Package {EFAtools}

Title:	Fast and Flexible Implementations of Exploratory Factor Analysis Tools
Version:	1.0.0
Description:	Provides a complete workflow for exploratory factor analysis (EFA). It covers data screening and factorability checks, a suite of factor retention criteria for choosing the number of factors, and factor extraction by principal axis factoring, maximum likelihood, unweighted least squares, or diagonally weighted least squares from Pearson, Spearman, Kendall, polychoric, tetrachoric, or two-stage full-information maximum likelihood correlations. A built-in rotation engine offers a range of orthogonal and oblique rotations, and standard errors for loadings and related quantities can be obtained by analytic, robust, or bootstrap methods. Further tools support model averaging across analytic choices, multigroup EFA with factor congruence, EFA on multiply imputed data, Schmid-Leiman transformation, reliability coefficients (including McDonald's omegas), factor score estimation, data simulation, and power analysis. Computationally intensive procedures are implemented in 'C++' for speed.
Depends:	R (≥ 4.1.0)
License:	GPL-3
Encoding:	UTF-8
LazyData:	true
URL:	https://github.com/mdsteiner/EFAtools, https://mdsteiner.github.io/EFAtools/
BugReports:	https://github.com/mdsteiner/EFAtools/issues
Imports:	psych, stats, ggplot2 (≥ 3.4.0), cli, Rcpp, future.apply, checkmate, progressr, rlang, clue, lifecycle
LinkingTo:	Rcpp, RcppArmadillo, roptim
Suggests:	testthat (≥ 3.0.0), future, GPArotation (≥ 2022.4-1), lavaan, lavaan.mi, MASS, mice, knitr, rmarkdown, vdiffr, polycor, mnormt, fungible, semTools, withr
VignetteBuilder:	knitr
Config/testthat/edition:	3
Config/roxygen2/version:	8.0.0
NeedsCompilation:	yes
Packaged:	2026-07-22 19:42:39 UTC; ma1159938
Author:	Markus Steiner [aut, cre], Silvia Steiner [aut], William Revelle [ctb], Max Auerswald [ctb], Morten Moshagen [ctb], John Ruscio [ctb], Brendan Roche [ctb], Urbano Lorenzo-Seva [ctb], David Navarro-Gonzalez [ctb], Johan Braeken [ctb], Andreas Soteriades [ctb]
Maintainer:	Markus Steiner <markus.d.steiner@gmail.com>
Repository:	CRAN
Date/Publication:	2026-07-23 10:50:46 UTC

EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools

Description

Provides a complete workflow for exploratory factor analysis (EFA). It covers data screening and factorability checks, a suite of factor retention criteria for choosing the number of factors, and factor extraction by principal axis factoring, maximum likelihood, unweighted least squares, or diagonally weighted least squares from Pearson, Spearman, Kendall, polychoric, tetrachoric, or two-stage full-information maximum likelihood correlations. A built-in rotation engine offers a range of orthogonal and oblique rotations, and standard errors for loadings and related quantities can be obtained by analytic, robust, or bootstrap methods. Further tools support model averaging across analytic choices, multigroup EFA with factor congruence, EFA on multiply imputed data, Schmid-Leiman transformation, reliability coefficients (including McDonald's omegas), factor score estimation, data simulation, and power analysis. Computationally intensive procedures are implemented in 'C++' for speed.

Author(s)

Maintainer: Markus Steiner markus.d.steiner@gmail.com (ORCID)

Authors:

• Markus Steiner markus.d.steiner@gmail.com (ORCID)

• Silvia Steiner silvia.steiner.grieder@gmail.com (ORCID)

Other contributors:

• William Revelle [contributor]

• Max Auerswald [contributor]

• Morten Moshagen [contributor]

• John Ruscio [contributor]

• Brendan Roche [contributor]

• Urbano Lorenzo-Seva [contributor]

• David Navarro-Gonzalez [contributor]

• Johan Braeken [contributor]

• Andreas Soteriades [contributor]

See Also

Useful links:

• https://github.com/mdsteiner/EFAtools

• https://mdsteiner.github.io/EFAtools/

• Report bugs at https://github.com/mdsteiner/EFAtools/issues

Average a list of matrices elementwise

Description

Average a list of matrices elementwise

Usage

.average_matrices(x)

Arguments

Value

A matrix with the same dimensions as the inputs.

Convert an "efa_loadings" table to matrix or a matrix to "efa_loadings"

Description

Convert an "efa_loadings" table to matrix or a matrix to "efa_loadings"

Usage

.change_class(x, cl = "matrix")

Arguments

Value

A table with the loadings, of class either "efa_loadings" or "matrix".

Author(s)

Andreas Soteriades

The loadings tables returned by efa_fit() are of class c("efa_loadings", "LOADINGS"), which prevents applying functions on them. This function allows to change their class to "matrix", and to change back to "efa_loadings" when done.

Compute explained variances from loadings

Description

From unrotated loadings compute the communalities and uniquenesses for total variance. Compute explained variances per factor from rotated loadings (and factor intercorrelations Phi if oblique rotation was used).

Usage

.compute_vars(L_unrot, L_rot, Phi = NULL)

Arguments

Value

A matrix with sum of squared loadings, proportion explained variance from total variance per factor, same as previous but cumulative, Proportion of explained variance from total explained variance, and same as previous but cumulative.

Mean squared discrepancy to a consensus target

Description

Mean squared discrepancy to a consensus target

Usage

.consensus_loss(aligned_loadings, target)

Arguments

Value

Mean sum of squared deviations from the target across matrices.

Internal single-start GPA-consensus engine

Description

Performs a single GPA-consensus run from one starting target. The multi-start wrapper .gpa_consensus_target() dispatches here.

Usage

.consensus_target_procrustes_single(
  unrotated_list,
  init_targets = NULL,
  rotation = c("orthogonal", "oblique"),
  start = 1,
  tol = 0.001,
  loss_tol = 1e-07,
  loss_patience = 5,
  convergence = c("either", "target", "loss", "both"),
  min_iter = 2,
  max_iter = 200,
  alpha = 1,
  match_target = TRUE,
  hyper_cutoff = 0.15,
  verbose = FALSE
)

Arguments

Extract a list object by its name

Description

Extract a list object by its name

Usage

.extract_list_object(alist, object)

Arguments

Value

A list of length m, with each element containing the extracted object for the kth element (k = 1,..., m).

Author(s)

Andreas Soteriades

Consider a list of named sub-lists. This function extracts, for each sub-list, the sub-list element that is specified by the user. This function is useful for extracting results from efa_fit() for each permutation run in efa_mi().

Compute number of non-matching indicator-to-factor correspondences

Description

Compute number of non-matching indicator-to-factor correspondences

Usage

.factor_corres(x, y, thresh = 0.3)

Arguments

Generalized Procrustes Analysis consensus target across loading matrices

Description

Internal helper that constructs a Generalized Procrustes Analysis (GPA) consensus target across a list of loading matrices and returns the aligned loadings, the centroid target, and convergence diagnostics. Used by efa_mi() under target_method = "consensus" to build a common rotation target across imputations. Oblique rotations are not supported here: the iteration is degenerate for oblique transforms with more than one factor (cf. Lorenzo-Seva & Van Ginkel 2016, who use a Promin step on top of the centroid rather than iterated oblique Procrustes); callers should pass the unrotated solutions of an orthogonal rotation, or use target_method = "first_target".

Usage

.gpa_consensus_target(
  unrotated_list,
  init_targets = NULL,
  rotation = c("orthogonal", "oblique"),
  start = 1,
  multi_start = FALSE,
  starts = NULL,
  tol = 0.001,
  loss_tol = 1e-06,
  loss_patience = 5,
  convergence = c("either", "target", "loss", "both"),
  min_iter = 2,
  max_iter = 200,
  alpha = 1,
  match_target = TRUE,
  hyper_cutoff = 0.15,
  verbose = FALSE
)

Arguments

Details

The iteration alternates two steps:

1. each loading matrix is aligned to the current target with efa_procrustes();

2. the target is updated to the elementwise centroid of the aligned matrices.

The outer loop stops when the target stabilises, when the consensus loss stabilises, or when both criteria are satisfied.

If multi_start = FALSE, one consensus run is performed. If multi_start = TRUE, the same engine is repeated for the selected starting targets and the run with the smallest final mean loss is returned as the main result; all runs and a between-run congruence summary are retained in the multi_start component.

Value

A list with the converged target, aligned matrices, pooled loadings, pooled Phi, convergence history, inner-alignment diagnostics, and hyperplane summaries. If multi_start = TRUE, the multi_start element also contains the per-start losses, convergence indicators, run summaries, all run objects, and between-run Tucker congruence matrices.

References

Gower, J. C. (1975). Generalized Procrustes analysis. Psychometrika, 40, 33-51.

Van Ginkel, J. R., & Kroonenberg, P. M. (2014). Using Generalized Procrustes Analysis for Multiple Imputation in Principal Component Analysis. Journal of Classification, 31, 242-269.

Lorenzo-Seva, U., & Van Ginkel, J. R. (2016). Multiple Imputation of missing values in exploratory factor analysis of multidimensional scales: estimating latent trait scores. Anales de Psicologia, 32, 596-608.

Count near-zero loadings

Description

Hyperplane count is the number of loadings with absolute value smaller than a user-specified cutoff.

Usage

.hyperplane_count(L, cutoff = 0.15)

Arguments

Value

A list with the total hyperplane count and counts by factor and item.

Oblique Procrustes target rotation using a k x k inner objective

Description

Compute an oblique target rotation for a loading matrix using a targetQ-compatible parameterization and a ⁠k x k⁠ objective.

Usage

.oblique_procrustes(
  A,
  B,
  S_r = NULL,
  T_init_r = NULL,
  eps = 1e-05,
  maxit = 1000L,
  max_line_search = 10L,
  step0 = 1,
  normalize = FALSE,
  random_starts = 0L,
  screen_keep = 2L,
  triage_maxit = 25L,
  triage_improve_tol = 0
)

Arguments

Details

The rotated loading matrix is defined as L = A %*% solve(t(T)), and the corresponding factor correlation matrix is Phi = t(T) %*% T. The optimization is carried out over the transformation matrix T under the oblique normalization constraint diag(t(T) %*% T) = 1.

Non-invertible candidate transformations are rejected rather than evaluated through a pseudo-inverse.

Additional random starts may be requested. To reduce runtime, the solver uses a two-stage strategy for extra starts: cheap objective screening, followed by short triage optimization, followed by full optimization only for starts that improve on the current incumbent by at least triage_improve_tol.

The routine is intended for repeated oblique target rotations in workflows such as bootstrap alignment or consensus alignment of exploratory factor solutions across multiply imputed datasets. It follows the same oblique transformation convention as GPArotation::targetQ().

Value

A named list containing the rotated loadings, transformation matrix, factor correlation matrix, target criterion value, convergence diagnostics, line-search diagnostics, and multi-start summaries.

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.

Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111-150.

Batched oblique Procrustes target rotation over a cube of loading matrices

Description

Align each slice of a loading-matrix cube to a single shared target using the same oblique target rotation as .oblique_procrustes(), in one call. This removes the per-replicate marshalling overhead of looping efa_procrustes() in R over bootstrap or multiple-imputation arrays.

Usage

.oblique_procrustes_batch(
  A,
  B,
  eps = 1e-05,
  maxit = 1000L,
  max_line_search = 10L,
  step0 = 1,
  normalize = FALSE,
  random_starts = 0L,
  screen_keep = 2L,
  triage_maxit = 25L,
  triage_improve_tol = 0
)

Arguments

Details

Each slice A[, , i] is aligned to B. For a single-factor cube the alignment reduces to the closed-form sign match T = sign(crossprod(A_i, B)) with factor correlation 1, matching the one-factor short-circuit in efa_procrustes(). For two or more factors the slice is warm-started from the closed-form orthogonal Procrustes solution (mirroring efa_procrustes()) and optimized with the same multi-start oblique solver as .oblique_procrustes(). Random starts are drawn serially with R::rnorm in the calling process.

Slices are aligned independently. A slice that cannot be aligned (a non-finite loading matrix, a failed warm-start decomposition, an invalid fit, or any linear-algebra exception) is reported with valid = FALSE and NA for the loadings, factor correlations, and all other per-slice diagnostics, rather than aborting the whole call, so one degenerate replicate does not discard the rest.

Value

A named list with the aligned-loadings array loadings (⁠n x m x b⁠), the factor-correlation array Phi (⁠m x m x b⁠), and the per-slice diagnostics valid, convergence, value, iterations, and line_search_failed.

Closed-form orthogonal Procrustes rotation

Description

Rotate A to the orthogonal target B by minimizing ⁠||A %*% T - B||_F^2⁠ subject to t(T) %*% T = I.

Usage

.orthogonal_procrustes(A, B)

Arguments

Value

A list with the rotated loadings, orthogonal transformation matrix, target criterion value, and basic diagnostics.

References

Schoenemann, P. H. (1966). A generalized solution of the orthogonal Procrustes problem. Psychometrika, 31, 1-10.

Perform the iterative PAF procedure

Description

Function called from within PAF so usually no call to this is needed by the user. Provides a C++ implementation of the PAF procedure

Usage

.paf_iter(h2, criterion, R, n_fac, abs_eig, crit_type, max_iter)

Arguments

Parallel analysis on simulated data.

