Hierarchical Factor Models For Analysis in Experimental Designs

Although factor models have been pivotal in understanding the relations among variables, they are difficult to apply to data from experimental designs. In experiments, data are from trials which are nested in conditions, tasks, and individuals. Data at the trial level is quite noisy, so much so that even averages on critical contrasts are subject to excessive measurement error. In this paper, we develop hierarchical factor models where the first level are linear models that account for trial noise and are parameterized with separate, latent individual-by-task scores. The second level consists of factor models on latent individual-by-task scores. Because the model is hierarchical, Bayesian analysis is convenient and conceptually straightforward. We provide development with good computational properties. The advatages of the model are that (a) correlations across tasks may be both disattenuated and localized fairly precisely; (b) uncertainty from both variability in individuals and across trials may be accurately assessed; and (c) factor structures may be recovered even in high-measurement-error contexts. Applications to visual illusions and to cognitive control are included.