An introduction to Bayesian hierarchical models with an application in the theory of signal detection.

Although many nonlinear models of cognition have been proposed in the past 50 years, there has been little consideration of corresponding statistical techniques for their analysis. In analyses with nonlinear models, unmodeled variability from the selection of items or participants may lead to asymptotically biased estimation. This asymptotic bias, in turn, renders inference problematic. We show, for example, that a signal detection analysis of recognition memory data leads to asymptotic underestimation of sensitivity. To eliminate asymptotic bias, we advocate hierarchical models in which participant variability, item variability, and measurement error are modeled simultaneously. By accounting for multiple sources of variability, hierarchical models yield consistent and accurate estimates of participant and item effects in recognition memory. This article is written in tutorial format; we provide an introduction to Bayesian statistics, hierarchical modeling, and Markov chain Monte Carlo computational techniques.