  Reshef & Reshef recently published a paper in which they present a method
called the Maximal Information Coefficient (MIC) that can detect all forms of
statistical dependence between pairs of variables as sample size goes to
infinity. While this method has been praised by some, it has also been
criticized for its lack of power in finite samples. We seek to modify MIC so
that it has higher power in detecting associations for limited sample sizes.
Here we present the Generalized Mean Information Coefficient (GMIC), a
generalization of MIC which incorporates a tuning parameter that can be used to
modify the complexity of the association favored by the measure. We define GMIC
and prove it maintains several key asymptotic properties of MIC. Its increased
power over MIC is demonstrated using a simulation of eight different functional
relationships at sixty different noise levels. The results are compared to the
Pearson correlation, distance correlation, and MIC. Simulation results suggest
that while generally GMIC has slightly lower power than the distance
correlation measure, it achieves higher power than MIC for many forms of
underlying association. For some functional relationships, GMIC surpasses all
other statistics calculated. Preliminary results suggest choosing a moderate
value of the tuning parameter for GMIC will yield a test that is robust across
underlying relationships. GMIC is a promising new method that mitigates the
power issues suffered by MIC, at the possible expense of equitability.
Nonetheless, distance correlation was in our simulations more powerful for many
forms of underlying relationships. At a minimum, this work motivates further
consideration of maximal information-based nonparametric exploration (MINE)
methods as statistical tests of independence.
