  In his seminal work, Chatterjee (2021) introduced a novel correlation measure
which is distribution-free, asymptotically normal, and consistent against all
alternatives. In this paper, we study the probabilistic relationships between
Chatterjee's correlation and the widely used Spearman's correlation. We show
that, under independence, the two sample-based correlations are asymptotically
joint normal and asymptotically independent. Under dependence, the magnitudes
of two correlations can be substantially different. We establish some extremal
cases featuring large differences between these two correlations. Motivated by
these findings, a new independence test is proposed by combining Chatterjee's
and Spearman's correlations into a maximal strength measure of variable
association. Our simulation study and real data application show the good
sensitivity of the new test to different correlation patterns.
