Varimax Rotation Based on Gradient Projection Is a Feasible Alternative to SPSS

Abstract

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.

Score Predictor Factor Analysis: Reproducing Observed Covariances by Means of Factor Score Predictors

“…Six items were from the knowledge domain mathematics and six items were from the knowledge domain of economics. Two PCA components, two MFA factors, and two SPFA factors were Varimax-rotated by means of the gradient projection method provided by Bernaards and Jennrich (2005) based on the R-script of Weide and Beauducel (2019). The empirical correlations and the script of PCA, MFA, and SPFA combined with GPR-Varimax rotation are available in the Supplementary Material (section 2).…”

Identification of Facet Models by Means of Factor Rotation: A Simulation Study and Data Analysis of a Test for the Berlin Model of Intelligence Structure

“…Two additional analyses were reported for the l = .50 ( n = 400, 1,000) condition of the orthogonal population models to investigate the robustness of the results of the overall simulation study. It has been shown that a number of different starting solutions might improve the performance of the gradient projection algorithm for factor rotation (Weide & Beauducel, 2019). As random starts need a substantial amount of computation time, an additional simulation was performed with only 200 runs for 10 conditions: The orthogonal population models with 2 + 2, 3 + 3, 4 + 4, 5 + 5, and 6 + 6 factors (= 5 population models × 2 sample sizes) to compare the sample average M (c) for 20 random start solutions with the sample average M (c) reached with a single start solution in the overall simulation.…”

“…It is therefore not surprising that simulation studies for the evaluation of methods of factor rotation were typically also based on simple structure models with salient loadings of each variable on only one factor (Velicer & Jackson, 1990). Although degraded simple structures with some substantial secondary loadings have sometimes been investigated (Schmitt & Sass, 2011; Weide & Beauducel, 2019) these degraded structures did not correspond to a faceted loading pattern. Since Guilford (1967) used target rotation, Jäger (1982) used a priori task aggregates, and simulation studies used only degraded simple structures one can conclude that population loading patterns representing more than one facet have not been explored systematically by means of EFA and subsequent (purely) explorative rotation toward simple structure.…”

Changes in landscape composition and configuration in the Beressa watershed, Blue Nile basin of Ethiopian Highlands: historical and future exploration

“…Important metrics were selected using an evaluation of the degree of redundancy based on a correlation matrix ( Riitters et al., 1995 ). Factors are subjected to varimax rotation to determine loading factors, and components that have an eigenvalue of >1 are retained ( Weide and Beauducel, 2019 ). Following data normality test, Spearman's product moment correlation between selected landscape metrics is used.…”