Let H I U and H I V be the fixed marginal entropies of the partition I : where C$_{x}$ ( I ) and C$_{y}$ ( I ) are the intervals induced by I in x and in y , respectively. Note that given I , the observed and expected margins of the partitions are fixed, and therefore The following simple derivation shows that the likelihood ratio score T I is a linear combination of ˆ H I $_{UV}$,H I U and H I $_{V}$: where the last equality follows from equations (B.2) and (B.4). 55 T I = ∑ C ∈C ( I ) o$_{C}$ log o$_{C}$ e$_{C}$ = ∑ C ∈C ( I ) o$_{C}$ log o$_{C}$ N − ∑ C ∈C ( I ) o$_{C}$ log e$_{C}$ N = − N ˆ H I UV − ∑ C ∈C ( I ) o$_{C}$ log$^{(}$ N r$_{h}$ ( C ) − r$_{l}$ ( C ) N s$_{h}$ ( C ) − s$_{l}$ ( C ) N 1 N$^{)}$ = − N ˆ H I UV + NH I U + NH I $_{V}$, H I U = − ∑ C ∈C ( I ) o$_{C}$ log$^{(}$ r$_{h}$ ( C ) − r$_{l}$ ( C ) N ) (B.2) = − ∑ C ∈C ( I ) Pr ( r$_{l}$ ( C ) ≤ U ≤ r$_{h}$ ( C ) ,s$_{l}$ ( C ) ≤ V ≤ s$_{h}$ ( C )) log$^{(}$ r$_{h}$ ( C ) − r$_{l}$ ( C ) N ) (B.3) H I V = − ∑ C ∈C ( I ) o$_{C}$ log$^{(}$ s$_{h}$ ( C ) − s$_{l}$ ( C ) N ) (B.4) H I U = − ∑ C$_{x}$ ∈C$_{x}$ ( I ) r$_{h}$ ( C$_{x}$ ) − r$_{l}$ ( C$_{x}$ ) N log$^{(}$ r$_{h}$ ( C$_{x}$ ) − r$_{l}$ ( C$_{x}$ ) N ) , H I V = − ∑ C$_{y}$ ∈C$_{y}$ ( I ) r$_{h}$ ( C$_{y}$ ) − r$_{l}$ ( C$_{y}$ ) N log$^{(}$ r$_{h}$ ( C$_{y}$ ) − r$_{l}$ ( C$_{y}$ ) N ) , = − ∑ C ∈C ( I ) Pr ( r$_{l}$ ( C ) ≤ U ≤ r$_{h}$ ( C ) ,s$_{l}$ ( C ) ≤ V ≤ s$_{h}$ ( C )) log$^{(}$ s$_{h}$ ( C ) − s$_{l}$ ( C ) N ) . (B.5)