it is close to the power of the optimal m . The choice of aggregation by maximization versus summation depends on how local the differences are between the distributions. In Figure 3 we see clearly that when the differences are in very local areas, maximization achieves the greatest power and the test based on minimum p -value has more power if the aggregation is by maximization rather than by summation (setups 1, 2, and 13), and aggregation by summation is better otherwise. Note that the optimal m for aggregation by summation is always larger than for aggregation by maximization. The reason is that in order to have a powerful statistic aggregated by maximization, it is enough to have one good parti- tion (i.e., contain cells where the distributions clearly differ) for a fixed m , whereas by summation it is necessary to have a large fraction of good partitions among all partitions of size m . We examined the power properties of the ADP and DDP statistics, aggregated by summation for m ∈ { 1 ,..., √ N } , aggregated by maximization for m ≤ 4, as well as the minimum p -value statistic min$_{{}$$_{m}$$_{∈}$$_{2}$$_{,...,m}$$_{max}$$_{}}$ p$_{m}$ based on aggregation by summa- tion. We display here the results for m$_{max}$ = 10 and N = 100. We compared these tests to seven tests of independence suggested in the literature. We compared to Spearman’s ρ , since it is perhaps the most widely used test to detect monotone associations. We also compared to previous tests suggested in the literature with the same two important properties as our suggested tests, namely proven consistency and distribution-freeness, as well as an available implementation: the test of Hoeffding (1948), referred to as Hoeffding; the test of Reshef et al. (2011), referred to as MIC; the tests of Sz´ ekely et al. (2007) and Heller et al. (2013) that 30 4.2 The independence problem