## 2 Notation: The Randomized Trial and the Uniformity Trial 

## 2.1 Blocked randomized trial with interference between units 

There are B GLYPH<21> 1 blocks, b = 1 ; : : : ; B , and N$_{b}$ GLYPH<21> 2 units bi in block b , i = 1 ; : : : ; N$_{b}$ , with N = P N$_{b}$ units in total. In §1, there are B = 58 GLYPH<2> 4 = 232 blocks consisting of the four sessions for each of 58 subjects, N = 22 ; 440 units or trials in total, with 87 GLYPH<20> N$_{b}$ GLYPH<20> 104 and a median N$_{b}$ of 97. In block b , n$_{b}$ units are picked at random for treatment, 1 GLYPH<20> n$_{b}$ < N$_{b}$ , the remaining m$_{b}$ = N$_{b}$ GLYPH<0> n$_{b}$ GLYPH<21> 1 units receive control. In §1, the probability that a unit was a GLYPH<147>stop trialGLYPH<148> was 1 $_{4}$, resulting in 13 GLYPH<20> n$_{b}$ GLYPH<20> 37 and a median n$_{b}$ of 24. If unit i in block b was assigned to treatment, write Z$_{bi}$ = 1 , and if this unit was assigned to control write Z$_{bi}$ = 0 , and let Z = ( Z$_{11}$; Z$_{12}$; : : : ; Z$_{B;N}$$_{B}$ ) T be the N -dimensional vector in the lexical order. For a GLYPH<133>nite set A , write j A j for the number of elements of A . Write GLYPH<10> for the set containing the j GLYPH<10> j = Q B b =1 GLYPH<0> N$_{b}$ n$_{b}$ GLYPH<1> possible values z of Z , so z 2 GLYPH<10> if and only if z$_{bi}$ = 0 or z$_{bi}$ = 1 and n$_{b}$ = P N$_{b}$ i $_{=1}$z$_{bi}$ for b = 1 ; : : : ; B . Write n = ( n$_{1}$; : : : ; n$_{B}$ ) T and m = ( m$_{1}$; : : : ; m$_{B}$ ) $^{T}$. 

Write r$_{bi}$$_{z}$ for the response that the i th unit in block b would have if the treatment assignment Z equalled z for z 2 GLYPH<10> . In §1.3, for trial i of subject/session b , the response r$_{bi}$$_{z}$ is the HRF weighting of either the unGLYPH<133>ltered or GLYPH<133>ltered activity in the subthalamic nucleus (STN). Each unit has j GLYPH<10> j potential responses, only one of which is observed, namely r$_{bi}$$_{Z}$ . Figure 2 plots r$_{bi}$$_{Z}$ for Z$_{bi}$ = 0 and Z$_{bi}$ = 1 for one b and i = 1 ; : : : ; N$_{b}$ . 

Unlike the notation of Neyman (1923) and Rubin (1974), the response of the i th unit in block b may depend on the treatments Z assigned to all the units; that is, this notation permits interference (Rosenbaum 2007a). In §1, it is quite plausible that a previous treatment for one subject may a⁄ect later responses of this same subject. Indeed, it is possible that interference extends across the four blocks or sessions for a given subject. 

