Throughout social and cognitive psychology, participants are routinely asked to respond in some way to experimental stimuli that are thought to represent categories of theoretical interest. For instance, in measures of implicit attitudes, participants are primed with pictures of specific African American and White stimulus persons sampled in some way from possible stimuli that might have been used. Yet seldom is the sampling of stimuli taken into account in the analysis of the resulting data, in spite of numerous warnings about the perils of ignoring stimulus variation (Clark, 1973; Kenny, 1985; Wells & Windschitl, 1999). Part of this failure to attend to stimulus variation is due to the demands imposed by traditional analysis of variance procedures for the analysis of data when both participants and stimuli are treated as random factors. In this article, we present a comprehensive solution using mixed models for the analysis of data with crossed random factors (e.g., participants and stimuli). We show the substantial biases inherent in analyses that ignore one or the other of the random factors, and we illustrate the substantial advantages of the mixed models approach with both hypothetical and actual, well-known data sets in social psychology (Bem, 2011; Blair, Chapleau, & Judd, 2005; Correll, Park, Judd, & Wittenbrink, 2002).

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