cell C , For a given partition I , the score is T I = ∑ C ∈C ( I $_{)}$t$_{C}$ (where if t$_{C}$ = o$_{C}$ log o$_{C}$ e$_{C}$ then T I is the likelihood ratio given the partition). As above, we consider as test statistics aggregation by summation and by maximization: We consider another test statistic based on DDP, where each set of m − 1 observed points in their turn define a partition (see Figure 1). This variant has a computational advantage over the ADP statistic for m < 5, see Remark 3.1. Since all observations have unique values, the remaining N − ( m − 1) points are inside cells defined by the partition. There are ( N m − 1 ) partitions, denote this set of partitions by Π DDP m . As before, since the cell membership of observations is the same regardless of whether the partition is defined on the original observations or on the ranked observations, and the statistics we suggest only depend on these cell memberships, we describe the proposed test statistics on the ranked observations. For a cell C ∈ C ( I ), where I ∈ Π DDP m , the boundaries of C are not necessarily defined by two sample points, as depicted at the bottom right panel of Figure 1. We refer to r$_{l}$ and r$_{h}$ as the lower and upper values of the ranks of X in C , and to s$_{l}$ and s$_{h}$ as the lower and upper values of the ranks of Y in C , where r$_{l}$, r$_{h}$, s$_{l}$, s$_{h}$ ∈ { 1 , . . . , N } . Let o$_{C}$ and e$_{C}$ be the observed and expected counts strictly inside the cell, respectively. The expected count in cell C with rank range [ r$_{l}$, r$_{h}$ ] × [ s$_{l}$, s$_{h}$ ] is e$_{C}$ = ( r$_{h}$ − r$_{l}$ − 1)( s$_{h}$ − s$_{l}$ − 1) / [ N − ( m − 1)]. We consider Pearson’s score or the likelihood ratio score for a given cell C , and define t$_{C}$ as in (2.3). For a given partition I , the score is T I = ∑ C ∈C ( I $_{)}$t$_{C}$ , and similarly to (2.4) we define 12 S ADP m × m = ∑ I∈ Π ADP m T $^{I}$, M ADP m × m = max I∈ Π ADP m T $^{I}$. (2.4) t$_{C}$ ∈ $^{{}$( o$_{C}$ − e$_{C}$ ) 2 e$_{C}$ , o$_{C}$ log o$_{C}$ e$_{C}$$^{}}$ . (2.3)