Gradient projection rotation (GPR) is an openly available and promising tool for factor and component rotation. We compare GPR toward the Varimax criterion in principal component analysis to the built-in Varimax procedure in SPSS. In a simulation study, we tested whether GPR-Varimax yielded multiple local solutions by creating population simple structure with a single optimum and with two optima, a global and a local one (double-optimum condition). The other conditions comprised the number of components ( k = 3, 6, 9, and 12), the number of variables per component ( m/k = 4, 6, and 8), the number of iterations per rotation ( i = 25 and 250), and whether loadings were Kaiser normalized before rotation or not. GPR-Varimax was conducted with unrotated and multiple ( q = 1, 10, 50, and 100) random start loadings. We found equal results for GPR-Varimax and SPSS-Varimax in most conditions. The few very small differences in favor of SPSS-Varimax were eliminated when Kaiser-normalized loadings and 250 iterations per rotation were used. Selecting the best solution out of multiple random starts in GPR-Varimax increased proximity to population components in the double-optimum condition with Kaiser normalized loadings, for which GPR-Varimax recovered population structure better than SPSS-Varimax. We also included an empirical example and found that GPR-Varimax and SPSS-Varimax yielded highly similar solutions for orthogonal simple structure in a real data set. We suggest that GPR-Varimax can be used as an alternative to Varimax rotation in SPSS. Users of GPR-Varimax should allow for at least 250 iterations, normalize loadings before rotation, and select the best solution from at least 10 random starts to ensure optimal results.

Difficulties in interpersonal behavior are often measured by the circumplex-based Inventory of Interpersonal Problems. Its eight scales can be represented by a three-factor structure with two circumplex factors, Dominance and Love, and a general problem factor, Distress. Bayesian confirmatory factor analysis is well-suited to evaluate the higher-level structure of interpersonal problems because circumplex loading priors allow for data-driven adjustments and a more flexible investigation of the ideal circumplex pattern than conventional maximum likelihood confirmatory factor analysis. Using a non-clinical sample from an online questionnaire study (N = 822), we replicated the three-factor structure of the IIP by maximum likelihood and Bayesian confirmatory factor analysis and found great proximity of the Bayesian loadings to perfect circumplexity. We found additional support for the validity of the three-factor model of the IIP by including external criteria-Agreeableness, Extraversion, and Neuroticism from the Big Five and subclinical grandiose narcissism-in the analysis. We also investigated higher-level scores for Dominance, Love, and Distress using traditional regression factor scores and weighted sum scores. We found excellent reliability (with Rtt ≥ 0.90) for Dominance, Love, and Distress for the two scoring methods. We found high congruence of the higher-level scores with the underlying factors and good circumplex properties of the scoring models. The correlational pattern with the external measures was in line with theoretical expectations and similar to the results from the factor analysis. We encourage the use of Bayesian modeling when dealing with circumplex structure and recommend the use of higher-level scores for interpersonal problems as parsimonious, reliable, and valid measures.

Difficulties in interpersonal behavior are often measured by the circumplex-based Inventory of Interpersonal Problems. Its eight scales can be represented by a three-factor structure with two circumplex factors, Dominance and Love, and a general problem factor, Distress. Bayesian confirmatory factor analysis is well-suited to evaluate the higher-level structure of interpersonal problems because circumplex loading priors allow for data-driven adjustments and a more flexible investigation of the ideal circumplex pattern than maximum likelihood confirmatory factor analysis. Using a nonclinical sample from an online questionnaire study (N = 822), we replicated the three-factor structure of the IIP by maximum likelihood and Bayesian confirmatory factor analysis and found great proximity of the Bayesian loadings to perfect circumplexity. We also investigated higher-level scores for Dominance, Love, and Distress using traditional regression factor scores, posterior mean factor scores from Bayesian confirmatory factor analysis, and weighted sum scores. We found excellent reliability (with Rtt ≥ .90) for Dominance, Love, and Distress for all scoring methods. We found high congruence of the higher-level scores with the underlying factors and good circumplex properties of the scoring models. The correlation pattern with external measures – Agreeableness, Extraversion, and Neuroticism from the Big Five and subclinical grandiose narcissism – were in line with theoretical expectations. We encourage the use of Bayesian modeling when dealing with circumplex structure and recommend the use of higher-level scores for interpersonal problems as parsimonious, reliable, and valid measures.

We introduce Cluster-Circ, a new method that finds item clusters as possible subscales for circumplex instruments. When developing circumplex instruments, sorting items into subscales can be difficult because of the intended conceptual overlap of subscales in the circular model. Cluster-Circ provides a statistical solution for sorting items into subscales by finding item clusters with optimal circumplex spacing of both items and clusters. In a simulation study, we found that Cluster-Circ outperformed conventional cluster analysis in revealing circumplex clusters, especially when within-cluster distances between items increased. Sorting accuracy was greater and effects of data complexity and sample size were more stable in Cluster-Circ than in conventional cluster analysis. We also found strong support for Cluster-Circ in empirical circumplex data. Cluster-Circ sorting resulted in subscales with good scale properties and greater circumplex fit than the original subscales and subscales based on cluster analysis. We recommend a sample size between n = 500 and 1,000 to ensure high sorting accuracy of Cluster-Circ. We provide SPSS codes for Cluster-Circ and for a tailored simulation study with perfect circumplexity and the specifications of the respective dataset for better interpretation of Cluster-Circ results (Supplement).