Description

Function called from within efa_parallel() so usually no call to this is needed by the user. Provides a C++ implementation of the efa_parallel() simulation procedure

Usage

.parallel_sim(n_datasets, n_vars, N, eigen_type, maxit = 10000L)

Arguments

Oblique Bentler factor rotation

Description

Rotate a loading matrix obliquely under Bentler's invariant pattern simplicity criterion using a gradient-projection optimizer along the oblique (column-normalized) manifold.

Usage

.rotate_bentler_oblq(
  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

Details

The criterion value f and its gradient dQ/dL at the rotated loadings L = A %*% solve(t(T)) define the search; the engine maps the gradient to the transformation T on the manifold diag(t(T) %*% T) = 1, projects it onto the tangent space, performs a non-monotone line search, and retracts back onto the manifold by column normalization. The Bentler criterion measures the departure of the cross-products of squared loadings from a diagonal pattern; it is prone to local minima, so additional random starts are recommended.

Additional random 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 transformation matrix Th (with L %*% t(solve(Th)) reproducing the rotated loadings), the factor correlation matrix Phi (t(Th) %*% Th), 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

Bentler, P. M. (1977). Factor simplicity index and transformations. Psychometrika, 42, 277-295.

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.

Orthogonal Bentler factor rotation

Description

Rotate a loading matrix orthogonally under Bentler's invariant pattern simplicity criterion using a gradient-projection optimizer along the orthogonal (Stiefel) manifold.

Usage

.rotate_bentler_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

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 Bentler criterion measures the departure of the cross-products of squared loadings from a diagonal pattern; it 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

Bentler, P. M. (1977). Factor simplicity index and transformations. Psychometrika, 42, 277-295.

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.

Oblique bifactor factor rotation

Description

Rotate a loading matrix obliquely under the Jennrich-Bentler bifactor criterion using a gradient-projection optimizer along the oblique (column-normalized) manifold.

Usage

.rotate_bifactor_oblq(
  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

Details

The criterion value f and its gradient dQ/dL at the rotated loadings L = A %*% solve(t(T)) define the search; the engine maps the gradient to the transformation T on the manifold diag(t(T) %*% T) = 1, projects it onto the tangent space, performs a non-monotone line search, and retracts back onto the manifold by column normalization. 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 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 transformation matrix Th (with L %*% t(solve(Th)) reproducing the rotated loadings), the factor correlation matrix Phi (t(Th) %*% Th), 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.

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

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.

Orthogonal Crawford-Ferguson factor rotation

Description

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

Usage

.rotate_cf_orth(
  L,
  kappa,
  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

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. kappa = 0 is the quartimax criterion and kappa = ncol(A) / (2 * nrow(A)) is the equamax criterion.

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.

Crawford, C. B., & Ferguson, G. A. (1970). A general rotation criterion and its use in orthogonal rotation. Psychometrika, 35, 321-332.

Oblique geomin factor rotation

Description

Rotate a loading matrix obliquely under the geomin criterion using a gradient-projection optimizer along the oblique (column-normalized) manifold.

Usage

.rotate_geomin_oblq(
  L,
  delta = 0.01,
  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

Details

The criterion value f and its gradient dQ/dL at the rotated loadings L = A %*% solve(t(T)) define the search; the engine maps the gradient to the transformation T on the manifold diag(t(T) %*% T) = 1, projects it onto the tangent space, performs a non-monotone line search, and retracts back onto the manifold by column normalization. The geomin criterion sums the per-variable geometric mean of the squared loadings offset by delta; it is prone to local minima, so additional random starts are recommended.

Additional random 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 transformation matrix Th (with L %*% t(solve(Th)) reproducing the rotated loadings), the factor correlation matrix Phi (t(Th) %*% Th), 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.

Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111-150.

Orthogonal geomin factor rotation

Description

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

Usage

.rotate_geomin_orth(
  L,
  delta = 0.01,
  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

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 geomin criterion sums the per-variable geometric mean of the squared loadings offset by delta; it 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.

Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111-150.

Oblique oblimin factor rotation

Description

Rotate a loading matrix obliquely under the oblimin criterion using a gradient-projection optimizer along the oblique (column-normalized) manifold.

Usage

.rotate_oblimin(
  L,
  gam = 0,
  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

Details

The criterion value f and its gradient dQ/dL at the rotated loadings L = A %*% solve(t(T)) define the search; the engine maps the gradient to the transformation T on the manifold diag(t(T) %*% T) = 1, projects it onto the tangent space, performs a non-monotone line search, and retracts back onto the manifold by column normalization. gam = 0 is the quartimin criterion.

Additional random 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 transformation matrix Th (with L %*% t(solve(Th)) reproducing the rotated loadings), the factor correlation matrix Phi (t(Th) %*% Th), 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., & Sampson, P. F. (1966). Rotation for simple loadings. Psychometrika, 31, 313-323.

Oblique simplimax factor rotation

Description

Rotate a loading matrix obliquely under the simplimax criterion using a gradient-projection optimizer along the oblique (column-normalized) manifold.

Usage

.rotate_simplimax_oblq(
  L,
  k,
  eps = 1e-05,
  normalize = TRUE,
  random_starts = 0L,
  maxit = 1000L,
  max_line_search = 10L,
  step0 = 1
)

Arguments

Details

The criterion value f and its gradient dQ/dL at the rotated loadings L = A %*% solve(t(T)) define the search; the engine maps the gradient to the transformation T on the manifold diag(t(T) %*% T) = 1, projects it onto the tangent space, performs a non-monotone line search, and retracts back onto the manifold by column normalization. The simplimax criterion sums the k smallest squared loadings, so it is minimized when the k "close-to-zero" loadings are driven toward zero; the count k is a tuning parameter. Because the set of k smallest loadings is reselected at every evaluation, the criterion is only piecewise smooth: its gradient jumps as loadings cross the kth-smallest threshold, so the line search accepts a step whenever it decreases the largest objective over a short window of recent iterations (a non-monotone test; Grippo, Lampariello, & Lucidi, 1986), letting the optimizer step across the kinks where a strictly monotone descent would stall.

The criterion is strongly prone to local minima, so the solver fully optimizes the identity start together with random_starts random orthogonal starts and keeps the solution with the lowest criterion value. Fully optimizing every start – rather than the screen-and-triage strategy used for the smooth criteria, which assumes the rational start lies in the global basin – is the standard remedy for the local minima of complexity-based rotation criteria (Kiers, 1994; Browne, 2001).

Value

A named list with the rotated loadings, the transformation matrix Th (with L %*% t(solve(Th)) reproducing the rotated loadings), the factor correlation matrix Phi (t(Th) %*% Th), 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.

Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111-150.

Grippo, L., Lampariello, F., & Lucidi, S. (1986). A nonmonotone line search technique for Newton's method. SIAM Journal on Numerical Analysis, 23, 707-716.

Kiers, H. A. L. (1994). Simplimax: Oblique rotation to an optimal target with simple structure. Psychometrika, 59, 567-579.

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
)

Arguments

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.

References

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

Reference eigenvalues for the efa_nest() simulation via the shared kernel.

Description

Internal helper called from efa_nest(). Simulates nreps datasets from an (nf - 1)-factor reference model, given that model's loadings Lambda and uniquenesses Psi, and returns the nf-th largest eigenvalue of each simulated correlation matrix. The data are drawn with the shared Z * M rule (see .simulate_cfm_mvn()) using the factor-score square root ⁠M = t([Lambda | diag(sqrt(Psi))])⁠, so a row randn(1, nf - 1 + p) * M is N(0, Lambda Lambda' + diag(Psi)). Drawing nf - 1 + p standard normals and post-multiplying by the factor-score matrix is faster than forming the model- implied correlation matrix and drawing from it, and matches the position at which efa_nest() reads the reference eigenvalue.

Usage

.simulate_cfm_eigen(nf, N, Lambda, Psi, nreps = 1000L)

Arguments

Draw multivariate-normal data from a population correlation matrix.

Description

Internal helper called from efa_simulate(). Draws N cases from a p-variate normal with correlation (or covariance) R by post-multiplying a matrix of standard normal deviates by a matrix square root M of R (with M' M = R, so the rows of Z * M are N(0, R)). This is the same Z * M rule used by the NEST reference simulation (.simulate_cfm_eigen): there M is the transposed factor-score matrix, here it is a Cholesky or eigen square root. A positive-definite R is factored by Cholesky; a positive-semidefinite but singular R (which makes the Cholesky fail although it is still a valid covariance, e.g. a no-factor block or a smoothed factor intercorrelation matrix) falls back to a symmetric eigen square root.

Usage

.simulate_cfm_mvn(R, N, tol = 1e-08)

Arguments

Tucker congruence between factors

Description

Compute the Tucker congruence matrix between the columns of two loading matrices.

Usage

.tucker_congruence(L1, L2)

Arguments

Value

A square matrix whose ⁠(i, j)⁠ entry is the Tucker congruence between column i of L1 and column j of L2.

References

Lorenzo-Seva, U., and ten Berge, J. M. F. (2006). Tucker's congruence coefficient as a meaningful index of factor similarity. Methodology, 2, 57-64.

Bartlett's test of sphericity

Description

BARTLETT() has been superseded by efa_bartlett(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

BARTLETT(
  x,
  N = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra")
)

Arguments

Value

A list of class c("efa_bartlett", "BARTLETT"), identical to the value of efa_bartlett(); see there for the components.

See Also

efa_bartlett()

Comparison data

Description

CD() has been superseded by efa_cd(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

CD(
  x,
  n_factors_max = NA,
  N_pop = 10000,
  N_samples = 500,
  alpha = 0.3,
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  max_iter = 50
)

Arguments

Value

An object of class efa_retention, identical to the value of efa_cd(); see there for the components.

See Also

efa_cd()

Compare two vectors or matrices (communalities or loadings)

Description

COMPARE() has been superseded by efa_compare(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

COMPARE(
  x,
  y,
  reorder = c("congruence", "names", "none"),
  corres = TRUE,
  thresh = 0.3,
  digits = 4,
  m_red = 0.001,
  range_red = 0.001,
  round_red = 3,
  print_diff = TRUE,
  na.rm = FALSE,
  x_labels = c("x", "y"),
  plot = TRUE,
  plot_red = 0.01
)

Arguments

Value

A list of class c("efa_compare", "COMPARE"), identical to the value of efa_compare(); see there for the components.

See Also

efa_compare()

DOSPERT

Description

A list containing the the bivariate correlations (cormat) of the 40 items of the Domain Specific Risk Taking Scale (DOSPERT; Weber, Blais, & Betz, 2002) and the sample size (N) based on the publicly available dataset at (https://osf.io/rce7g) of the Basel-Berlin Risk Study (Frey et al., 2017). The items measure risk-taking propensity on six different domains: social, recreational, gambling, health/ safety, investment, and ethical.

Usage

DOSPERT

Format

A list of 2 with elements "cormat" (40 x 40 matrix of bivariate correlations) and "N" (scalar).

cormat

(matrix) - Bivariate correlations of the 40 DOSPERT items, which span the six risk domains (social, recreational, gambling, health/safety, investment, and ethical).

N

(numeric) - The sample size the correlations are based on.

Source

Weber, E. U., Blais, A.-R., & Betz, N. E. (2002). A domain specific risk-attitude scale: Measuring risk perceptions and risk behaviors. Journal of Behavioral Decision Making, 15(4), 263–290. doi: 10.1002/bdm.414

Frey, R., Pedroni, A., Mata, R., Rieskamp, J., & Hertwig, R. (2017). Risk preference shares the psychometric structure of major psychological traits. Science Advances, 3, e1701381.

https://osf.io/rce7g

DOSPERT_raw

Description

A data.frame containing responses to the risk subscale of the Domain Specific Risk Taking Scale (DOSPERT; Weber, Blais, & Betz, 2002) based on the publicly available dataset (at https://osf.io/pjt57/) by Frey, Duncan, and Weber (2020). The items measure risk-taking propensity on five different domains: social, recreational, financial, health/ safety, and ethical.

Usage

DOSPERT_raw

Format

A data.frame with 3,123 rows (participants) and 30 columns, named by a domain prefix and item number, with six items in each of five risk domains:

ethR_1 to ethR_6

(numeric) - Ethical-domain risk-taking items.

finR_1 to finR_6

(numeric) - Financial-domain risk-taking items.

heaR_1 to heaR_6

(numeric) - Health/safety-domain risk-taking items.

recR_1 to recR_6

(numeric) - Recreational-domain risk-taking items.

socR_1 to socR_6

(numeric) - Social-domain risk-taking items.

Source

Blais, A.-R., & Weber, E. U. (2006). A domain-specific risk-taking (DOSPERT) scale for adult populations. Judgment and Decision Making, 1(1), 33–47. doi: 10.1017/S1930297500000334

Frey, R., Duncan, S. M., & Weber, E. U. (2020). Towards a typology of risk preference: Four risk profiles describe two thirds of individuals in a large sample of the U.S. population. PsyArXiv Preprint. doi:10.31234/osf.io/yjwr9

Exploratory factor analysis (EFA)

Description

EFA() has been superseded by efa_fit(), which is the recommended interface going forward. efa_fit() keeps the primary choices (data, factors, estimator, rotation, standard errors) as top-level arguments and collects the estimation and rotation tuning knobs into two control objects, estimate_control() and rotate_control(). EFA() remains available and unchanged – its full flat argument list still works exactly as before – so existing code keeps running.

