When two variables are related by a known function, the coefficient of determination (denoted R2) measures the proportion of the total variance in the observations explained by that function. For linear relationships, this is equal to the square of the correlation coefficient, ρ. When the parametric form of the relationship is unknown, however, it is unclear how to estimate the proportion of explained variance equitably—assigning similar values to equally noisy relationships. Here we demonstrate how to directly estimate a generalised R2 when the form of the relationship is unknown, and we consider the performance of the Maximal Information Coefficient (MIC)—a recently proposed information theoretic measure of dependence. We show that our approach behaves equitably, has more power than MIC to detect association between variables, and converges faster with increasing sample size. Most importantly, our approach generalises to higher dimensions, estimating the strength of multivariate relationships (Y against A, B, …) as well as measuring association while controlling for covariates (Y against X controlling for C). An R package named matie (“Measuring Association and Testing Independence Efficiently”) is available (http://cran.r-project.org/web/packages/matie/).

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Kinney and Atwal (1) make excellent points about mutual information, the maximal information coefficient (2, 3), and "equitability." One of their central claims, however, is that, "No nontrivial dependence measure can satisfy R 2 -equitability." We argue that this is the result of a poorly constructed definition, which we quote:

The area of the medial wall is related to ethnic/gender groups.