.rotation_se_jacobian: Rotation Jacobians for analytic rotation standard errors

Description

Forward-difference the warm-started re-rotation map A -> (rotated loadings, Phi) over the unrotated loadings A to obtain the rotation Jacobians used by the analytic standard errors for rotated loadings (se = "information" in efa_fit()). The full nrow(A) * ncol(A) finite- difference loop runs in compiled code, re-solving the rotation from the converged transformation T_init at each perturbation; the caller forms J V J' in R.

Usage

.rotation_se_jacobian(
  A,
  T_init,
  method,
  param,
  normalize,
  oblique,
  eps,
  general_col = 0L
)

Value

A named list with the Jacobian J_L (pk x pk), the re-rotated base_loadings, a validity flag, and -- when oblique -- the Jacobian J_Phi (k^2 x pk) and base_Phi.

Arguments

A: Numeric matrix. The unrotated loading matrix at the solution.
T_init: Numeric matrix. The converged transformation that warm-starts each re-rotation.
method: Character scalar. The criterion family: one of "cf", "oblimin", "geomin", "bentler", "bifactor".
param: Numeric scalar. The criterion's tuning argument (kappa for "cf", gam for "oblimin", delta for "geomin"); ignored for "bentler" and "bifactor".
normalize: Logical scalar. Apply Kaiser normalization before rotation and reverse it after.
oblique: Logical scalar. Use the oblique (column-normalized) manifold; otherwise orthogonal.
eps: Numeric scalar. The forward-difference step on the loadings.
general_col: Integer scalar. For method = "bifactor", the zero-based column holding the general factor in the (factor-reordered) reported solution; ignored by the other criteria.

References

Jennrich, R. I. (1973). Standard errors for obliquely rotated factor loadings. Psychometrika, 38, 593-604.