Usage

EFA(
  x,
  n_factors,
  N = NA,
  method = c("PAF", "ML", "ULS", "MINRES", "DWLS"),
  rotation = c("none", "varimax", "equamax", "quartimax", "geominT", "bentlerT",
    "bifactorT", "promax", "oblimin", "quartimin", "simplimax", "bentlerQ", "geominQ",
    "bifactorQ"),
  se = c("none", "information", "sandwich", "np-boot"),
  type = c("EFAtools", "psych", "SPSS", "none"),
  max_iter = NA,
  init_comm = NA,
  criterion = NA,
  criterion_type = NA,
  abs_eigen = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  varimax_type = NA,
  k = NA,
  normalize = TRUE,
  p_type = NA,
  precision = 1e-05,
  order_type = NA,
  start_method = "psych",
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra", "fiml"),
  b_boot = 1000,
  ci = 0.95,
  random_starts = 100,
  seed = NULL,
  P_type = lifecycle::deprecated(),
  randomStarts = lifecycle::deprecated(),
  ...
)

Arguments

Value

The value of efa_fit(), a list of class c("efa", "EFA"); see there for the components.

See Also

efa_fit(), estimate_control(), rotate_control()

Model averaging across different EFA methods and types

Description

EFA_AVERAGE() has been superseded by efa_average(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

EFA_AVERAGE(
  x,
  n_factors,
  N = NA,
  method = "PAF",
  rotation = "promax",
  type = "none",
  averaging = c("mean", "median"),
  trim = 0,
  salience_threshold = 0.3,
  max_iter = 10000,
  init_comm = c("smc", "mac", "unity"),
  criterion = c(0.001),
  criterion_type = c("sum", "max_individual"),
  abs_eigen = c(TRUE),
  varimax_type = c("svd", "kaiser"),
  normalize = TRUE,
  k_promax = 2:4,
  k_simplimax = ncol(x),
  P_type = c("norm", "unnorm"),
  precision = 1e-05,
  start_method = c("psych", "factanal"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra", "fiml"),
  show_progress = TRUE
)

Arguments

Value

The value of efa_average(), normally a list of class c("efa_average", "EFA_AVERAGE"); see there for the components.

See Also

efa_average()

Exploratory factor analysis on multiple data imputations

Description

EFA_POOLED() has been superseded by efa_mi(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

EFA_POOLED(
  data_list,
  p = 0.05,
  target_method = c("first_target", "consensus"),
  align_unrotated = c("signed_tucker_congruence", "none", "procrustes"),
  fit_pool_method = c("D2"),
  consensus_args = list(),
  procrustes_args = list(),
  rmsea_ci_level = 0.9,
  rmsr_upper = TRUE,
  ...
)

Arguments

Value

The value of efa_mi(), normally a list of class c("efa_mi", "EFA_POOLED", "efa", "EFA"); see there for the components.

See Also

efa_mi()

Empirical Kaiser criterion

Description

EKC() has been superseded by efa_ekc(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

EKC(
  x,
  N = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  type = "BvA2017"
)

Arguments

Value

An object of class efa_retention, identical to the value of efa_ekc(); see there for the components.

See Also

efa_ekc()

Estimate factor scores for an EFA model

Description

FACTOR_SCORES() has been superseded by efa_scores(), which is the recommended interface going forward. It remains available so existing code keeps working. Note that R2 is now the squared factor-score determinacy (the value psych::factor.scores() returns with Grice = TRUE); earlier versions returned psych's default Grice = FALSE validity coefficient, so the slot is not comparable across versions.

A convenience wrapper around efa_scores() that returns factor scores and weights in a compact list. Factor scores are calculated according to the specified method if raw data are provided, and only factor weights if a correlation matrix is provided.

Usage

FACTOR_SCORES(
  x,
  f,
  Phi = NULL,
  rho = NULL,
  method = c("Thurstone", "tenBerge", "Anderson", "Bartlett", "Harman", "components")
)

Arguments

Value

A list of class FACTOR_SCORES containing the following:

See Also

efa_scores() for the factor-score weights together with the full set of score-quality diagnostics (determinacy, univocality, and Guttman indeterminacy index) and a print/summary method.

Examples

# Example with raw data with method "Bartlett"
EFA_raw <- efa_fit(DOSPERT_raw, n_factors = 10, estimator = "PAF",
                   rotation = "oblimin",
                   rotate_control = rotate_control(random_starts = 1))
fac_scores_raw <- FACTOR_SCORES(DOSPERT_raw, f = EFA_raw, method = "Bartlett")

# Same as above, but with raw data AND a correlation matrix
cor_pearson <- cor(DOSPERT_raw)
EFA_cor_pearson <- efa_fit(cor_pearson, n_factors = 10, N = nrow(DOSPERT_raw),
                           estimator = "PAF", rotation = "oblimin",
                           rotate_control = rotate_control(random_starts = 1))
fac_scores_cor_pearson <- FACTOR_SCORES(DOSPERT_raw, f = EFA_cor_pearson,
                                        rho = cor_pearson,
                                        method = "Bartlett")

# Scores between two alternatives above are identical
isTRUE(all.equal(fac_scores_raw$scores, fac_scores_cor_pearson$scores,
                 check.attributes = FALSE))

# Example with a correlation matrix only (does not return factor scores)
EFA_cor <- efa_fit(test_models$baseline$cormat, n_factors = 3, N = 500,
                   estimator = "PAF", rotation = "oblimin")
fac_scores_cor <- FACTOR_SCORES(test_models$baseline$cormat, f = EFA_cor)

GRiPS_raw

Description

A data.frame containing responses to the General Risk Propensity Scale (GRiPS, Zhang, Highhouse & Nye, 2018) of 810 participants of Study 1 of Steiner and Frey (2020). The original data can be accessed via https://osf.io/kxp8t/.

Usage

GRiPS_raw

Format

A data.frame with 810 rows (participants) and 8 columns, one per GRiPS item. Each item is a self-report indicator of general risk propensity, labelled by a keyword from the item:

fun

(numeric) - Risk-taking makes life more fun.

friends

(numeric) - Friends would describe the respondent as a risk taker.

enjoy

(numeric) - Enjoyment of taking risks.

hurt

(numeric) - Willingness to take a risk even if it might hurt.

part

(numeric) - Risk-taking as an important part of life.

commonly

(numeric) - Commonly takes risks.

chances

(numeric) - Belief in taking chances.

attracted

(numeric) - Attracted, rather than scared, by risk.

Source

Zhang, D. C., Highhouse, S., & Nye, C. D. (2019). Development and validation of the general risk propensity scale (GRiPS). Journal of Behavioral Decision Making, 32, 152–167. doi: 10.1002/bdm.2102

Steiner, M., & Frey, R. (2020). Representative design in psychological assessment: A case study using the Balloon Analogue Risk Task (BART). PsyArXiv Preprint. doi:10.31234/osf.io/dg4ks

Hull method

Description

HULL() has been superseded by efa_hull(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

HULL(
  x,
  N = NA,
  n_fac_theor = NA,
  method = c("PAF", "ULS", "ML"),
  gof = c("CAF", "CFI", "RMSEA"),
  eigen_type = c("SMC", "PCA", "EFA"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  n_datasets = 1000,
  percent = 95,
  decision_rule = c("means", "percentile", "crawford"),
  n_factors = 1,
  ...
)

Arguments

Value

An object of class efa_retention, identical to the value of efa_hull(); see there for the components.

See Also

efa_hull()

Intelligence subtests from the Intelligence and Development Scales–2

Description

A matrix containing the bivariate correlations of the 14 intelligence subtests from the Intelligence and Development Scales–2 (IDS-2; Grob & Hagmann-von Arx, 2018), an intelligence and development test battery for children and adolescents aged 5 to 20 years, for the standardization and validation sample (N = 1,991). Details can be found in Grieder & Grob (2019).

Usage

IDS2_R

Format

A 14 x 14 matrix of bivariate correlations

GS

(numeric) - Geometric shapes.

PL

(numeric) - Plates.

TC

(numeric) - Two characteristics.

CB

(numeric) - Crossing out boxes.

NL

(numeric) - Numbers / letters.

NLM

(numeric) - Numbers / letter mixed.

GF

(numeric) - Geometric figures.

RGF

(numeric) - Rotated geometric figures.

CM

(numeric) - Completing matrices.

EP

(numeric) - Excluding pictures.

CA

(numeric) - Categories.

OP

(numeric) - Opposites.

RS

(numeric) - Retelling a story.

DP

(numeric) - Describing pictures.

Source

Grieder, S., & Grob, A. (2019). Exploratory factor analyses of the intelligence and development scales–2: Implications for theory and practice. Assessment. Advance online publication. doi:10.1177/1073191119845051

Grob, A., & Hagmann-von Arx, P. (2018). Intelligence and Development Scales–2 (IDS-2). Intelligenz- und Entwicklungsskalen für Kinder und Jugendliche. [Intelligence and Development Scales for Children and Adolescents.]. Bern, Switzerland: Hogrefe.

Kaiser-Guttman criterion

Description

KGC() has been superseded by efa_kgc(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

KGC(
  x,
  eigen_type = c("PCA", "SMC", "EFA"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  n_factors = 1,
  ...
)

Arguments

Value

An object of class efa_retention, identical to the value of efa_kgc(); see there for the components.

See Also

efa_kgc()

Kaiser-Meyer-Olkin criterion

Description

KMO() has been superseded by efa_kmo(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

KMO(
  x,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra")
)

Arguments

Value

A list of class c("efa_kmo", "KMO"), identical to the value of efa_kmo(); see there for the components.

See Also

efa_kmo()

Minimum average partial

Description

MAP() has been superseded by efa_map(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

MAP(
  x,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra")
)

Arguments

Value

An object of class efa_retention, identical to the value of efa_map(); see there for the components.

See Also

efa_map()

Next eigenvalue sufficiency test

Description

NEST() has been superseded by efa_nest(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

NEST(
  x,
  N = NA,
  alpha = 0.05,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  n_datasets = 1000,
  ...
)

Arguments

Value

An object of class efa_retention, identical to the value of efa_nest(); see there for the components.

See Also

efa_nest()

Various factor retention criteria

Description

N_FACTORS() has been superseded by efa_retain(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

N_FACTORS(
  x,
  criteria = c("CD", "EKC", "HULL", "MAP", "NEST", "PARALLEL"),
  suitability = TRUE,
  N = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  n_factors_max = NA,
  N_pop = 10000,
  N_samples = 500,
  alpha = 0.3,
  max_iter_CD = 50,
  n_fac_theor = NA,
  method = c("ML", "PAF", "ULS"),
  gof = c("CAF", "CFI", "RMSEA"),
  eigen_type_HULL = c("SMC", "PCA", "EFA"),
  eigen_type_other = c("SMC"),
  n_factors = 1,
  n_datasets = 1000,
  percent = 95,
  decision_rule = c("means", "percentile", "crawford"),
  ekc_type = c("BvA2017"),
  n_datasets_nest = 1000,
  alpha_nest = 0.05,
  show_progress = FALSE,
  ...
)

Arguments

Value

A list of class c("efa_retain", "N_FACTORS"), identical to the value of efa_retain(); see there for the components.

See Also

efa_retain()

McDonald's omega

Description

OMEGA() has been superseded by efa_reliability(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

This function finds omega total, hierarchical, and subscale, as well as additional model-based indices of interpretive relevance (H index, ECV, PUC) from a Schmid-Leiman (SL) solution or lavaan single factor, second-order (see below), or bifactor solution. The SL-based omegas can either be found from a psych::schmid(), efa_schmid_leiman(), or, in a more flexible way, by leaving model = NULL and specifying additional arguments. By setting the type argument, results from psych::omega() can be reproduced.

Usage

OMEGA(
  model = NULL,
  type = c("EFAtools", "psych"),
  g_name = "g",
  group_names = NULL,
  add_ind = TRUE,
  factor_corres = NULL,
  var_names = NULL,
  fac_names = NULL,
  g_load = NULL,
  s_load = NULL,
  u2 = NULL,
  cormat = NULL,
  pattern = NULL,
  Phi = NULL,
  variance = c("correlation", "sums_load")
)

Arguments

Details

What this function does

This function calculates McDonald's omegas (McDonald, 1978, 1985, 1999), the H index (Hancock & Mueller, 2001), the explained common variance (ECV; Rodriguez et al., 2016a, 2016b), and the percent of uncontaminated correlations (PUC; Bonifay et al., 2015; Reise et al., 2013).

All types of omegas (total, hierarchical, and subscale) are calculated for the general factor as well as for the subscales / group factors (see, e.g., Gignac, 2014; Rodriguez et al., 2016a, 2016b). Omegas refer to the correlation between a factor and a unit-weighted composite score and thus the true score variance in a unit-weighted composite based on the respective indicators. Omega total is the total true score variance in a composite. Omega hierarchical is the true score variance in a composite that is attributable to the general factor, and omega subscale is the true score variance in a composite attributable to all subscales / group factors (for the whole scale) or to the specific subscale / group factor (for subscale composites).

