Bias by Variance: How Common Parameter Transformations in Hierarchical Models Distort Group-Level Estimates

Hierarchical models are widely used to estimate group- and individual-level parameters in cognitive and behavioral research. When model parameters are bounded within a particular range, it is common practice to estimate the group-level parameters on the unbounded real line and then map the estimated mean onto the desired bounded parameter range using a nonlinear transformation. We show that the way group-level estimates are commonly computed with this approach can produce systematic biases because it ignores within-group variability. We derive correct expressions for computing the group-level mean that take within-group variability into account. Using formal derivation, simulation, and reanalyses of empirical data, we demonstrate (using cumulative prospect theory as an example) how the commonly used but incorrect approach distorts parameter estimates. In a reanalysis of age differences in risky choice, we find that although the distortions do not alter qualitative conclusions about group differences, they can affect the size of such differences and lead to different psychological interpretations of the groups' behavior. We point out and elaborate that the underlying issue is general and can arise in a wide range of hierarchical models. Our results highlight a subtle but consequential issue in hierarchical modeling and we provide concrete solutions for ensuring the validity of group-level inferences about behavior.