Varimax rotation based on gradient projection needs between 10 and more than 500 random start loading matrices for optimal performance

Gradient projection rotation (GPR) is a promising method to rotate factor or component loadings by different criteria. Since the conditions for optimal performance of GPR-Varimax are widely unknown, this simulation study investigates GPR towards the Varimax criterion in principal component analysis. The conditions of the simulation study comprise two sample sizes (n = 100, n = 300), with orthogonal simple structure population models based on four numbers of components (3, 6, 9, 12), with- and without Kaiser-normalization, and six numbers of random start loading matrices for GPR-Varimax rotation (1, 10, 50, 100, 500, 1,000). GPR-Varimax rotation always performed better when at least 10 random matrices were used for start loadings instead of the identity matrix. GPR-Varimax worked better for a small number of components, larger (n = 300) as compared to smaller (n = 100) samples, and when loadings were Kaiser-normalized before rotation. To ensure optimal (stationary) performance of GPR-Varimax in recovering orthogonal simple structure, we recommend using at least 10 iterations of start loading matrices for the rotation of up to three components and 50 iterations for up to six components. For up to nine components, rotation should be based on a sample size of at least 300 cases, Kaiser-normalization, and more than 50 different start loading matrices. For more than nine components, GPR-Varimax rotation should be based on at least 300 cases, Kaiser-normalization, and at least 500 different start loading matrices.Comment: 19 pages, 8 figures, 2 tables, 4 figures in the Supplemen

ClusterCirc: Finding item clusters for circumplex instruments

We introduce ClusterCirc, a new method that finds item clusters for circumplex instruments as an alternative to conventional cluster analysis. When developing circumplex instruments, sorting items into subscales can be difficult because of the intended conceptual overlap of subscales in the circular model. ClusterCirc 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 ClusterCirc outperformed conventional cluster analysis in revealing circumplex clusters, especially for large within-cluster distances between items. Sorting accuracy was greater, and effects of data complexity and sample size were more stable in ClusterCirc than in conventional cluster analysis. We also found strong support for ClusterCirc in empirical circumplex data. ClusterCirc 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 ClusterCirc. We provide an R package for ClusterCirc (https://github.com/ancleo/ClusterCirc) with two main functions: ClusterCirc-Data (cc_data) performs ClusterCirc on empirical data. ClusterCirc-Simu (cc_simu) performs a tailored simulation study with the specifications of the data under the assumption of perfect circumplex spacing to assess circumplex fit of the data. We also provide the corresponding SPSS codes for ClusterCirc (https://github.com/ancleo/ClusterCirc_SPSS)