Circumplex Analysis: Addressing Methodological Challenges of Data With Circular Structure

Many psychological traits, such as interpersonal behavior, values, and affect, can be described by a circular structure (e.g., Gurtman, 1993; Russell et al., 1989; Schwartz et al., 2012; Wiggins, 1979). Circumplex models assume specific geometric arrangements of variables, reflected in correlational patterns or factor loadings on two orthogonal axes (L. Guttman, 1954; Tracey, 1997). Factor analysis and principal component analysis (PCA) are widely used to examine such structures. For example, the interpersonal circumplex is defined by the two factors Dominance and Love and underlies popular measures such as the Interpersonal Adjective Scales (IAS; Jacobs &amp; Scholl, 2005) and the Inventory of Interpersonal Problems (IIP; Horowitz et al., 2017). Many models are based on evenly spaced subscales around the circle, and some additionally incorporate an overarching general factor (e.g., overall interpersonal distress in the IIP). <br> A key challenge in developing circumplex instruments is assigning items to subscales, as adjacent subscales are closely related both conceptually and geometrically. Existing approaches often rely on subjective visual inspection and do not guarantee optimal circumplex spacing. To address methodological challenges in circumplex research, this thesis examined factor analytic approaches as widely used methods for circumplex analysis and developed a new clustering method tailored to circular data. <br> Study 1 investigated local optima in factor rotation procedures, a well-known issue for various rotation methods (Weide &amp; Beauducel, 2019). A simulation study compared Varimax rotation based on the gradient projection algorithm (GPA) as a newer method (Bernaards &amp; Jennrich, 2005) to the traditional Kaiser algorithm (Kaiser, 1958) in PCA. Results showed that GPA-Varimax performed comparably to the Kaiser algorithm but was vulnerable to local optima under circumplex conditions. Using multiple random starts effectively overcame this issue and resulted in better performance than the Kaiser algorithm. These findings indicate that GPA-Varimax, combined with multiple starts, is a valuable alternative to established methods, particularly in the presence of local optima, as found in circumplex structures. <br> Study 2 modeled the higher-level structure of the IIP based on two circumplex factors (Dominance, Love) and a general factor of interpersonal difficulties (Distress) in data from 822 participants (Weide et al., 2021). Bayesian confirmatory factor analysis outperformed traditional frequentist confirmatory and exploratory approaches in terms of model fit and parameter robustness. Different higher-level scoring methods produced reliable scores and preserved circumplex properties. External validity of the factor models and higher-level scores was supported by expected associations with Big Five traits Agreeableness, Extraversion, and Neuroticism as well as with narcissism. The findings support both Bayesian modeling in circumplex analysis and the use of higher-level scores for the IIP. <br> Study 3 addressed the lack of objective methods for clustering items into circumplex subscales (Weide et al., 2025). I developed ClusterCirc, a clustering approach that simultaneously optimizes circumplexity at the item and subscale levels while promoting even spacing around the circle. Simulations showed that ClusterCirc consistently outperformed traditional cluster analysis based on Wards method and k-means clustering in the identification of circumplex clusters. Applied to a German IAS dataset, ClusterCirc reproduced most original subscales while improving circumplex fit in empirical data. To facilitate the use of ClusterCirc, I developed an R package and SPSS syntax implementing the method (https://github.com/ancleo/ClusterCirc; https://github.com/ancleo/ClusterCirc_SPSS). <br> Despite these contributions, limitations of the thesis research include the use of nonclinical convenience samples and relatively short measures for external variables. Future research should examine additional circular constructs, more complex violations of circumplex assumptions, and combinations of factor rotation procedures with ClusterCirc to further improve circumplex modeling.