Accordingly, on a subscale row the hier column reports the share of that subscale's composite variance due to the general factor and the sub column the share due to the subscale-specific factor; the latter corresponds to the omega hierarchical subscale of Rodriguez et al. (2016a, 2016b).

The H index (also construct reliability or replicability index) is the correlation between an optimally-weighted composite score and a factor (Hancock & Mueller, 2001; Rodriguez et al., 2016a, 2016b). It, too, can be calculated for the whole scale / general factor as well as for the subscales / group factors. Low values indicate that a latent variable is not well defined by its indicators.

The ECV (Rodriguez et al., 2016a, 2016b) is the ratio of the variance explained by the general factor and the variance explained by the general factor and the group factors.

The PUC (Bonifay et al., 2015; Reise et al., 2013, Rodriguez et al., 2016a, 2016b) refers to the proportion of correlations in the underlying correlation matrix that is not contaminated by variance of both the general factor and the group factors (i.e., correlations between indicators from different group factors, which reflect only general factor variance). The higher the PUC, the more similar a general factor from a multidimensional model will be to the single factor from a unidimensional model.

How to use this function

If model is a lavaan second-order or bifactor solution, only the name of the general factor from the lavaan model needs to be specified additionally with the g_name argument. It is then determined whether this general factor is a second-order factor (second-order model with one second-order factor assumed) or a breadth factor (bifactor model assumed). Please note that this function only works for second-order models if they contain no more than one second-order factor. In case of a second-order solution, a Schmid-Leiman transformation is performed on the first- and second-order loadings and omega coefficients are obtained from the transformed (orthogonalized) solution (see efa_schmid_leiman() for more information on Schmid-Leiman transformation). There is also the possibility to enter a lavaan single factor solution. In this case, g_name is not needed. Finally, if a solution from a lavaan multiple group analysis is entered, the indices are computed for each group. For lavaan input the composite variances entering the omegas are model-implied (computed from the fitted loadings and residual variances), so the coefficients coincide with the observed-score versions when the model fits perfectly. The type argument is not evaluated if model is of class lavaan.

If model is of class efa_schmid_leiman() or psych::schmid() only the type and, depending on the type (see below), the factor_corres arguments need to be specified additionally. If model is of class psych::schmid() and variance = "correlation" (default), it is recommended to also provide the original correlation matrix in cormat to get more accurate results. Otherwise, the correlation matrix will be found based on the pattern matrix and Phi from the psych::schmid() output using the psych::factor.model() function.

If model = NULL, the arguments type, factor_corres (depending on the type, see below), var_names, g_load, s_load, and u2 and either cormat (recommended) or Phi and pattern need to be specified. If Phi and pattern are specified instead of cormat, the correlation matrix is found using the psych::factor.model() function.

The only difference between type = "EFAtools" and type = "psych" is the determination of variable-to-factor correspondences. type = "psych" reproduces the psych::omega() results, where variable-to-factor correspondences are found by taking the highest group factor loading for each variable as the relevant group factor loading. To do this, factor_corres must be left NULL.

The variance argument controls how the total variances (for the whole scale and the subscale composites) are computed. "correlation" (default) finds them from the correlation matrix – the observed-variance form of omega, which reproduces psych::omega(). "sums_load" instead uses the model-implied variances from the squared sums of the general and group factor loadings and the sum of the uniquenesses, so the whole-scale omega total is McDonald's model-implied total and partitions exactly into omega hierarchical plus omega subscale. The two settings agree on the whole-scale omega total and omega hierarchical up to model misfit, and differ mainly in the whole-scale omega subscale, which counts all group-factor variance under "sums_load" but only the assigned subscale composites under "correlation". On the subscale rows the two conventions agree when simple structure is well-achieved.

Value

If found for an SL or lavaan second-order or bifactor solution without multiple groups: A matrix with omegas for the whole scale and for the subscales and (only if add_ind = TRUE) with the H index, ECV, and PUC.

If found for a lavaan single factor solution without multiple groups: A (named) vector with omega total and (if add_ind = TRUE) the H index for the single factor.

If found for a lavaan output from a multiple group analysis: A list containing the output described above for each group.

Source

McDonald, R. P. (1978). Generalizability in factorable domains: ‘‘Domain validity and generalizability’’. Educational and Psychological Measurement, 38, 75–79.

McDonald, R. P. (1985). Factor analysis and related methods. Hillsdale, NJ: Erlbaum.

McDonald, R. P. (1999). Test theory: A unified treatment. Mahwah, NJ: Erlbaum.

Rodriguez, A., Reise, S. P., & Haviland, M. G. (2016a). Applying bifactor statistical indices in the evaluation of psychological measures. Journal of Personality Assessment, 98, 223-237.

Rodriguez, A., Reise, S. P., & Haviland, M. G. (2016b). Evaluating bifactor models: Calculating and interpreting statistical indices. Psychological Methods, 21, 137-150.

Hancock, G. R., & Mueller, R. O. (2001). Rethinking construct reliability within latent variable systems. In R. Cudeck, S. du Toit, & D. Sörbom (Eds.), Structural equation modeling: Present and future—A Festschrift in honor of Karl Jöreskog (pp. 195–216). Lincolnwood, IL: Scientific Software International.

Reise, S. P., Scheines, R., Widaman, K. F., & Haviland, M. G. (2013). Multidimensionality and structural coefficient bias in structural equation modeling: A bifactor perspective. Educational and Psychological Measurement, 73, 5–26.

Bonifay, W. E., Reise, S. P., Scheines, R., & Meijer, R. R. (2015). When are multidimensional data unidimensional enough for structural equation modeling?: An evaluation of the DETECT multidimensionality index. Structural Equation Modeling, 22, 504—516.

Gignac, G. E. (2014). On the Inappropriateness of Using Items to Calculate Total Scale Score Reliability via Coefficient Alpha for Multidimensional Scales. European Journal of Psychological Assessment, 30, 130-139.

See Also

efa_reliability() for the same coefficients in a tidy, long-format result.

Examples

## Use with lavaan outputs
if (requireNamespace("lavaan", quietly = TRUE)) {

# Create and fit bifactor model in lavaan (assume all variables have SDs of 1)
mod <- 'F1 =~ V1 + V2 + V3 + V4 + V5 + V6
        F2 =~ V7 + V8 + V9 + V10 + V11 + V12
        F3 =~ V13 + V14 + V15 + V16 + V17 + V18
        g =~ V1 + V2 + V3 + V4 + V5 + V6 + V7 + V8 + V9 + V10 + V11 + V12 +
             V13 + V14 + V15 + V16 + V17 + V18'
fit_bi <- lavaan::cfa(mod, sample.cov = test_models$baseline$cormat,
                      sample.nobs = 500, estimator = "ml", orthogonal = TRUE)

# Compute omegas and additional indices for bifactor solution
OMEGA(fit_bi, g_name = "g")

# Compute only omegas
OMEGA(fit_bi, g_name = "g", add_ind = FALSE)

# Create and fit second-order model in lavaan (assume all variables have SDs of 1)
mod <- 'F1 =~ V1 + V2 + V3 + V4 + V5 + V6
        F2 =~ V7 + V8 + V9 + V10 + V11 + V12
        F3 =~ V13 + V14 + V15 + V16 + V17 + V18
        g =~ F1 + F2 + F3'
fit_ho <- lavaan::cfa(mod, sample.cov = test_models$baseline$cormat,
                      sample.nobs = 500, estimator = "ml")

# Compute omegas and additional indices for second-order solution
OMEGA(fit_ho, g_name = "g")
}


## Use with an output from the SL function, with type EFAtools
efa_mod <- efa_fit(test_models$baseline$cormat, N = 500, n_factors = 3,
                   estimator = "PAF", rotation = "promax")
sl_mod <- efa_schmid_leiman(efa_mod, estimator = "PAF")

# Two examples how to specify the indicator-to-factor correspondences:

# Based on a specific salience threshold for the loadings (here: .20):
factor_corres_1 <- sl_mod$sl[, c("F1", "F2", "F3")] >= .2

# Or more flexibly (could also be TRUE and FALSE instead of 0 and 1):
factor_corres_2 <- matrix(c(rep(0, 12), rep(1, 6), rep(0, 6), rep(1, 6),
                         rep(0, 6), rep(1, 6), rep(0, 12)), ncol = 3,
                         byrow = FALSE)

OMEGA(sl_mod, type = "EFAtools", factor_corres = factor_corres_1)

## Use with an output from the psych::schmid function, with type psych for
## OMEGA
schmid_mod <- psych::schmid(test_models$baseline$cormat, nfactors = 3,
                            n.obs = 500, fm = "pa", rotate = "Promax")
# Find correlation matrix from phi and pattern matrix from psych::schmid output
OMEGA(schmid_mod, type = "psych")
# Use specified correlation matrix
OMEGA(schmid_mod, type = "psych", cormat = test_models$baseline$cormat)

## Manually specify components (useful if omegas should be computed for a SL
## or bifactor solution found with another program)
## As an example, we extract the elements from an SL output here. This gives
## the same results as in the second example above.

efa_mod <- efa_fit(test_models$baseline$cormat, N = 500, n_factors = 3,
                   estimator = "PAF", rotation = "promax")
sl_mod <- efa_schmid_leiman(efa_mod, estimator = "PAF")

factor_corres <- matrix(c(rep(0, 12), rep(1, 6), rep(0, 6), rep(1, 6),
                        rep(0, 6), rep(1, 6), rep(0, 12)), ncol = 3,
                        byrow = FALSE)

OMEGA(model = NULL, type = "EFAtools", var_names = rownames(sl_mod$sl),
      g_load = sl_mod$sl[, "g"], s_load = sl_mod$sl[, c("F1", "F2", "F3")],
      u2 = sl_mod$sl[, "u2"], cormat = test_models$baseline$cormat,
      factor_corres = factor_corres)

Parallel analysis

Description

PARALLEL() has been superseded by efa_parallel(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

PARALLEL(
  x = NULL,
  N = NA,
  n_vars = NA,
  n_datasets = 1000,
  percent = 95,
  eigen_type = c("PCA", "SMC", "EFA"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  decision_rule = c("means", "percentile", "crawford"),
  n_factors = 1,
  ...
)

Arguments

Value

An object of class efa_retention, identical to the value of efa_parallel(); see there for the components.

See Also

efa_parallel()

Rotate a loading matrix to a target using Procrustes alignment

Description

PROCRUSTES() has been superseded by efa_procrustes(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

PROCRUSTES(
  A,
  Target,
  rotation = c("orthogonal", "oblique"),
  S = NULL,
  T_init = NULL,
  oblique_eps = 1e-05,
  oblique_maxit = 1000,
  oblique_max_line_search = 10,
  oblique_step0 = 1,
  oblique_normalize = FALSE,
  oblique_random_starts = 0,
  oblique_screen_keep = 2,
  oblique_triage_maxit = 25,
  oblique_triage_improve_tol = 0
)

Arguments

Value

A list identical to the value of efa_procrustes(); see there for the components.

See Also

efa_procrustes()

RiskDimensions

Description

A list containing the bivariate correlations (cormat) of the 9 dimensions on which participants in Fischhoff et al. (1978) rated different activities and technologies as well as the sample size (N). This was then analyzed together with ratings of the risks and benefits of these activities and technologies.

Usage

RiskDimensions

Format

A list of 2 with elements "cormat" (9 x 9 matrix of bivariate correlations) and "N" (scalar). The correlation matrix contains the following risk dimensions:

Voluntariness

(numeric) - Voluntariness of exposure to the risk.

Immediacy

(numeric) - Immediacy of the risk's effect.

Known to exposed

(numeric) - How well the risk is known to those exposed to it.

Known to science

(numeric) - How well the risk is known to science.

Controllability

(numeric) - Controllability of the risk.

Newness

(numeric) - Newness of the risk.

Chronic

(numeric) - Whether the risk is chronic rather than catastrophic.

Common

(numeric) - Whether the risk is common rather than dreaded.

Severity of consequences

(numeric) - Severity of the consequences.

Source

Fischhoff, B, Slovic, P, Lichtenstein, S, Read, S, and Combs, B. (1978). How safe is safe enough? A psychometric study of attitudes towards technological risks and benefits. Policy Sciences, 9, 127-152. doi: 10.1007/BF00143739

Scree plot

Description

SCREE() has been superseded by efa_scree(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

SCREE(
  x,
  eigen_type = c("PCA", "SMC", "EFA"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  n_factors = 1,
  ...
)

Arguments

Value

An object of class efa_retention, identical to the value of efa_scree(); see there for the components.

See Also

efa_scree()

Schmid-Leiman transformation

Description

SL() has been superseded by efa_schmid_leiman(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

SL(
  x,
  Phi = NULL,
  type = c("EFAtools", "psych", "SPSS", "none"),
  method = c("PAF", "ML", "ULS", "MINRES"),
  g_name = "g",
  ...
)

Arguments

Value

A list of class c("efa_schmid_leiman", "SL"), identical to the value of efa_schmid_leiman(); see there for the components.

See Also

efa_schmid_leiman()

Sequential model tests

Description

SMT() has been superseded by efa_smt(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

Usage

SMT(
  x,
  N = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra")
)

Arguments

Value

An object of class efa_retention, identical to the value of efa_smt(); see there for the components.

