1 Introduction: An Application of Inference with Interference 1.1 What is interference between units? If treatment e⁄ects are deGLYPH<133>ned as comparisons of the two potential responses that an ex- perimental unit would exhibit under treatment or under control (Neyman 1923, Welch 1937, Rubin 1974, Lindquist and Sobel 2011), then an implicit premise of this deGLYPH<133>nition is GLYPH<147>no interference between units,GLYPH<148>as discussed by Cox (1958, p. 19): GLYPH<147>There is no GLYPH<145>interfer- enceGLYPH<146>between di⁄erent units if the observation on one unit [is] una⁄ected by the particular assignment of treatments to the other units;GLYPH<148> see also Rubin (1986). For instance, wide- spread use of a vaccine may beneGLYPH<133>t unvaccinated individuals because they are less likely to encounter an infected individual, a form of interference known as herd immunity; see Hudgens and Halloran (2008). In agriculture, the treatment applied to one plot may also a⁄ect adjacent plots; see David and Kempton (1996). In social experiments, people talk, and changing the treatment applied to one person may change what she says to someone else, altering his response to treatment; see Sobel (2006). In some contexts, interference is of central interest in itself GLYPH<151> this can be true of herd immunity or of social interaction, for example GLYPH<151> but in many if not most contexts, interference is principally an inconvenience, depriving us of both independent observations and a familiar deGLYPH<133>nition of treatment e⁄ects. We apply and extend a recent, general approach to inference with interference (Rosenbaum 2007a) in the context of a cognitive neuroscience experiment in which the brains of a moderate number of subjects are studied using fMRI while faced with a rapid GLYPH<133>re sequence of randomized stimuli. In this context, interference is likely to be widespread and di¢ cult to model with precision. The goal is a simple, sturdy, valid method of inference whose conclusions about the magnitude of treatment e⁄ects are intelligible when the interference may be complex in form. 2