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The maximal correlation coefficient is a well-established generalization ofthe Pearson correlation coefficient for measuring non-linear dependence betweenrandom variables. It is appealing from a theoretical standpoint, satisfyingR\'enyi's axioms for a measure of dependence. It is also attractive from acomputational point of view due to the celebrated alternating conditionalexpectation algorithm, allowing to compute its empirical version directly fromobserved data. Nevertheless, from the outset, it was recognized that themaximal correlation coefficient suffers from some fundamental deficiencies,limiting its usefulness as an indicator of estimation quality. Anotherwell-known measure of dependence is the correlation ratio which also suffersfrom some drawbacks. Specifically, the maximal correlation coefficient equalsone too easily whereas the correlation ratio equals zero too easily. Thepresent work recounts some attempts that have been made in the past to alterthe definition of the maximal correlation coefficient in order to overcome itsweaknesses and then proceeds to suggest a natural variant of the maximalcorrelation coefficient as well as a modified conditional expectation algorithmto compute it. The proposed dependence measure at the same time resolves themajor weakness of the correlation ratio measure and may be viewed as a bridgebetween these two classical measures.

On the Importance of Asymmetry and Monotonicity Constraints in Maximal Correlation Analysis