A stepwise principal component regression method for localizing effects and individual differences in hierarchical structures

Psychological studies often seek to test the degrees of associations between several independent variables and a dependent variable. In many instances, the psychological constructs measured by independent variables form the foundation of a hierarchical structure, where they are linked to one or more levels of higher-order latent factors. In the presence of hierarchical structures, the most commonly used statistical method to test association effects – multiple regression – could encounter problems. Collinearity between independent variables can distort the associations being tested. In some cases, the dependent variable is associated with a higher-order latent factor rather than the independent variables themselves, which the multiple regression method cannot localize. Multiple regression results could also be highly inconsistent across different replications. In this study, we used large-scale simulations to demonstrate that a stepwise principal component regression (step-PCR) method can mitigate these problems. In the step-PCR, we first use a parallel analysis to identify the maximum number of factors N in independent variables, then perform model selection across 1- to N-component PCRs to identify the optimal model. The optimal model reflects association effects and their locations in the hierarchical structure. Bayesian Information Criterion and Bayes Factor were identified as optimal methods for model selection. Simulations showed that the step-PCR could accurately identify and localize the associations between independent variables and the dependent variable. Additionally, results of the step-PCR were highly consistent across different replications. We illustrated the utility of the step-PCR method using an empirical dataset assessing associations between the Dark Triad personality and hypersensitive narcissism.