit square, i.e. calculating Spearman's rho, leads to a
distribution-free coefficient of determination (a.k.a. copula correlation).
Several real data examples illustrate the importance of the introduced
methodology.