See Also

efa_smt()

Various outputs from SPSS (version 23) FACTOR

Description

Various outputs from SPSS (version 23) FACTOR for the IDS-2 (Grob & Hagmann-von Arx, 2018), the WJIV (3 to 5 and 20 to 39 years; McGrew, LaForte, & Schrank, 2014), the DOSPERT (Frey et al., 2017; Weber, Blais, & Betz, 2002), the NEO-PI-R (Costa, & McCrae, 1992), and four simulated datasets (baseline, case_1a, case_6b, and case_11b, see test_models and population_models) used in Grieder and Steiner (2022).

Usage

SPSS_23

Format

A list of 9 containing EFA results for each of the data sets mentioned above. Each of these nine entries is a list of 4 or 8 (see details), of the following structure:

paf_comm

(vector) - The final communalities obtained with the FACTOR algorithm with PAF and no rotation. For details, see Grieder and Grob (2019).

paf_load

(matrix) - F1 to FN = unrotated factor loadings obtained with the FACTOR algorithm with PAF. Rownames are the abbreviated subtest names.

paf_iter

(numeric) - Number of iterations needed for the principal axis factoring to converge.

var_load

(matrix) - F1 to FN = varimax rotated factor loadings obtained with the FACTOR algorithm with PAF. Rownames are the abbreviated subtest names.

pro_load

(matrix) - F1 to FN = promax rotated factor loadings obtained with the FACTOR algorithm with PAF. Rownames are the abbreviated subtest names.

pro_phi

(matrix) - F1 to FN = intercorrelations of the promax rotated loadings.

sl

(matrix) - g = General / second order factor of the Schmid-Leiman solution. F1 to FN = First order factors of the Schmid-Leiman solution. h2 = Communalities of the Schmid-Leiman solution. This Schmid-Leiman solution was found using the SPSS Syntax provided by Wolff and Preising (2005).

L2

(matrix) - Second order loadings used for the Schmid-Leiman transformation. This Schmid-Leiman solution was found using the SPSS Syntax provided by Wolff and Preising (2005).

Details

The IDS-2, the two WJIV, the DOSPERT, and the NEO-PI-R contain all the above entries, while the four simulated datasets contain only paf_load, var_load, pro_load, and pro_phi.

Source

Grieder, S., & Steiner, M. D. (2022). Algorithmic jingle jungle: A comparison of implementations of principal axis factoring and promax rotation in R and SPSS. Behavior Research Methods, 54, 54–74. doi: 10.3758/s13428-021-01581-x

Wolff, H.G., & Preising, K. (2005). Exploring item and higher order factor structure with the Schmid-Leiman solution: Syntax codes for SPSS and SAS. Behavior Research Methods, 37, 48–58. doi: 10.3758/BF03206397

Grieder, S., & Grob, A. (2019). Exploratory factor analyses of the intelligence and development scales–2: Implications for theory and practice. Assessment. Advance online publication. doi:10.1177/1073191119845051

Grob, A., & Hagmann-von Arx, P. (2018). Intelligence and Development Scales–2 (IDS-2). Intelligenz- und Entwicklungsskalen für Kinder und Jugendliche. [Intelligence and Development Scales for Children and Adolescents.]. Bern, Switzerland: Hogrefe.

Frey, R., Pedroni, A., Mata, R., Rieskamp, J., & Hertwig, R. (2017). Risk preference shares the psychometric structure of major psychological traits. Science Advances, 3, e1701381.

McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Schrank, F. A., McGrew, K. S., & Mather, N. (2014). Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Costa, P. T., & McCrae, R. R. (1992). NEO PI-R professional manual. Odessa, FL: Psychological Assessment Resources, Inc.

Various outputs from SPSS (version 27) FACTOR

Description

Various outputs from SPSS (version 27) FACTOR for the IDS-2 (Grob & Hagmann-von Arx, 2018), the WJIV (3 to 5 and 20 to 39 years; McGrew, LaForte, & Schrank, 2014), the DOSPERT (Frey et al., 2017; Weber, Blais, & Betz, 2002), the NEO-PI-R (Costa, & McCrae, 1992), and four simulated datasets (baseline, case_1a, case_6b, and case_11b, see test_models and population_models) used in Grieder and Steiner (2022).

Usage

SPSS_27

Format

A list of 9 containing EFA results for each of the data sets mentioned above. Each of these nine entries is a list of 4 or 8 (see details), of the following structure:

paf_comm

(vector) - The final communalities obtained with the FACTOR algorithm with PAF and no rotation. For details, see Grieder and Grob (2019).

paf_load

(matrix) - F1 to FN = unrotated factor loadings obtained with the FACTOR algorithm with PAF. Rownames are the abbreviated subtest names.

paf_iter

(numeric) - Number of iterations needed for the principal axis factoring to converge.

var_load

(matrix) - F1 to FN = varimax rotated factor loadings obtained with the FACTOR algorithm with PAF. Rownames are the abbreviated subtest names.

pro_load

(matrix) - F1 to FN = promax rotated factor loadings obtained with the FACTOR algorithm with PAF. Rownames are the abbreviated subtest names.

pro_phi

(matrix) - F1 to FN = intercorrelations of the promax rotated loadings.

sl

(matrix) - g = General / second order factor of the Schmid-Leiman solution. F1 to FN = First order factors of the Schmid-Leiman solution. h2 = Communalities of the Schmid-Leiman solution. This Schmid-Leiman solution was found using the SPSS Syntax provided by Wolff and Preising (2005).

L2

(matrix) - Second order loadings used for the Schmid-Leiman transformation. This Schmid-Leiman solution was found using the SPSS Syntax provided by Wolff and Preising (2005).

Details

The IDS-2, the two WJIV, the DOSPERT, and the NEO-PI-R contain all the above entries, while the four simulated datasets contain only paf_load, var_load, pro_load, and pro_phi.

Source

Grieder, S., & Steiner, M. D. (2022). Algorithmic jingle jungle: A comparison of implementations of principal axis factoring and promax rotation in R and SPSS. Behavior Research Methods, 54, 54–74. doi: 10.3758/s13428-021-01581-x

Wolff, H.G., & Preising, K. (2005). Exploring item and higher order factor structure with the Schmid-Leiman solution: Syntax codes for SPSS and SAS. Behavior Research Methods, 37, 48–58. doi: 10.3758/BF03206397

Grieder, S., & Grob, A. (2019). Exploratory factor analyses of the intelligence and development scales–2: Implications for theory and practice. Assessment. Advance online publication. doi:10.1177/1073191119845051

Grob, A., & Hagmann-von Arx, P. (2018). Intelligence and Development Scales–2 (IDS-2). Intelligenz- und Entwicklungsskalen für Kinder und Jugendliche. [Intelligence and Development Scales for Children and Adolescents.]. Bern, Switzerland: Hogrefe.

Frey, R., Pedroni, A., Mata, R., Rieskamp, J., & Hertwig, R. (2017). Risk preference shares the psychometric structure of major psychological traits. Science Advances, 3, e1701381.

McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Schrank, F. A., McGrew, K. S., & Mather, N. (2014). Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Costa, P. T., & McCrae, R. R. (1992). NEO PI-R professional manual. Odessa, FL: Psychological Assessment Resources, Inc.

UPPS_raw

Description

A dataframe containing responses to the UPPS personality scale (Whiteside & Lynam, 2005) of 645 participants of Study 2 of Steiner and Frey (2020). Each column are the ratings to one of 45 items to assess urgency, premeditation, perseverance, and sensation seeking. The original data can be accessed via https://osf.io/kxp8t/.

Usage

UPPS_raw

Format

A data.frame with 645 rows (participants) and 45 columns, named by a subscale prefix and item number, covering the four UPPS subscales:

perseverance_1 to perseverance_10

(numeric) - Perseverance-subscale items.

premeditation_1 to premeditation_11

(numeric) - Premeditation-subscale items.

ss_1 to ss_12

(numeric) - Sensation-seeking-subscale items.

urgency_1 to urgency_12

(numeric) - Urgency-subscale items.

Source

Whiteside, S. P., Lynam, D. R., Miller, J. D., & Reynolds, S. K. (2005). Validation of the UPPS impulsive behaviour scale: A four-factor model of impulsivity. European Journal of Personality, 19 (7), 559–574.

Steiner, M., & Frey, R. (2020). Representative design in psychological assessment: A case study using the Balloon Analogue Risk Task (BART). PsyArXiv Preprint. doi:10.31234/osf.io/dg4ks

Woodcock Johnson IV: ages 14 to 19

Description

A list containing the bivariate correlations (N = 1,685) of the 47 cognitive and achievement subtests from the WJ IV for 14- to 19-year-olds from the standardization sample obtained from the WJ-IV technical manual (McGrew, LaForte, & Schrank, 2014). Tables are reproduced with permission from the publisher.

Usage

WJIV_ages_14_19

Format

A list of 2 with elements "cormat" (47 x 47 matrix of bivariate correlations) and "N" (scalar). The correlation matrix contains the following variables:

ORLVOC

(numeric) - Oral Vocabulary.

NUMSER

(numeric) - Number Series.

VRBATN

(numeric) - Verbal Attention.

LETPAT

(numeric) - Letter-Pattern Matching.

PHNPRO

(numeric) - Phonological Processing.

STYREC

(numeric) - Story Recall.

VISUAL

(numeric) - Visualization.

GENINF

(numeric) - General Information.

CONFRM

(numeric) - Concept Formation.

NUMREV

(numeric) - Numbers Reversed.

NUMPAT

(numeric) - Number-Pattern Matching.

NWDREP

(numeric) - Nonword Repetition.

VAL

(numeric) - Visual-Auditory Learning.

PICREC

(numeric) - Picture Recognition.

ANLSYN

(numeric) - Analysis-Synthesis.

OBJNUM

(numeric) - Object-Number Sequencing.

PAIRCN

(numeric) - Pair Cancellation.

MEMWRD

(numeric) - Memory for Words.

PICVOC

(numeric) - Picture Vocabulary.

ORLCMP

(numeric) - Oral Comprehension.

SEGMNT

(numeric) - Segmentation.

RPCNAM

(numeric) - Rapid Picture Naming.

SENREP

(numeric) - Sentence Repetition.

UNDDIR

(numeric) - Understanding Directions.

SNDBLN

(numeric) - Sound Blending.

RETFLU

(numeric) - Retrieval Fluency.

SNDAWR

(numeric) - Sound Awareness.

LWIDNT

(numeric) - Letter-Word Identification.

APPROB

(numeric) - Applied Problems.

SPELL

(numeric) - Spelling.

PSGCMP

(numeric) - Passage Comprehension.

CALC

(numeric) - Calculation.

WRTSMP

(numeric) - Writing Samples.

WRDATK

(numeric) - Word Attack.

ORLRDG

(numeric) - Oral Reading.

SNRDFL

(numeric) - Sentence Reading Fluency.

MTHFLU

(numeric) - Math Facts Fluency.

SNWRFL

(numeric) - Sentence Writing Fluency.

RDGREC

(numeric) - Reading Recall.

NUMMAT

(numeric) - Number Matrices.

EDIT

(numeric) - Editing.

WRDFLU

(numeric) - Word Reading Fluency.

SPLSND

(numeric) - Spelling of Sounds.

RDGVOC

(numeric) - Reading Vocabulary.

SCI

(numeric) - Science.

SOC

(numeric) - Social Studies.

HUM

(numeric) - Humanities.

Source

McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Schrank, F. A., McGrew, K. S., & Mather, N. (2014). Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Woodcock Johnson IV: ages 20 to 39

Description

A list containing the bivariate correlations (N = 1,251) of the 47 cognitive and achievement subtests from the WJ IV for the 20- to 39-year-olds from the standardization sample obtained from the WJ-IV technical manual (McGrew, LaForte, & Schrank, 2014). Tables are reproduced with permission from the publisher.

Usage

WJIV_ages_20_39

Format

A list of 2 with elements "cormat" (47 x 47 matrix of bivariate correlations) and "N" (scalar). The correlation matrix contains the following variables:

ORLVOC

(numeric) - Oral Vocabulary.

NUMSER

(numeric) - Number Series.

VRBATN

(numeric) - Verbal Attention.

LETPAT

(numeric) - Letter-Pattern Matching.

PHNPRO

(numeric) - Phonological Processing.

STYREC

(numeric) - Story Recall.

VISUAL

(numeric) - Visualization.

GENINF

(numeric) - General Information.

CONFRM

(numeric) - Concept Formation.

NUMREV

(numeric) - Numbers Reversed.

NUMPAT

(numeric) - Number-Pattern Matching.

NWDREP

(numeric) - Nonword Repetition.

VAL

(numeric) - Visual-Auditory Learning.

PICREC

(numeric) - Picture Recognition.

ANLSYN

(numeric) - Analysis-Synthesis.

OBJNUM

(numeric) - Object-Number Sequencing.

PAIRCN

(numeric) - Pair Cancellation.

MEMWRD

(numeric) - Memory for Words.

PICVOC

(numeric) - Picture Vocabulary.

ORLCMP

(numeric) - Oral Comprehension.

SEGMNT

(numeric) - Segmentation.

RPCNAM

(numeric) - Rapid Picture Naming.

SENREP

(numeric) - Sentence Repetition.

UNDDIR

(numeric) - Understanding Directions.

