the absence of interference. For k GLYPH<21> 2 , if n$_{b}$ = 1 and N$_{b}$ = N=B does not vary with b , then T$_{Z}$ is the statistic discussed in Rosenbaum (2007b). Taking k > 2 tends to increase power when only a subset of treated units respond to treatment, as seems likely here for reasons discussed in §1.3. Indeed, with k > 2 , the ranks are scored in a manner that closely approximates Conover and SalsburgGLYPH<146>s (1988) locally most powerful ranks for an alternative in which only a fraction of treated units respond, and the scores are identical to those proposed by Stephenson (1981). In these special cases of the two previous paragraphs, T$_{Z}$ is a stratiGLYPH<133>ed linear rank sta- tistic. In general, T$_{Z}$ is a function of the ranks, but not a linear function; however, it is a sum of B linear functions of the placements within blocks in the sense of Orban and Wolfe (1982). For B = 1 , the statistic with k GLYPH<21> 2 has been discussed by Deshpande and Kochar (1980) and Stephenson and Gosh (1985) as an instance of Hoe⁄dingGLYPH<146>s (1948) U-statistics under independent sampling of two distributions. Because interference precludes indepen- dent observations, inferences must be based on the random assignment of treatments, and for this the combinatorial development in Orban and Wolfe (1982) is particularly helpful. where GLYPH<0> ‘ $^{k}$GLYPH<0> 1 GLYPH<1> is deGLYPH<133>ned to equal zero for ‘ < k GLYPH<0> 1 . Consider testing FisherGLYPH<146>s null hypothesis H$_{0}$ of no e⁄ect which asserts that r$_{bi}$$_{z}$ = r$_{bi}$$_{z}$ 0 for all z , z 0 2 GLYPH<10> and all b; i . If H$_{0}$ were true, then the observed response r$_{bi}$$_{Z}$ is the Orban and Wolfe (1982) deGLYPH<133>ne the placement m$_{b}$U$_{bj}$ of the j th treated unit in block b to be the number of controls in block b who have a response less than or equal to the re- sponse of this treated unit. A linear placement statistic for one block b is then of the form P N$_{b}$ i $_{=1}$Z$_{bi}$ GLYPH<30>$_{n}$$_{b}$$_{;m}$$_{b}$ ( U$_{bj}$ ) for some function GLYPH<30>$_{n}$$_{b}$$_{;m}$$_{b}$ ( GLYPH<1> ) . If there are no ties among responses within blocks, then taking GLYPH<30>$_{n}$$_{b}$$_{;m}$$_{b}$ ( u ) = w$_{b}$ GLYPH<0> m$_{b}$ u $^{k}$GLYPH<0> 1 expresses T$_{Z}$ = P B b $_{=1}$w$_{b}$ P S b Z 2K b z v ( S b $_{Z}$) as a sum of linear functions of placements of the treated units, T$_{Z}$ = B b =1 n$_{b}$ j $_{=1}$GLYPH<30>$_{n}$$_{b}$$_{;m}$$_{b}$ ( U$_{bj}$ ) , 17