  For analysis of a high-dimensional dataset, a common approach is to test a
null hypothesis of statistical independence on all variable pairs using a
non-parametric measure of dependence. However, because this approach attempts
to identify any non-trivial relationship no matter how weak, it often
identifies too many relationships to be useful. What is needed is a way of
identifying a smaller set of relationships that merit detailed further
analysis.
  Here we formally present and characterize equitability, a property of
measures of dependence that aims to overcome this challenge. Notionally, an
equitable statistic is a statistic that, given some measure of noise, assigns
similar scores to equally noisy relationships of different types [Reshef et al.
2011]. We begin by formalizing this idea via a new object called the
interpretable interval, which functions as an interval estimate of the amount
of noise in a relationship of unknown type. We define an equitable statistic as
one with small interpretable intervals.
  We then draw on the equivalence of interval estimation and hypothesis testing
to show that under moderate assumptions an equitable statistic is one that
yields well powered tests for distinguishing not only between trivial and
non-trivial relationships of all kinds but also between non-trivial
relationships of different strengths. This means that equitability allows us to
specify a threshold relationship strength $x_0$ and to search for relationships
of all kinds with strength greater than $x_0$. Thus, equitability can be
thought of as a strengthening of power against independence that enables
fruitful analysis of data sets with a small number of strong, interesting
relationships and a large number of weaker ones. We conclude with a
demonstration of how our two equivalent characterizations of equitability can
be used to evaluate the equitability of a statistic in practice.