SNDBLN

(numeric) - Sound Blending.

RETFLU

(numeric) - Retrieval Fluency.

SNDAWR

(numeric) - Sound Awareness.

LWIDNT

(numeric) - Letter-Word Identification.

APPROB

(numeric) - Applied Problems.

SPELL

(numeric) - Spelling.

PSGCMP

(numeric) - Passage Comprehension.

CALC

(numeric) - Calculation.

WRTSMP

(numeric) - Writing Samples.

WRDATK

(numeric) - Word Attack.

ORLRDG

(numeric) - Oral Reading.

SNRDFL

(numeric) - Sentence Reading Fluency.

MTHFLU

(numeric) - Math Facts Fluency.

SNWRFL

(numeric) - Sentence Writing Fluency.

RDGREC

(numeric) - Reading Recall.

NUMMAT

(numeric) - Number Matrices.

EDIT

(numeric) - Editing.

WRDFLU

(numeric) - Word Reading Fluency.

SPLSND

(numeric) - Spelling of Sounds.

RDGVOC

(numeric) - Reading Vocabulary.

SCI

(numeric) - Science.

SOC

(numeric) - Social Studies.

HUM

(numeric) - Humanities.

Source

McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Schrank, F. A., McGrew, K. S., & Mather, N. (2014). Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Woodcock Johnson IV: ages 3 to 5

Description

A list containing the bivariate correlations (N = 435) of the 29 cognitive and achievement subtests from the WJ IV for 3- to 5-year-olds from the standardization sample obtained from the WJ IV technical Manual (McGrew, LaForte, & Schrank, 2014). Tables are reproduced with permission from the publisher.

Usage

WJIV_ages_3_5

Format

A list of 2 with elements "cormat" (29 x 29 matrix of bivariate correlations) and "N" (scalar). The correlation matrix contains the following variables:

ORLVOC

(numeric) - Oral Vocabulary.

VRBATN

(numeric) - Verbal Attention.

LETPAT

(numeric) - Phonological Processing.

STYREC

(numeric) - Story Recall.

VISUAL

(numeric) - Visualization.

GENINF

(numeric) - General Information.

CONFRM

(numeric) - Concept Formation.

NUMREV

(numeric) - Numbers Reversed.

NUMPAT

(numeric) - Number-Pattern Matching.

NWDREP

(numeric) - Nonword Repetition.

VAL

(numeric) - Visual-Auditory Learning.

PICREC

(numeric) - Picture Recognition.

MEMWRD

(numeric) - Memory for Words.

PICVOC

(numeric) - Picture Vocabulary.

ORLCMP

(numeric) - Oral Comprehension.

SEGMNT

(numeric) - Segmentation.

RPCNAM

(numeric) - Rapid Picture Naming.

SENREP

(numeric) - Sentence Repetition.

UNDDIR

(numeric) - Understanding Directions.

SNDBLN

(numeric) - Sound Blending.

RETFLU

(numeric) - Retrieval Fluency.

SNDAWR

(numeric) - Sound Awareness.

LWIDNT

(numeric) - Letter-Word Identification.

APPROB

(numeric) - Applied Problems.

SPELL

(numeric) - Spelling.

PSGCMP

(numeric) - Passage Comprehension.

SCI

(numeric) - Science.

SOC

(numeric) - Social Studies.

HUM

(numeric) - Humanities.

Source

McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Schrank, F. A., McGrew, K. S., & Mather, N. (2014). Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Woodcock Johnson IV: ages 40 to 90 plus

Description

A list containing the bivariate correlations (N = 1,146) of the 47 cognitive and achievement subtests from the WJ IV for 40- to 90+-year-olds from the standardization sample obtained from the WJ-IV technical manual (McGrew, LaForte, & Schrank, 2014). Tables are reproduced with permission from the publisher.

Usage

WJIV_ages_40_90

Format

A list of 2 with elements "cormat" (47 x 47 matrix of bivariate correlations) and "N". The correlation matrix contains the following variables:

ORLVOC

(numeric) - Oral Vocabulary.

NUMSER

(numeric) - Number Series.

VRBATN

(numeric) - Verbal Attention.

LETPAT

(numeric) - Letter-Pattern Matching.

PHNPRO

(numeric) - Phonological Processing.

STYREC

(numeric) - Story Recall.

VISUAL

(numeric) - Visualization.

GENINF

(numeric) - General Information.

CONFRM

(numeric) - Concept Formation.

NUMREV

(numeric) - Numbers Reversed.

NUMPAT

(numeric) - Number-Pattern Matching.

NWDREP

(numeric) - Nonword Repetition.

VAL

(numeric) - Visual-Auditory Learning.

PICREC

(numeric) - Picture Recognition.

ANLSYN

(numeric) - Analysis-Synthesis.

OBJNUM

(numeric) - Object-Number Sequencing.

PAIRCN

(numeric) - Pair Cancellation.

MEMWRD

(numeric) - Memory for Words.

PICVOC

(numeric) - Picture Vocabulary.

ORLCMP

(numeric) - Oral Comprehension.

SEGMNT

(numeric) - Segmentation.

RPCNAM

(numeric) - Rapid Picture Naming.

SENREP

(numeric) - Sentence Repetition.

UNDDIR

(numeric) - Understanding Directions.

SNDBLN

(numeric) - Sound Blending.

RETFLU

(numeric) - Retrieval Fluency.

SNDAWR

(numeric) - Sound Awareness.

LWIDNT

(numeric) - Letter-Word Identification.

APPROB

(numeric) - Applied Problems.

SPELL

(numeric) - Spelling.

PSGCMP

(numeric) - Passage Comprehension.

CALC

(numeric) - Calculation.

WRTSMP

(numeric) - Writing Samples.

WRDATK

(numeric) - Word Attack.

ORLRDG

(numeric) - Oral Reading.

SNRDFL

(numeric) - Sentence Reading Fluency.

MTHFLU

(numeric) - Math Facts Fluency.

SNWRFL

(numeric) - Sentence Writing Fluency.

RDGREC

(numeric) - Reading Recall.

NUMMAT

(numeric) - Number Matrices.

EDIT

(numeric) - Editing.

WRDFLU

(numeric) - Word Reading Fluency.

SPLSND

(numeric) - Spelling of Sounds.

RDGVOC

(numeric) - Reading Vocabulary.

SCI

(numeric) - Science.

SOC

(numeric) - Social Studies.

HUM

(numeric) - Humanities.

Source

McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Schrank, F. A., McGrew, K. S., & Mather, N. (2014). Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Woodcock Johnson IV: ages 6 to 8

Description

A list containing the bivariate correlations (N = 825) of the 47 cognitive and achievement subtests from the WJ IV for 6- to 8-year-olds from the standardization sample obtained from the WJ-IV technical manual (McGrew, LaForte, & Schrank, 2014). Tables are reproduced with permission from the publisher.

Usage

WJIV_ages_6_8

Format

A list of 2 with elements "cormat" (47 x 47 matrix of bivariate correlations) and "N". The correlation matrix contains the following variables:

ORLVOC

(numeric) - Oral Vocabulary.

NUMSER

(numeric) - Number Series.

VRBATN

(numeric) - Verbal Attention.

LETPAT

(numeric) - Letter-Pattern Matching.

PHNPRO

(numeric) - Phonological Processing.

STYREC

(numeric) - Story Recall.

VISUAL

(numeric) - Visualization.

GENINF

(numeric) - General Information.

CONFRM

(numeric) - Concept Formation.

NUMREV

(numeric) - Numbers Reversed.

NUMPAT

(numeric) - Number-Pattern Matching.

NWDREP

(numeric) - Nonword Repetition.

VAL

(numeric) - Visual-Auditory Learning.

PICREC

(numeric) - Picture Recognition.

ANLSYN

(numeric) - Analysis-Synthesis.

OBJNUM

(numeric) - Object-Number Sequencing.

PAIRCN

(numeric) - Pair Cancellation.

MEMWRD

(numeric) - Memory for Words.

PICVOC

(numeric) - Picture Vocabulary.

ORLCMP

(numeric) - Oral Comprehension.

SEGMNT

(numeric) - Segmentation.

RPCNAM

(numeric) - Rapid Picture Naming.

SENREP

(numeric) - Sentence Repetition.

UNDDIR

(numeric) - Understanding Directions.

SNDBLN

(numeric) - Sound Blending.

RETFLU

(numeric) - Retrieval Fluency.

SNDAWR

(numeric) - Sound Awareness.

LWIDNT

(numeric) - Letter-Word Identification.

APPROB

(numeric) - Applied Problems.

SPELL

(numeric) - Spelling.

PSGCMP

(numeric) - Passage Comprehension.

CALC

(numeric) - Calculation.

WRTSMP

(numeric) - Writing Samples.

WRDATK

(numeric) - Word Attack.

ORLRDG

(numeric) - Oral Reading.

SNRDFL

(numeric) - Sentence Reading Fluency.

MTHFLU

(numeric) - Math Facts Fluency.

SNWRFL

(numeric) - Sentence Writing Fluency.

RDGREC

(numeric) - Reading Recall.

NUMMAT

(numeric) - Number Matrices.

EDIT

(numeric) - Editing.

WRDFLU

(numeric) - Word Reading Fluency.

SPLSND

(numeric) - Spelling of Sounds.

RDGVOC

(numeric) - Reading Vocabulary.

SCI

(numeric) - Science.

SOC

(numeric) - Social Studies.

HUM

(numeric) - Humanities.

Source

McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Schrank, F. A., McGrew, K. S., & Mather, N. (2014). Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Woodcock Johnson IV: ages 9 to 13

Description

A list containing the bivariate correlations (N = 1,572) of the 47 cognitive and achievement subtests from the WJ IV for 9- to 13-year-olds from the standardization sample obtained from the WJ-IV technical manual (McGrew, LaForte, & Schrank, 2014). Tables are reproduced with permission from the publisher.

Usage

WJIV_ages_9_13

Format

A list of 2 with elements "cormat" (47 x 47 matrix of bivariate correlations) and "N". The correlation matrix contains the following variables:

ORLVOC

(numeric) - Oral Vocabulary.

NUMSER

(numeric) - Number Series.

VRBATN

(numeric) - Verbal Attention.

LETPAT

(numeric) - Letter-Pattern Matching.

PHNPRO

(numeric) - Phonological Processing.

STYREC

(numeric) - Story Recall.

VISUAL

(numeric) - Visualization.

GENINF

(numeric) - General Information.

CONFRM

(numeric) - Concept Formation.

NUMREV

(numeric) - Numbers Reversed.

NUMPAT

(numeric) - Number-Pattern Matching.

NWDREP

(numeric) - Nonword Repetition.

VAL

(numeric) - Visual-Auditory Learning.

PICREC

(numeric) - Picture Recognition.

ANLSYN

(numeric) - Analysis-Synthesis.

OBJNUM

(numeric) - Object-Number Sequencing.

PAIRCN

(numeric) - Pair Cancellation.

MEMWRD

(numeric) - Memory for Words.

PICVOC

(numeric) - Picture Vocabulary.

ORLCMP

(numeric) - Oral Comprehension.

SEGMNT

(numeric) - Segmentation.

RPCNAM

(numeric) - Rapid Picture Naming.

SENREP

(numeric) - Sentence Repetition.

UNDDIR

(numeric) - Understanding Directions.

SNDBLN

(numeric) - Sound Blending.

RETFLU

(numeric) - Retrieval Fluency.

SNDAWR

(numeric) - Sound Awareness.

LWIDNT

(numeric) - Letter-Word Identification.

APPROB

(numeric) - Applied Problems.

SPELL

(numeric) - Spelling.

PSGCMP

(numeric) - Passage Comprehension.

CALC

(numeric) - Calculation.

WRTSMP

(numeric) - Writing Samples.

WRDATK

(numeric) - Word Attack.

ORLRDG

(numeric) - Oral Reading.

SNRDFL

(numeric) - Sentence Reading Fluency.

MTHFLU

(numeric) - Math Facts Fluency.

SNWRFL

(numeric) - Sentence Writing Fluency.

RDGREC

(numeric) - Reading Recall.

NUMMAT

(numeric) - Number Matrices.

EDIT

(numeric) - Editing.

WRDFLU

(numeric) - Word Reading Fluency.

SPLSND

(numeric) - Spelling of Sounds.

RDGVOC

(numeric) - Reading Vocabulary.

SCI

(numeric) - Science.

SOC

(numeric) - Social Studies.

HUM

(numeric) - Humanities.

Source

McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Schrank, F. A., McGrew, K. S., & Mather, N. (2014). Woodcock-Johnson IV. Rolling Meadows, IL: Riverside.

Model averaging across different EFA estimators and types

Description

Not all EFA procedures always arrive at the same solution. This function allows you perform a number of EFAs from different estimators (e.g., Maximum Likelihood and Principal Axis Factoring), with different implementations (e.g., the SPSS and psych implementations of Principal Axis Factoring), and across different rotations of the same type (e.g., multiple oblique rotations, like promax and oblimin). efa_average() will then run all these EFAs (using the efa_fit() function) and provide a summary across the different solutions.

