uniformity trial was not performed and none of the e r$_{bi}$ are observed. Nonetheless, in the absence of ties, the distribution of e T$_{Z}$ is known, because the null distribution in §3.1 depends upon n and m but not on R ; speciGLYPH<133>cally, it is the convolution of B random variables whose exact distributions are given by Orban and Wolfe (1982, Theorem 2.1, and expressions (2.1) and (2.2)), with expectation and variance E GLYPH<16> e T$_{Z}$ GLYPH<17> = B X b =1 n$_{b}$ GLYPH<30>$_{n}$$_{b}$$_{;m}$$_{b}$ where GLYPH<30>$_{n}$$_{b}$$_{;m}$$_{b}$ = 1 m$_{b}$ + 1 m$_{b}$ X j =0 w$_{b}$ GLYPH<18> j k GLYPH<0> 1 GLYPH<19> ; var GLYPH<16> e T$_{Z}$ GLYPH<17> = B X b =1 n$_{b}$ ( n$_{b}$ + m$_{b}$ + 1) ( m$_{b}$ + 1) ( m$_{b}$ + 2) 2 4 8 < : m$_{b}$ X j =0 w 2 b GLYPH<18> j k GLYPH<0> 1 GLYPH<19> 2 9 = ; GLYPH<0> ( m$_{b}$ + 1) GLYPH<30> 2 n$_{b}$;m$_{b}$ 3 5 ; moreover, for reasonable choices of weights, w$_{b}$ , as B ! 1 , n e T$_{Z}$ GLYPH<0> E GLYPH<16> e T$_{Z}$ GLYPH<17>o = r var GLYPH<16> e T$_{Z}$ GLYPH<17> D ! GLYPH<8> ( GLYPH<1> ) (2) where GLYPH<8> ( GLYPH<1> ) is the standard Normal cumulative distribution. To emphasize, because the null hypothesis of no e⁄ect is known to be true in the uniformity experiment, and because the null distribution of e T$_{Z}$ depends upon n and m but not on the e r$_{bi}$ GLYPH<146>s, it follows that we know the distribution of e T$_{Z}$ in the uniformity trial even though we did not perform the uniformity trial and even if the treatment did have an e⁄ect with interference in the actual randomized experiment. This fact turns out to be useful with the aid of the concept of attributable e⁄ects (Rosenbaum 2001, 2007a, 2007b). Consider a speciGLYPH<133>c treatment assignment z 2 GLYPH<10> and the speciGLYPH<133>c comparison S b z = f i$_{1}$; : : : ; i$_{k}$ g 2 K b z of one treated unit, say i$_{1}$ with z$_{bi}$$_{1}$ = 1 , and k GLYPH<0> 1 control units, i$_{j}$ with z$_{bi}$$_{j}$ = 0 for j = 2 ; : : : ; k . If r$_{bi}$$_{1}$$_{z}$ > $^{max$_{j}$}$2f i$_{2}$$^{;:::;i$_{k}$}$g r$_{bj}$$_{z}$ then i$_{1}$ had the highest response in this compar- 19 3.3 Attributable e⁄ects