Improving hierarchical models of individual differences: An extension of Goldberg’s bass-ackward method

Goldberg’s (2006) bass-ackward approach to elucidating the hierarchical structure of individual differences data has been used widely to improve our understanding of the relationships among constructs of varying levels of granularity. The traditional approach has been to extract a single component or factor on the first level of the hierarchy, two on the second level, and so on, treating the correlations between adjoining levels akin to path coefficients in a hierarchical structure. This article proposes three modifications to the traditional approach with a particular focus on examining associations among all levels of the hierarchy: 1) identify and remove redundant elements that perpetuate through multiple levels of the hierarchy; 2) (optionally) identify and remove artefactual elements; and 3) plot the strongest correlations among the remaining elements to identify their hierarchical associations. Together these steps can offer a simpler and more complete picture of the underlying hierarchical structure among a set of observed variables. The rationale for each step is described, illustrated in a hypothetical example and three basic simulations, and then applied in real data. The results are compared to the traditional bass-ackward approach together with agglomerative hierarchical cluster analysis, and a basic tutorial with code is provided to apply the extended bass-ackward approach in other data.