Usage

efa_average(
  x,
  n_factors,
  N = NA,
  estimator = "PAF",
  rotation = "promax",
  type = "none",
  averaging = c("mean", "median"),
  trim = 0,
  salience_threshold = 0.3,
  max_iter = 10000,
  init_comm = c("smc", "mac", "unity"),
  criterion = c(0.001),
  criterion_type = c("sum", "max_individual"),
  abs_eigen = c(TRUE),
  varimax_type = c("svd", "kaiser"),
  normalize = TRUE,
  k_promax = 2:4,
  k_simplimax = ncol(x),
  p_type = c("norm", "unnorm"),
  precision = 1e-05,
  start_method = c("psych", "factanal"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra", "fiml"),
  show_progress = TRUE,
  seed = NULL,
  P_type = lifecycle::deprecated()
)

Arguments

Details

As a first step in this function, a grid is produced containing the setting combinations for the to-be-performed EFAs. These settings are then entered as arguments to the efa_fit() function and the EFAs are run in a second step. After all EFAs are run, the factor solutions are averaged and their variability determined in a third step.

When raw data are supplied, the correlation matrix is computed once before the grid is run and reused for every EFA in it. Under cor_method = "fiml" this means the saturated multivariate-normal moments are EM-estimated a single time (from the raw data with missing values, assuming the data are missing at random) and the resulting two-stage correlation is analysed by every solution in the grid; the EM is not re-run per solution. Under cor_method = "fiml", use does not select cases (every case contributes to the EM). The averaged loadings and communalities are then the two-stage FIML estimates, but the averaged Chi-Square and the indices derived from it (CFI, TLI, RMSEA, AIC, BIC, ECVI) are the ordinary ML/ULS discrepancy statistics on the EM correlation, not the corrected two-stage (Satorra-Bentler) statistics that a standalone efa_fit() with cor_method = "fiml" reports; in particular the averaged AIC and BIC are finite here rather than NA.

The grid containing the setting combinations is produced based on the entries to the respective arguments. To this end, all possible combinations resulting in unique EFA models are considered: combinations that resolve to the same model are run only once. Two combinations are the same model only if every setting the fit consumes agrees, and for "PAF" that includes the iteration cap max_iter. Since a named type brings its own cap, a type of c("none", "SPSS") whose specific settings match the SPSS combination in every other respect still gives two "PAF" models, unless max_iter is also set to the cap of the SPSS implementation. We include here a list of arguments that are only evaluated under specific conditions:

The arguments init_comm, criterion, criterion_type, abs_eigen, and max_iter are only evaluated if "PAF" is included in estimator and "none" is included in type.

The argument varimax_type is only evaluated if "varimax", "promax", "oblique", or "orthogonal" is included in rotation and "none" is included in type.

The argument normalize is only evaluated if rotation is not set to "none" and "none" is included in type.

The argument k_simplimax is only evaluated if "simplimax" or "oblique" is included in rotation.

The arguments k_promax and p_type are only evaluated if "promax" or "oblique" is included in rotation and "none" is included in type.

The argument start_method is only evaluated if "ML" is included in estimator.

To avoid a bias in the averaged factor solutions from problematic solutions, these are excluded prior to averaging. A solution is deemed problematic if at least one of the following is true: an error occurred, the model did not converge, or there is at least one Heywood (improper) case (a communality at or above 1, or, for ML/ULS, a uniqueness pinned at the estimator's lower bound). Information on errors, convergence, and Heywood cases are returned in the implementations_grid and a summary of these is given when printing the output. In addition to these, information on the admissibility of the factor solutions is also included. A solution was deemed admissible if (1) no error occurred, (2) the model converged, (3) no Heywood cases are present, and (4) there are at least two salient loadings (i.e., loadings exceeding the specified salience_threshold) for each factor. So, solutions failing one of the first three of these criteria of admissibility are also deemed problematic and therefore excluded from averaging. However, solutions failing only the fourth criterion of admissibility are still included for averaging. Finally, if all solutions are problematic (e.g., all solutions contain Heywood cases), no averaging is performed and the respective outputs are NA. In this case, the implementations_grid should be inspected to see if there are any error messages, and the separate EFA solutions that are also included in the output can be inspected as well, for example, to see where Heywood cases occurred.

A core output of this function includes the average, minimum, and maximum loadings derived from all non-problematic (see above) factor solutions. Please note that these are not entire solutions, but the matrices include the average, minimum, or maximum value for each cell (i.e., each loading separately). This means that, for example, the matrix with the minimum loadings will contain the minimum value in any of the factor solutions for each specific loading, and therefore most likely contains loadings from different factor solutions. The matrices containing the minimum and maximum factor solutions can therefore not be interpreted as whole factor solutions.

The output also includes information on the average, minimum, maximum, and variability of the fit indices across the non-problematic factor solutions. It is important to note that not all fit indices are computed for all fit methods: For ML and ULS, all fit indices can be computed, while for PAF the chi-square-based indices (the chi-square statistic and its significance, CFI, TLI, RMSEA, AIC, BIC, and ECVI) are NA. The common part accounted for (CAF) index (Lorenzo-Seva, Timmerman, & Kiers, 2011) and the residual-based SRMR and RMSR are still computed for PAF. As a consequence, if only "PAF" is included in the estimator argument, averaging is performed for the CAF, SRMR, and RMSR, while the chi-square-based indices are NA. If a combination of "PAF" and "ML" and/or "ULS" are included in the estimator argument, the CAF, SRMR, and RMSR are averaged across all non-problematic factor solutions, while the chi-square-based indices are only averaged across the ML and ULS solutions. The user should therefore keep in mind that the number of EFAs across which the fit indices are averaged can diverge for the CAF, SRMR, and RMSR compared to the chi-square-based indices.

Each reported fit index is summarised across the (non-problematic) solutions in the same descriptive way: the average, standard deviation, minimum, and maximum of the per-solution values. This includes the chi-square significance level (p_chi), which is therefore the mean (or median) of the per-solution p-values and is purely descriptive; it is not the p-value of any pooled chi-square test.

Value

A list of class c("efa_average", "EFA_AVERAGE") containing

If the supplied arguments admit only a single EFA, there is nothing to average across: that one efa_fit() object is returned instead, with a warning.

Source

Grieder, S., & Steiner, M. D. (2022). Algorithmic jingle jungle: A comparison of implementations of principal axis factoring and promax rotation in R and SPSS. Behavior Research Methods, 54, 54–74. doi: 10.3758/s13428-021-01581-x

Hendrickson, A. E., & White, P. O. (1964). Promax: A quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 17 , 65–70. doi: 10.1111/j.2044-8317.1964.tb00244.x

Lorenzo-Seva, U., Timmerman, M. E., & Kiers, H. A. L. (2011). The Hull Method for Selecting the Number of Common Factors, Multivariate Behavioral Research, 46, 340-364, doi: 10.1080/00273171.2011.564527

Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200. doi: 10.1007/BF02289233

See Also

Other factor analysis: efa_fit(), efa_group(), efa_mi(), plot.efa_group(), print.efa_group()

Examples

# Averaging across one implementation each of PAF (EFAtools type), ULS, and
# ML with one implementation of promax (EFAtools type) (3 EFAs)
Aver_meth <- efa_average(test_models$baseline$cormat, n_factors = 3, N = 500,
                         estimator = c("PAF", "ULS", "ML"), type = "EFAtools",
                         start_method = "psych")


# Averaging across different implementations of PAF and promax rotation (72 EFAs)
Aver_PAF <- efa_average(test_models$baseline$cormat, n_factors = 3, N = 500)

# Use median instead of mean for averaging (72 EFAs)
Aver_PAF_md <- efa_average(test_models$baseline$cormat, n_factors = 3, N = 500,
                           averaging = "median")

# Averaging across different implementations of PAF and promax rotation,
# and across ULS and different versions of ML (108 EFAs)
Aver_meth_ext <- efa_average(test_models$baseline$cormat, n_factors = 3, N = 500,
                             estimator = c("PAF", "ULS", "ML"))

# Averaging across different oblique rotation methods, using one implementation
# of ML and one implementation of promax (EFAtools type) (7 EFAs)
Aver_rot <- efa_average(test_models$baseline$cormat, n_factors = 3, N = 500,
                         estimator = "ML", rotation = "oblique", type = "EFAtools",
                         start_method = "psych")



# Two-stage FIML correlations from raw data with missing values: the EM
# saturated moments are estimated once and the resulting correlation is
# averaged across the grid of EFAs.
x_miss <- GRiPS_raw
x_miss[cbind(1:20, 1)] <- NA
Aver_fiml <- efa_average(x_miss, n_factors = 1, estimator = c("PAF", "ML"),
                         cor_method = "fiml")

Bartlett's test of sphericity

Description

This function tests whether a correlation matrix is significantly different from an identity matrix (Bartlett, 1951). If the Bartlett's test is not significant, the correlation matrix is not suitable for factor analysis because the variables show too little covariance.

Usage

efa_bartlett(
  x,
  N = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra")
)

Arguments

Details

Bartlett (1951) proposed this statistic to determine a correlation matrix' suitability for factor analysis. The statistic is approximately chi square distributed with df = \frac{p(p - 1)}{2} and is given by

chi^2 = -log(det(R)) (N - 1 - (2 * p + 5)/6)

where det(R) is the determinant of the correlation matrix, N is the sample size, and p is the number of variables.

This tests requires multivariate normality. If this condition is not met, the Kaiser-Meyer-Olkin criterion (efa_kmo()) can still be used.

This function was heavily influenced by the psych::cortest.bartlett() function from the psych package.

The efa_bartlett function can also be called together with the (efa_kmo()) function and with factor retention criteria in the efa_retain() function.

Value

A list containing

Source

Bartlett, M. S. (1951). The effect of standardization on a Chi-square approximation in factor analysis. Biometrika, 38, 337-344.

See Also

efa_kmo() for another measure to determine suitability for factor analysis.

efa_retain() as a wrapper function for this function, efa_kmo() and several factor retention criteria.

Other factor analysis suitability: efa_kmo(), efa_screen(), print.efa_screen()

Examples

efa_bartlett(test_models$baseline$cormat, N = 500)

Comparison data

Description

Factor retention method introduced by Ruscio and Roche (2012). The code was adapted from the CD code by Auerswald and Moshagen (2019) available at https://osf.io/x5cz2/?view_only=d03efba1fd0f4c849a87db82e6705668

Usage

efa_cd(
  x,
  n_factors_max = NA,
  N_pop = 10000,
  N_samples = 500,
  alpha = 0.3,
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  max_iter = 50
)

Arguments

Details

Comparison data (CD) extends parallel analysis by reproducing the observed correlation matrix rather than generating random data: datasets with a known factor structure are generated with an increasing number of factors, and the smallest number for which adding a further factor no longer significantly improves the reproduction of the observed eigenvalues is retained (Ruscio & Roche, 2012).

Note that if the data contains missing values, these will be removed for the comparison data procedure using stats::na.omit(). If missing data should be treated differently, e.g., by imputation, do this outside efa_cd() and then pass the complete data.

Value

An object of class efa_retention (see print.efa_retention() and plot.efa_retention() for the print and plot methods). Its main fields are:

Source

Auerswald, M., & Moshagen, M. (2019). How to determine the number of factors to retain in exploratory factor analysis: A comparison of extraction methods under realistic conditions. Psychological Methods, 24(4), 468–491. https://doi.org/10.1037/met0000200

Ruscio, J., & Roche, B. (2012). Determining the number of factors to retain in an exploratory factor analysis using comparison data of known factorial structure. Psychological Assessment, 24, 282–292. doi: 10.1037/a0025697

See Also

efa_retain() as a wrapper function for this and the other factor retention criteria.

Other factor retention criteria: efa_ekc(), efa_hull(), efa_kgc(), efa_map(), efa_nest(), efa_parallel(), efa_retain(), efa_scree(), efa_smt()

Examples

# determine n factors of the GRiPS
efa_cd(GRiPS_raw)

# determine n factors of the DOSPERT risk subscale
efa_cd(DOSPERT_raw)

Compare two vectors or matrices (communalities or loadings)

Description

The function takes two objects of the same dimensions containing numeric information (loadings or communalities) and returns a list of class efa_compare containing summary information of the differences of the objects.

Usage

efa_compare(
  x,
  y,
  reorder = c("congruence", "names", "none"),
  corres = TRUE,
  thresh = 0.3,
  digits = 4,
  m_red = 0.001,
  range_red = 0.001,
  round_red = 3,
  print_diff = TRUE,
  na.rm = FALSE,
  x_labels = c("x", "y"),
  plot = TRUE,
  plot_red = 0.01
)

Arguments

Value

A list of class efa_compare containing summary statistics on the differences of x and y.

See Also

efa_fit() for the solutions being compared, and efa_procrustes() to rotate one solution onto another before comparing.

Examples

# A type SPSS EFA to mimick the SPSS implementation
EFA_SPSS_6 <- efa_fit(test_models$case_11b$cormat, n_factors = 6,
                      estimate_control = estimate_control(type = "SPSS"),
                      rotate_control = rotate_control(type = "SPSS"))

# A type psych EFA to mimick the psych::fa() implementation
EFA_psych_6 <- efa_fit(test_models$case_11b$cormat, n_factors = 6,
                       estimate_control = estimate_control(type = "psych"),
                       rotate_control = rotate_control(type = "psych"))

# compare the two
efa_compare(EFA_SPSS_6$unrot_loadings, EFA_psych_6$unrot_loadings,
            x_labels = c("SPSS", "psych"))

Empirical Kaiser criterion

Description

The empirical Kaiser criterion incorporates random sampling variations of the eigenvalues from the Kaiser-Guttman criterion (efa_kgc(); see Auerswald & Moshagen , 2019; Braeken & van Assen, 2017). The code is based on Braeken & van Assen, (2017) and on Auerswald and Moshagen (2019).

