Local minima and factor rotations in exploratory factor analysis.

Abstract

In exploratory factor analysis, factor rotation algorithms can converge to local solutions (i.e., local minima) when they are initiated from different starting points. To better understand this problem, we performed three studies that investigated the prevalence and correlates of local solutions with five factor rotation algorithms: varimax, oblimin, entropy, and geomin (orthogonal and oblique). In total, we simulated 16,000 data sets and performed more than 57 million factor rotations to examine the influence of (a) factor loading size, (b) number of factor indicators, (c) factor cross loadings, (d) factor correlation size, (e) factor loading standardization, (f) sample size, and (g) model approximation error on the frequency of local solutions in factor rotation. We also examined local solutions in an exploratory factor analysis of an open source data set that included 54 personality items. Across three studies, all five algorithms converged to local solutions under some conditions with geomin (orthogonal and oblique) producing the highest number of local solutions. Follow-up analyses showed that, when factor rotations produced multiple solutions, the factor pattern with the maximum hyperplane count (rather than the lowest complexity value) was typically closest in mean squared error to the population factor pattern.

Thinking thrice about sum scores, and then some more about measurement and analysis

“…Even there, when simulating sample data from such populations, a single factor will fail to fit item covariances perfectly in a given sample due to sampling error (MacCallum & Tucker, 1991), leading to some qualifications of alpha vs. omega claims. Furthermore, most empirical data are not strictly unidimensional, but are at best only approximately or essentially unidimensional, so have additional small perturbations of item covariances that preclude perfect fit of a one-factor model even in the population (e.g., minor domain factors too numerous to model, as posited by Tucker et al, 1969; see also Nguyen & Waller, 2022). Calculation of coefficient alpha using Eq.…”

A tutorial on the use of temporal principal component analysis in developmental ERP research – Opportunities and challenges

“…Gradient projection can be used to rotate the factor loadings optimizing any mathematical criterion calculated from the factor loadings – enabling researchers to choose from a variety of rotation criteria. However, the user must be aware of the occurrence of local optima ( Hattori et al, 2017 , Nguyen and Waller, 2022 , Weide and Beauducel, 2019 ). In simplified terms, Gradient projection proceeds as follows: (1) Choose a random rotation matrix for a start; (2) Compute the rotated factor loadings for this starting rotation matrix; (3) Evaluate the rotation criterion for these intermediate rotated factor loadings; (4) Compute a new rotation matrix to achieve a more optimal value of the rotation criterion; (5) Re-iterate through steps (2) to (4) until the value of the rotation criterion does not change substantially any more between iterations; (6) The rotated factor loadings from the last iteration are the final rotated solution.…”

Is exploratory factor analysis always to be preferred? A systematic comparison of factor analytic techniques throughout the confirmatory–exploratory continuum.

“…Fourth, we only explored the performance of EFA with Geomin rotation, although it should be highlighted again that other rotation procedures might lead to different results (Browne, 2001; Hakstian, 1971; Hakstian & Abell, 1974; Sass & Schmitt, 2010; Schmitt & Sass, 2011). Recently, Nguyen and Waller (2022) found that Geomin is more prone to converge to local minima than other rotation procedures. For the sake of the present investigation, and following the suggestion from one of the reviewers, we replicated the simulation study using EFA with Oblimin rotation and found that the results did not substantially differ from those of EFA with Geomin.…”