In Chapter 5, we considered how best to account for data from multiple participants in our modeling. We proposed two solutions: At one end of the spectrum, fitting a model to individual participants avoids the averaging artifacts that we discussed in Section 5.2. At the other end of the spectrum, fitting aggregate data runs the risk of introducing artifacts, but it takes advantage of the stability introduced by averaging data. There is, however, a third approach that we are now ready to discuss. This approach is known as hierarchical or multilevel modeling (the terms are typically used interchangeably), and like fitting of aggregate data, it takes into account the data from all participants simultaneously. However, unlike fits of individual participants, we do not consider the data from different participants independently; instead, hierarchical models postulate – and exploit – some degree of dependence between participants. This chapter first conceptualizes hierarchical models at a general level before presenting three examples of Bayesian hierarchical models of cognition in some detail. We then briefly touch on maximum-likelihood techniques for hierarchical models before concluding with some recommendations.

Conceptualizing Hierarchical Modeling

The key aspect of a hierarchical model is that although it recognizes individual variation, it also assumes that there is an orderly distribution governing this variation. This distribution across individuals is frequently known as the parent distribution. The parent distribution characterizes the distribution of the parameters that determine the priors for each individual. For that reason, the parent distribution is also sometimes known as a “hyperprior distribution” (Gelman et al., 2013) because it determines the priors for each individual. A hierarchical model therefore always embodies a theory of individual differences, however rudimentary.

Each participant's performance is described by whatever model is being considered – in principle, any cognitive model can be instantiated as a hierarchical model. When the model is fitted to data, the individual parameters are estimated for each subject together with the parameters of the parent distribution. The latter are sometimes called hyperhyperparameters (Gelman, 2006), although we find it simpler and more intuitive to refer to them as “parameters of the parent distribution.”

The hierarchical approach necessarily creates a tension between fitting each participant as well as possible (by estimating optimal individual parameters) and fitting the group of participants as a whole (by seeking to minimize the variance of the parent distribution).