In order to test each pair of genes for co-expression, it is necessary to account for multiplicity of M = 2 × 10 7 hypotheses. For the permutation tests of Heller et al. (2013) and Sz´ ekely et al. (2007), the number of permutations required for deriving a p -value that is below 0 . 05 /M is therefore of the order of 10 $^{10}$. Since these test statistics are relatively costly to compute for each hypothesis, e.g., O ( N $^{2}$) in Sz´ ekely et al. (2007), and O ( N $^{2}$log N ) in Heller et al. (2013), applying them to the family of M = 2 × 10 7 hypotheses is computationally very challenging, even with sophisticated resampling approaches such as that of Yu et al. (2011). Distribution-free tests have the advantage over non-distribution-free tests, that quantiles of the null distribution of the test statistic can be tabulated once per sample size, and repeating the test on new data for the same sample size will not require recomputing the null distribution. Therefore, the computational cost is only that of computing the test statistic for each of the hypotheses. We note that for univariate random variables Sz´ ekely and Rizzo (2009) considered using the ranks of each random variable instead of the actual values in the test of dis- tance covariance (Sz´ ekely et al., 2007), resulting in a distribution-free test. Similarly, for the test of Heller et al. (2013) replacing data with ranks results in a distribution- free test. An earlier work by Feuerverger (1993) defined another test based on the empirical characteristic functions for univariate random variables. The test statistic of Feuerverger (1993) was based on a different distance metric of the joint empiri- cal characteristic function from the product of marginal characteristic functions than that of Sz´ ekely et al. (2007). Moreover, in Feuerverger’s test the X ’s and Y ’s are first replaced by their normal scores, where the normal scores of the X ’s depend on the data only through their ranks among the X ’s, and similarly the normal scores of the Y ’s depend on the data only through their ranks among the Y ’s, thus making this test distribution-free. 3