dot-rotate_bifactor_orth: Orthogonal bifactor factor rotation

In EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools

Orthogonal bifactor factor rotation

Description

Rotate a loading matrix orthogonally under the Jennrich-Bentler bifactor criterion using a gradient-projection optimizer along the orthogonal (Stiefel) manifold.

Usage

.rotate_bifactor_orth(
  L,
  eps = 1e-05,
  normalize = TRUE,
  random_starts = 0L,
  maxit = 1000L,
  max_line_search = 10L,
  step0 = 1,
  screen_keep = 5L,
  triage_maxit = 25L,
  triage_improve_tol = 0
)

Arguments

L	Numeric matrix. The unrotated loading matrix (variables by factors).
eps	Numeric scalar. Convergence tolerance for the projected-gradient norm.
normalize	Logical scalar. If TRUE, apply Kaiser normalization before rotation and reverse it afterwards.
random_starts	Integer scalar. Number of additional random orthogonal starts.
maxit	Integer scalar. Maximum number of projected-gradient updates.
max_line_search	Integer scalar. Maximum number of step-halving attempts after the initial trial step in each line-search phase.
step0	Numeric scalar. Initial step size used in the projected-gradient update.
screen_keep	Integer scalar. Number of screened random starts retained for triage optimization.
triage_maxit	Integer scalar. Number of short optimization iterations used in the triage stage.
triage_improve_tol	Numeric scalar. Relative improvement required for a triaged start to be promoted to full optimization.

Details

The criterion value f and its gradient dQ/dL at the rotated loadings L = A %*% T define the search; the engine maps the gradient to the orthogonal transformation T, projects it onto the tangent space, performs a non-monotone line search, and retracts back onto the orthogonal group via a polar (singular value) projection. The first factor is treated as a general factor and is exempt from the penalty; the criterion measures the between-group-factor cross-products of the squared loadings, so it is minimized when each variable loads on the general factor plus at most one group factor. The criterion is prone to local minima, so additional random starts are recommended.

Additional random orthogonal starts may be requested. To bound runtime the solver screens each random start by its objective, runs a short triage optimization on the best-screened starts, and fully optimizes only those that improve on the current incumbent by at least triage_improve_tol.

Value

A named list with the rotated loadings, the orthogonal rotation matrix Th (with L %*% Th reproducing the rotated loadings), the attained criterion value, and the convergence and validity flags. The list additionally reports the criterion value reached at each optimized start in all_values, with a per-start convergence flag in all_converged.

References

Bernaards, C. A., & Jennrich, R. I. (2005). Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis. Educational and Psychological Measurement, 65, 676-696.

Jennrich, R. I., & Bentler, P. M. (2011). Exploratory bi-factor analysis. Psychometrika, 76, 537-549.

EFAtools documentation built on Aug. 21, 2026, 5:16 p.m.