The attributable e⁄ect A$_{Z}$ depends upon the sample sizes, n and m , and the choice of k . Dividing A$_{Z}$ by E GLYPH<16> e T$_{Z}$ GLYPH<17> can aid interpretation. Then 100 GLYPH<2> A$_{Z}$= E GLYPH<16> e T$_{Z}$ GLYPH<17> is the (weighted) percent increase above chance in the number of times a treated unit had a higher response than k GLYPH<0> 1 controls due to e⁄ects caused by the treatment, and with conGLYPH<133>dence 1 GLYPH<0> GLYPH<11> , the unobserved 100 GLYPH<2> A$_{Z}$= E GLYPH<16> T$_{Z}$ GLYPH<17> is at least 100 GLYPH<2> GLYPH<0> T$_{Z}$ GLYPH<0> t$_{GLYPH<11>}$ GLYPH<1> = E GLYPH<16> T$_{Z}$ GLYPH<17> . We are suggesting that the unobservable random variable A$_{Z}$= E T$_{Z}$ is a useful mea- 

e e e GLYPH<16> e GLYPH<17> sure of the magnitude of the treatment e⁄ect when interference may be present; however, it is a new measure, and its magnitude is unfamiliar. To build some intuition about mag- nitudes of A$_{Z}$= E GLYPH<16> e T$_{Z}$ GLYPH<17> , consider its behavior in a familiar context, namely a single block, B = 1 , independent observations without interference and a treatment e⁄ect that is an additive shift, GLYPH<14> . In this case, as n$_{1}$ ! 1 and m$_{1}$ ! 1 , the quantity T$_{Z}$= n n$_{1}$ GLYPH<0> m$_{1}$ $^{k}$GLYPH<0> 1 GLYPH<1> o converges in probability to the probability that a treated response exceeds k GLYPH<0> 1 control responses and A$_{Z}$= E GLYPH<16> e T$_{Z}$ GLYPH<17> converges in probability the percent increase in this probability above the chance level of 1 =k . Table 1 evaluates these limits for the standard Normal distribution and the t -distribution with 2 degrees of freedom. For instance, with a shift GLYPH<14> in a Normal that equals a full standard deviation, GLYPH<14> = 1 , the probability that a treated response exceeds nine control responses in 0.341 which is 241% above the chance level of 0.1 for GLYPH<14> = 1 . The quantity A$_{Z}$= E GLYPH<16> e T$_{Z}$ GLYPH<17> has the advantage that it continues to be meaningful with interference where shift models are inapplicable. 

Proposition 1 refers to an analysis of responses, but it is possible in a randomized exper- iment to use the same approach with residuals from a robust covariance adjustment which controls for measured disturbing covariates such as head motion. See Rosenbaum (2002) for general discussion of randomization inference for covariance adjustment in randomized experiments, and see Rosenbaum (2007a, §6) for its application with interference. This procedure is illustrated in §4 in Table 4. 

