Extending Goldberg’s bass-ackwards method for delineating hierarchical structural models in individual differences data

Goldberg’s (2006) bass-ackwards 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 on the first level of the hierarchy, two components on the second level, and so on, treating the correlations between the components on adjoining levels akin to path coefficients in a hierarchical structure. This article proposes three modifications to the current approach with a particular focus on examining associations among all levels of the hierarchy: 1) identify and remove redundant components that perpetuate through multiple levels of the hierarchy; 2) (optionally) identify and remove artefactual components; and 3) plot the strongest component correlations among the remaining components 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 then applied in real data with specific methodological recommendations. The results are compared to the traditional bass-ackwards approach, and basic code is provided to apply the proposed modifications in other data.