  Chatterjee (2021) introduced a simple new rank correlation coefficient that
has attracted much recent attention. The coefficient has the unusual appeal
that it not only estimates a population quantity first proposed by Dette et al.
(2013) that is zero if and only if the underlying pair of random variables is
independent, but also is asymptotically normal under independence. This paper
compares Chatterjee's new correlation coefficient to three established rank
correlations that also facilitate consistent tests of independence, namely,
Hoeffding's $D$, Blum-Kiefer-Rosenblatt's $R$, and Bergsma-Dassios-Yanagimoto's
$\tau^*$. We contrast their computational efficiency in light of recent
advances, and investigate their power against local rotation and mixture
alternatives. Our main results show that Chatterjee's coefficient is
unfortunately rate sub-optimal compared to $D$, $R$, and $\tau^*$. The
situation is more subtle for a related earlier estimator of Dette et al.
(2013). These results favor $D$, $R$, and $\tau^*$ over Chatterjee's new
correlation coefficient for the purpose of testing independence.
