we developed, detailed in Section 3, we are able to test for any m ≤ N in the K - sample problem, and for aggregation by summation in the test of independence. If the aggregation is by maximization in the test of independence, the algorithm, detailed in Section 3, is exponential in m and thus the computations are feasible only for m ≤ 4. We shall show in Section 4 that the power of the test based on a summation statistic can be different from the power of the test based on a maximization statistic, and which is more powerful depends on the joint distribution. However, for both ag- gregation methods, using m > 3 partitions improves power considerably for complex settings. Therefore, in complex settings our tests with m > 3 have a power advantage over the classical distribution-free tests, which focused on rough partitions, typically m = 2. Connection to the MI An attractive feature of the statistics S$_{m}$ and S ADP m × $_{m}$, for m large enough, is that they are directly associated with the MI. MI (defined as I$_{XY}$ = ∫ h ( x,y ) log[ h ( x,y ) / { f ( x ) g ( y ) } ] dxdy for continuous X and Y ) is a useful measure of statistical dependence. The variables X and Y are independent if and only if the MI is zero. Estimated MI is used in many applications to quantify the relationships between variables, see Steuer et al. (2002), Paninski (2003), Kinney and Atwal (2014) and references within. Although many works on MI estimation exist, no single one has been accepted as a state-of-the-art solution in all situations (Kinney and Atwal, 2014). A popular estimator among practitioners due to its simplicity and consistency is the histogram estimator, where the data are binned according to some scheme and the empirical mutual information of the resulting partition, i.e, the likelihood ratio score, is computed. Intuitively, one can expect that the statistic S ADP m × $_{m}$, properly normalized, can also serve as a consistent estimator of the mutual information, when the contingency tables are summarized by the likelihood ratio statistic, since it is the average of histogram estimators, over all partitions. This 14