How hierarchical models improve point estimates of model parameters at the individual level

Computational models have been used to analyze the data from behavioral experiments. One objective of the use of computational models is to estimate model parameters or internal variables for individual subjects from behavioral data. The estimates are often correlated with other variables that characterize subjects in order to investigate which computational processes are associated with specific personal or physiological traits. Although the accuracy of the estimates is important for these purposes, the parameter estimates obtained from individual subject data are often unreliable. To solve this problem, researchers have begun to use hierarchical modeling approaches to estimate parameters of computational models from multiple-subject data. It is widely accepted that the hierarchical model provides reliable estimates compared to other non-hierarchical approaches. However, how and under what conditions the hierarchical models provide better estimates than other approaches has yet to be systematically investigated. This study attempts to investigate these issues, focusing on two measures of estimation accuracy: the correlation between estimates of individual parameters and subject trait variables and the absolute measures of error (root mean squared error, RMSE) of the estimates. An analytical calculation based on a simple Gaussian model clarifies how the hierarchical model improves the point estimates of these two measures. We also performed simulation studies employing several realistic computational models based on the synthesized data to confirm that the theoretical properties hold in realistic situations.