ison, contributing a 1 rather than a 0 to T$_{z}$ ; however, this might or might not be an e⁄ect caused by the treatment, because even under the null hypothesis of no e⁄ect H$_{0}$ , one of the k units will have the highest response among the k units. If r$_{bi}$$_{1}$$_{z}$ > $^{max$_{j}$}$2f i$_{2}$$^{;:::;i$_{k}$}$g r$_{bj}$$_{z}$ but e r$_{bi}$$_{1}$ GLYPH<20> $^{max$_{j}$}$2f i$_{2}$$^{;:::;i$_{k}$}$g e r$_{bj}$ then treatment assignment z in the actual experiment does cause unit i$_{1}$ in block b to have a higher response than units f i$_{2}$; : : : ; i$_{k}$ g in block b in the sense that unit i$_{1}$ in block b would not have had the highest response in this comparison in the uniformity trial of §2.2 in which no unit was treated. In §1.3, this would mean that in block b , stop trial i$_{1}$ caused activity in the STN region to exceed the level in go trials i$_{2}$; : : : ; i$_{k}$ in the sense that the activity was higher in the actual experiment and would not have been higher in the uniformity trial. Conversely, if r$_{bi}$$_{1}$$_{z}$ GLYPH<20> $^{max$_{j}$}$2f i$_{2}$$^{;:::;i$_{k}$}$g r$_{bj}$$_{z}$ but e r$_{bi}$$_{1}$ > $^{max$_{j}$}$2f i$_{2}$$^{;:::;i$_{k}$}$g e r$_{bj}$ then treatment assignment z in the actual experiment prevented treated unit i$_{1}$ from having the highest response in S b $_{z}$, in the sense that i$_{1}$ would have had the highest response in the uniformity trial but did not have the highest response in the actual experiment. The third possibility is that treatment assignment z does not alter whether or not i$_{1}$ has the highest response in S b $_{z}$. Concisely, these three situations are: (i) v ( S b $_{z}$) = 1 and e v ( S b $_{z}$) = 0 , (ii) v ( S b $_{z}$) = 0 and e v ( S b $_{z}$) = 1 , and (iii) v ( S b $_{z}$) = e v ( S b $_{z}$) . For treatment assignment z 2 GLYPH<10> , the attributable e⁄ect A$_{z}$ = T$_{z}$ GLYPH<0> e T$_{z}$ = B X b =1 w$_{b}$ X S b z 2K b z f v ( S b $_{z}$) GLYPH<0> e v ( S b $_{z}$) g is the net increase in the number of times (weighted by w$_{b}$ ) that a treated response in the actual experiment exceeded k GLYPH<0> 1 control responses because of e⁄ects caused by using treatment assignment z . So A$_{z}$ is a real valued function of z , e r and R . In contrast, A$_{Z}$ is the attributable e⁄ect for the Z randomly chosen according to (1), so A$_{Z}$ is the di⁄erence between an observed statistic, T$_{Z}$ , that describes the actual experiment, and an 20