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

Anneke Cleopatra Weide,André Beauducel

doi:10.3389/fpsyg.2019.00645

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.

Highlights

Exploratory factor analysis (EFA) and principal component analysis (PCA) are of major relevance in behavioral research, and many extraction and rotation methods have been proposed for them
Future research should investigate the advantage of random starts in GPR-Varimax we found for Kaiser normalized loadings in the double-optimum case in more detail
If congruence with population components thereby reaches the level that we found for GPR-Varimax, the effect can be attributed to random starts rather than the Varimax algorithm applied

Summary

Introduction

Exploratory factor analysis (EFA) and principal component analysis (PCA) are of major relevance in behavioral research, and many extraction and rotation methods have been proposed for them. We provide a simulation study on GPR toward the Varimax criterion (Kaiser, 1958) because it is one of the most popular and accepted rotation criteria toward orthogonal simple structure (Fabrigar et al, 1999; Browne, 2001) and it is implemented in most statistical software packages. As one may assume that the relative popularity of Varimax rotation is due to the lack of a backward limit for the time frame in this search, we performed an additional Google Scholar search for publications in the time frame between 2014 and 2019 (searched on 02/13/2019) For this time frame, we got 24,700 hits when entering “Varimax.” For “Geomin,” a more recent method for oblique and orthogonal rotation (Browne, 2001), we got 2,460 hits. Entering the combination of “Varimax” and “SPSS” yielded 85,400 hits, and we got 21,800 hits for the combination of “Varimax” and “SAS” in Google Scholar

Results

Discussion

Conclusion