35 Table 2: Power of competitors (columns 3–7), along with the DDP minimum p -value statistic for N = 100. The standard error is at most 0 . 011. The score per partition was the likelihood ratio test statitsic. The DDP minimum p -value statistic performs very well in comparison to the other tests. Although the competitors may have greater power in some examples, the advantage is usually small. By far the best competitor is HHG, yet it has a disadvantage when the relationship is more complex, thus benefiting from the finer partition of the minimum p -value test, especially in the Sine, Heavisine, Spiral and Circles examples. The highest power per row is underlined. Table 3: The power of different variants aggregated by maximization (columns 3– 6), along with the DDP minimum p -value statistic (column 2) for N = 100. The standard error is at most 0 . 011. The score per partition was the likelihood ratio test statitsic. Although maximization is better than summation in some examples, the advantage is usually small. The advantage of the minimum p -value statistic, which is based on aggregation by summation, is large in the Cubic, Cross, Spiral, and Circles relationships. In most examples, power increases with m . The power differences between the ADP and DDP variants are small. We present only the maximum variants that take at most O ( N $^{3}$) to compute, therefore for ADP only results with m = 2 are presented. The highest power per row is underlined. Setup min$_{{}$$_{m}$$_{∈}$$_{2}$$_{,...,}$$_{10}$$_{}}$ p$_{m}$ Spearman Hoeffding MIC dCov HHG W 0.655 0.000 0.414 0.526 0.361 0.798 Diamond 0.919 0.013 0.116 0.074 0.074 0.965 Parabola 0.847 0.028 0.413 0.211 0.386 0.784 2Parabolas 0.844 0.095 0.135 0.048 0.124 0.723 Circle 0.886 0.000 0.033 0.046 0.002 0.850 Cubic 0.731 0.344 0.655 0.515 0.627 0.768 Sine 0.995 0.368 0.494 1.000 0.415 0.804 Wedge 0.595 0.064 0.360 0.303 0.338 0.673 Cross 0.704 0.089 0.160 0.069 0.130 0.706 Spiral 0.949 0.112 0.141 0.251 0.140 0.337 Circles 0.999 0.048 0.084 0.093 0.061 0.354 Heavisine 0.710 0.396 0.493 0.532 0.492 0.585 Doppler 0.949 0.513 0.784 0.975 0.744 0.912 5Clouds 0.996 0.000 0.000 0.561 0.004 0.904 4Clouds 0.051 0.050 0.057 0.050 0.051 0.050 Setup min$_{{}$$_{m}$$_{∈}$$_{2}$$_{,...,}$$_{10}$$_{}}$ p$_{m}$ M DDP 2 × 2 M DDP 3 × 3 M DDP 4 × 4 M ADP 2 × 2 W 0.655 0.190 0.637 0.574 0.155 Diamond 0.919 0.272 0.931 0.912 0.247 Parabola 0.847 0.533 0.855 0.803 0.597 2Parabolas 0.844 0.466 0.907 0.897 0.578 Circle 0.886 0.222 0.880 0.884 0.170 Cubic 0.731 0.496 0.654 0.653 0.496 Sine 0.995 0.768 0.958 0.998 0.774 Wedge 0.595 0.410 0.536 0.478 0.455 Cross 0.704 0.268 0.680 0.673 0.341 Spiral 0.949 0.116 0.489 0.764 0.189 Circles 0.999 0.085 0.606 0.844 0.088 Heavisine 0.710 0.519 0.642 0.692 0.534 Doppler 0.949 0.828 0.969 0.977 0.833 5Clouds 0.996 0.062 0.999 0.999 0.076 4Clouds 0.051 0.052 0.051 0.051 0.052