  The maximal correlation coefficient is a well-established generalization of
the Pearson correlation coefficient for measuring non-linear dependence between
random variables. It is appealing from a theoretical standpoint, satisfying
R\'enyi's axioms for a measure of dependence. It is also attractive from a
computational point of view due to the celebrated alternating conditional
expectation algorithm, allowing to compute its empirical version directly from
observed data. Nevertheless, from the outset, it was recognized that the
maximal correlation coefficient suffers from some fundamental deficiencies,
limiting its usefulness as an indicator of estimation quality. Another
well-known measure of dependence is the correlation ratio which also suffers
from some drawbacks. Specifically, the maximal correlation coefficient equals
one too easily whereas the correlation ratio equals zero too easily. The
present work recounts some attempts that have been made in the past to alter
the definition of the maximal correlation coefficient in order to overcome its
weaknesses and then proceeds to suggest a natural variant of the maximal
correlation coefficient as well as a modified conditional expectation algorithm
to compute it. The proposed dependence measure at the same time resolves the
major weakness of the correlation ratio measure and may be viewed as a bridge
between these two classical measures.
