Integrating covariates into circumplex structures: an extension procedure for Browne's circular stochastic process model

Circumplex structures are elements of various psychological domains. Most work focuses on assessing the circular ordering of circumplex indicators and their relationships with covariates. In this article, an extension procedure for Browne's circumplex model is presented. Our approach models the relationships among circumplex indicators and the relationships of covariates with a latent circumplex simultaneously without affecting the circumplex indicators' positions on the circumplex. The approach builds upon Browne's Fourier series parameterization of a correlation function, which is used to model the latent circumplex correlation structure. It extends the shape of the correlation function to the profile of each covariate's correlations with the circumplex. The model is specified in the framework of structural equation modeling, thereby making it possible to test various hypotheses. Procedures are presented for deriving interval estimates for the parameters that relate the covariates to the circumplex. The model is compared to other approaches for assessing the relationships of a circumplex with covariates. The results of the exemplary applications and a simulation study were in favor of the suggested model. The approach is furthermore illustrated with a real-data example, focusing on the relationships between the interpersonal circumplex and the rivalry and admiration aspects of narcissism.