If the randomized treatment assignment Z replaces the speciGLYPH<133>c treatment assignment z , then needed parts of R are observed, and T$_{Z}$ is a statistic that can be computed from the data. Indeed, if B = 1 , k = 2 , and w$_{1}$ = 1 , then T$_{Z}$ is the Mann-Whitney U-statistic and is linearly related to WilcoxonGLYPH<146>s rank sum statistic. More generally, for k = 2 and B > 2 , T$_{Z}$ is a weighted sum of B Mann-Whitney statistics; see Lehmann (1975, §3.3) and Puri (1965) who discuss weights intended to increase power against shift alternatives in The set S b z compares one treated unit to k GLYPH<0> 1 control units. Write v ( S b $_{z}$) = 1 if the treated unit, say i$_{1}$ , in S b z = f i$_{1}$; : : : ; i$_{k}$ g has the largest response under assignment z , that is, if r$_{bi}$$_{1}$$_{z}$ > $^{max$_{j}$}$2f i$_{2}$$^{;:::;i$_{k}$}$g r$_{bj}$$_{z}$ , and write v ( S b $_{z}$) = 0 otherwise. For k = 2 , the set S b z = f i$_{1}$; i$_{2}$ g has one treated unit and one control and v ( S b $_{z}$) = 1 if the treated unit has a higher response than the control under assignment z . Also, let w$_{b}$ be a weight to be attached to block b where w$_{b}$ is a function of n and m . In the current paper, w$_{b}$ = 1 for all b , but another reasonable deGLYPH<133>nition of w$_{b}$ will be given in a moment. For this treatment assignment z , the quantity T$_{z}$ = P B b $_{=1}$w$_{b}$ P S b z 2K b z v ( S b $_{z}$) is a weighted count of the number of sets S b z such that the treated unit had a higher response than k GLYPH<0> 1 controls. With w$_{b}$ = 1 , the quantity T$_{z}$ is a count, and a count is reasonable if the N$_{b}$ and n$_{b}$ do not vary much, as is true in §1.3. If the n$_{b}$ and m$_{b}$ varied greatly with b , then w$_{b}$ given by 1 =w$_{b}$ = B n$_{b}$ GLYPH<0> m$_{b}$ $^{k}$GLYPH<0> 1 GLYPH<1> makes T$_{z}$ the unweighted average over the B blocks, b = 1 ; : : : ; B , of the proportion of sets S b z 2 K b z in which the treated unit had a higher response than k GLYPH<0> 1 controls, v ( S b $_{z}$) = 1 . For most z , the quantity T$_{z}$ depends upon parts of R that are not observed, so T$_{z}$ cannot be computed from the observed data. For a speciGLYPH<133>c treatment assignment, z 2 GLYPH<10> , consider a subset S b z = f i$_{1}$; : : : ; i$_{k}$ g of k units from the same block b with one treated unit, z$_{bi}$$_{1}$ = 1 , and k GLYPH<0> 1 control units, z$_{bi}$$_{j}$ = 0 , j = 2 ; : : : ; k . Write K b z for the collection of all n$_{b}$ GLYPH<0> m$_{b}$ $^{k}$GLYPH<0> 1 GLYPH<1> such subsets S b z for block b , so S b z 2 K b z if and only if S b z GLYPH<18> f 1 ; : : : ; N$_{b}$ g with jS b z j = k and 1 = P $^{i}$2S b z z$_{bi}$ . The set compares one treated unit to k 1 control units. Write v ( $_{z}$) = 1 if 16