compared, because a broader range of validity is being weighed against possibly reduced power. In contrast, using the randomization distribution of T$_{Z}$ to test H$_{0}$ , the test is valid, with correct level, for both rapid-GLYPH<133>re designs with many trials and widely-spaced designs with fewer trials, and a comparison of power is possible. For instance, a smaller number of trials with more successful trials and less interference ( N = 250 , GLYPH<21> = : 5 , case A in Table 3) yields greater power than more trials with fewer successful trials and more interference ( N = 1000 , GLYPH<21> = : 1 , case D in Table 4), so it is clear that increasing the number of trials must be weighed against potential harms from increasing the pace at which trials are conducted. 

## 6 Summary 

Randomized experiments in cognitive neuroscience of the type described in §1.3 have three attributes that were important in the current discussion. First, with about 100 random- ized stimuli for a single brain in a session of 600 seconds, interference is likely: the stimulus applied in one trial is likely to a⁄ect the response measured for other trials. Interference that is local in time is almost inevitable because the measurable response to one stimu- lus lasts for more than six seconds, but additionally as the trial progresses a subject is growing more familiar and experienced with the tasks and equipment, so interference may have a complex form that can extend across di⁄erent sessions for the same subject. The use of the HRF function in passing from Figure 1 to Figure 2 is a standard attempt to pick out the response to a particular stimulus, and useful though this is, it is at best an approximation. Second, with rapid GLYPH<133>re trials of this sort, not every trial will be successful in eliciting the intended cognitive activity. This is quite evident in the experiment in §1.3, because subjects responded inappropriately to some go or stop trials, but inattention, distraction or confusion can also occur without visible evidence. Some exposures to a 

