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. 

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$_{bj}$ ) , 

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 