Usage

efa_ekc(
  x,
  N = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  type = "BvA2017"
)

Arguments

Details

The Kaiser-Guttman criterion was defined with the intend that a factor should only be extracted if it explains at least as much variance as a single factor (see efa_kgc()). However, this only applies to population-level correlation matrices. Due to sampling variation, the KGC strongly overestimates the number of factors to retrieve (e.g., Zwick & Velicer, 1986). To account for this and to introduce a factor retention method that performs well with small number of indicators and correlated factors (cases where the performance of parallel analysis, see efa_parallel(), is known to deteriorate) Braeken and van Assen (2017) introduced the empirical Kaiser criterion in which a series of reference eigenvalues is created as a function of the variables-to-sample-size ratio and the observed eigenvalues.

Braeken and van Assen (2017) showed that "(a) EKC performs about as well as parallel analysis for data arising from the null, 1-factor, or orthogonal factors model; and (b) clearly outperforms parallel analysis for the specific case of oblique factors, particularly whenever factor intercorrelation is moderate to high and the number of variables per factor is small, which is characteristic of many applications these days" (p.463-464).

In EFAtools version <= 0.5.0 only the implementation of Auerswald and Moshagen (2019) was implemented (now available with type = "AM2019"). However, this implementation, that was probably also used in Caron (2025), differs from the original implementation by Braeken and van Assen (2017) in that it corrects by the reference values, i.e., without using the empirical eigenvalues used in the original implementation. Thanks to Luis Eduardo Garrido for pointing this out and to Johan Braeken for sharing sample code, based on which the original version is now implemented and used by default with type = "BvA2017".

While the adapted version performed relatively well in the simulation studies by Auerswald and Moshagen (2019) and Caron (2025), the theoretical derivations of the EKC as introduced by Braeken and van Assen (2017) may no longer hold. Currently we are unaware of studies comparing the two implementations, but based on our own brief comparisons across multiple datasets, the two implementations appear to often differ substantially regarding the number of factors suggested.

As both implementations exist in different packages and studies, we provide both versions here. Be sure to state clearly which version you use when reporting your results to avoid confusion and ensure reproducibility.

Value

An object of class efa_retention (see print.efa_retention() and plot.efa_retention() for the print and plot methods). Its main fields are:

Source

Auerswald, M., & Moshagen, M. (2019). How to determine the number of factors to retain in exploratory factor analysis: A comparison of extraction methods under realistic conditions. Psychological Methods, 24(4), 468–491. https://doi.org/10.1037/met0000200

Braeken, J., & van Assen, M. A. (2017). An empirical Kaiser criterion. Psychological Methods, 22, 450 – 466. http://dx.doi.org/10.1037/met0000074

Caron, P.-O. (2025). A Comparison of the Next Eigenvalue Sufficiency Test to Other Stopping Rules for the Number of Factors in Factor Analysis. Educational and Psychological Measurement, Online-first publication. https://doi.org/10.1177/00131644241308528

Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99, 432–442. http://dx.doi.org/10.1037/0033-2909.99.3.432

See Also

efa_retain() as a wrapper function for this and the other factor retention criteria.

Other factor retention criteria: efa_cd(), efa_hull(), efa_kgc(), efa_map(), efa_nest(), efa_parallel(), efa_retain(), efa_scree(), efa_smt()

Examples

# original implementation
efa_ekc(test_models$baseline$cormat, N = 500)

Exploratory factor analysis (EFA)

Description

This function does an EFA with either PAF, ML, ULS/MINRES, or DWLS with or without subsequent rotation. All arguments with default value NA can be left to default if type is set to one of "EFAtools", "SPSS", or "psych". The respective specifications are then handled according to the specified type (see details).

Usage

efa_fit(
  x,
  n_factors,
  N = NA,
  estimator = c("PAF", "ML", "ULS", "MINRES", "DWLS"),
  rotation = c("none", "varimax", "equamax", "quartimax", "geominT", "bentlerT",
    "bifactorT", "promax", "oblimin", "quartimin", "simplimax", "bentlerQ", "geominQ",
    "bifactorQ"),
  se = c("none", "information", "sandwich", "np-boot"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra", "fiml"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  estimate_control = NULL,
  rotate_control = NULL,
  b_boot = 1000,
  ci = 0.95,
  seed = NULL,
  ...
)

Arguments

Details

There are two main ways to use this function. The easiest way is to use it with a specified type (see above), which sets most of the other arguments accordingly. Another way is to use it more flexibly by explicitly specifying all arguments used and set type to "none" (see examples). A mix of the two can also be done by specifying a type as well as additional arguments. However, this will throw warnings to avoid unintentional deviations from the implementations according to the specified type.

Estimators

The estimator is chosen with estimator.

• PAF (principal axis factoring) iteratively estimates the communalities and makes no distributional assumptions, which makes it robust and a good general-purpose default. Because it minimises no likelihood or weighted discrepancy it provides no model chi-square, and hence no chi-square-based fit indices (see Fit indices). The PAF iteration is governed by init_comm, criterion, criterion_type, max_iter, and abs_eigen (set by type; see Using the type presets).

• ML (maximum likelihood) maximises the normal-theory likelihood. It yields the full set of fit indices and is the only estimator with analytic expected-information standard errors (se = "information"), but it assumes multivariate normality and is the most prone to Heywood (improper) cases. Its starting values are set by start_method.

• ULS (unweighted least squares) minimises the sum of squared correlation residuals. "MINRES" (minimum residual) is the same estimator under a different name and returns identical results. It makes no normality assumption, is robust to mild non-normality, and yields the full set of fit indices.

• DWLS (diagonally weighted least squares) is the recommended estimator for ordinal data. It weights each off-diagonal correlation residual by the inverse asymptotic variance of the corresponding polychoric correlation (Muthén, 1984), reproducing the loadings of a diagonally weighted least squares fit (e.g. lavaan::efa(..., estimator = "DWLS")). It therefore requires raw ordinal data with cor_method = "poly" or "tetra" and has no fallback for a supplied correlation matrix or a continuous cor_method. Because the weighting follows the polychoric asymptotic covariance, the matrix and the weights are estimated on the listwise-complete cases. Its fit-index behaviour is described under Fit indices.

Correlation methods

When raw data are supplied, cor_method selects how the correlation matrix is computed (it is ignored when a correlation matrix is entered directly).

• "pearson" (default), "spearman", and "kendall" are passed to stats::cor() for continuous or rank data.

• "poly" / "tetra" compute polychoric / tetrachoric correlations for ordinal / binary data, assuming an underlying bivariate-normal latent variable. They use a two-step estimator with no empty-cell continuity correction, matching polycor::polychor() and lavaan. The polychoric asymptotic covariance that underlies both the DWLS weights and the scaled (sandwich) statistic relies on large-sample theory that degrades for empty or near-empty response-category combinations; with very sparse cells the resulting weights and standard errors can be unreliable (a warning is issued when empty cells are present), so interpret them with caution and consider collapsing rare categories.

• "fiml" estimates a two-stage full-information maximum-likelihood correlation. The saturated multivariate-normal mean and covariance are estimated from raw data with missing values by an EM algorithm assuming the data are missing at random (Yuan, Marshall, & Bentler, 2002; Little & Rubin, 2002), and the standardized covariance is then analysed. This reproduces psych::corFiml() followed by psych::fa() and lavaan(missing = "two.stage"), not lavaan::efa(missing = "ml"), so the point estimates are not expected to match the latter. The model fit indices are corrected two-stage statistics (see Fit indices). "fiml" uses every case and handles the missingness itself, so use is ignored; it supplies a continuous (Pearson-type) correlation only and is therefore not compatible with estimator = "DWLS". Standard errors are available analytically for estimator = "ML" or "ULS" and, for any estimator, by the non-parametric bootstrap (see Standard errors). For multiply imputed data, efa_mi() is the alternative route to handling missingness.

Rotations

A rotation transforms the unrotated loadings toward a simpler, more interpretable pattern; all rotations are performed by rotation engines built into the package. Orthogonal rotations keep the factors uncorrelated, whereas oblique rotations let them correlate (returning a pattern matrix, a structure matrix, and the factor intercorrelations Phi) and are usually more realistic for psychological constructs.

Orthogonal rotations:

• varimax maximises the variance of the squared loadings within each factor (column simplicity). It is the most widely used orthogonal rotation and spreads variance across factors rather than concentrating it in a general factor.

• quartimax simplifies the variables (rows) so that each loads mainly on one factor; it tends to produce a strong general factor.

• equamax is a Crawford-Ferguson compromise between varimax (column) and quartimax (row) simplicity.

• geominT uses a geometric-mean criterion that rewards a sparse pattern and tolerates variables with cross-loadings; a smaller offset delta gives a sparser solution but sharper local minima.

• bentlerT uses Bentler's invariant pattern simplicity criterion.

• bifactorT is the Jennrich-Bentler orthogonal bifactor criterion: a general factor plus group factors (bifactor simple structure).

Oblique rotations:

• promax is a fast two-step rotation: a varimax solution is raised to a power (controlled by k and p_type) to form a target that is then fitted obliquely. It is the common, inexpensive oblique default.

• oblimin is a flexible oblique family controlled by gam (default 0); a good general-purpose criterion.

• quartimin is oblimin pinned at gam = 0; a robust default oblique criterion.

• simplimax drives the k smallest loadings toward zero. Its criterion is only piecewise smooth, so it is the most prone to local minima and relies on several random_starts.

• bentlerQ is the oblique Bentler invariant pattern simplicity criterion.

• geominQ is the oblique geomin criterion; it handles complex (cross-loading) structure well but is multimodal, so it benefits from more random_starts (and uses a more thorough multi-start search internally).

• bifactorQ is the oblique (correlated) Jennrich-Bentler bifactor criterion.

The criterion-based rotations (all except varimax and promax) are fitted by gradient projection with random_starts random starts to guard against local minima; the complexity criteria (simplimax and geominQ in particular) are the most multimodal. The starts are drawn from the random-number generator, so different starts can reach genuinely different optima and such a fit is reproducible only when the generator is controlled: pass seed, or call set.seed() beforehand. The type argument changes the varimax and promax settings (see Using the type presets) and, for every rotation, the factor order_type. A single factor cannot be rotated.

Standard errors

se selects whether and how standard errors (and matching confidence intervals) are computed. Which quantities they cover depends on the method. The analytic methods ("information" and "sandwich") cover the unrotated loadings, the uniquenesses and the communalities and, when a rotation is applied, the rotated loadings and – for oblique rotations – the factor correlations and the structure coefficients. The bootstrap ("np-boot") covers the unrotated loadings, the residuals, and the fit indices and, when a rotation is applied, the rotated loadings and – for oblique rotations – the factor correlations and the structure coefficients; it reports no uniqueness or communality standard errors (see the SE and CI entries in Value).

• "none" (default) computes no standard errors.

• "information" returns analytic standard errors from the expected (Fisher) information matrix of the maximum-likelihood solution, and therefore requires estimator = "ML" and cor_method = "pearson" (or "fiml", see below). The rotated standard errors are obtained by propagating the unrotated-loading covariance through the rotation by the delta method (Jennrich, 1973); because rotated quantities are identification-invariant they are directly comparable across programs. Unlike the bootstrap it also works from a correlation matrix as long as N is supplied.

  efa_fit() analyses a correlation structure – the diagonal of the analysed matrix is fixed at 1 and the uniquenesses \psi_i = 1 - \sum_j \lambda_{ij}^2 are a function of the loadings rather than free parameters – so the information is the correlation-structure one, \Delta' \Gamma^{-1} \Delta over the off-diagonal correlations, with \Gamma the normal-theory asymptotic covariance of the sample correlations (Olkin & Siotani, 1976) at the model-implied \Sigma (Cudeck, 1989). It is scaled by 1 / (N - 1) and the confidence intervals are Wald intervals (estimate \pm z * SE). These standard errors assume multivariate normality and a correctly specified model; under heavy-tailed data or model misfit they can understate the sampling variability, where "sandwich" or "np-boot" are more robust.

  The rotated quantities, the uniquenesses, and the communalities are identification-invariant and so are comparable across programs. The unrotated loading standard errors are not: they are reported in the orientation the solution itself is identified in (for ML, \Lambda' \Psi^{-1} \Lambda diagonal), and a program using a different identification will report different unrotated loading standard errors for the same fit.

  That orientation can also fail to be determined. When two of the canonical variances \mathrm{diag}(\Lambda' \Psi^{-1} \Lambda) nearly coincide – which happens with a weakly determined factor, or two factors of near-equal strength – the loadings can be rotated within that plane without leaving the identifying constraint, so the unrotated loadings have no well-defined sampling distribution and their standard errors diverge. efa_fit() detects this and returns NA for the unrotated loading standard errors with an efa_se_unreliable warning, rather than reporting a number that looks like a standard error but is an artefact of the orientation. Everything that does not depend on the orientation – the rotated loadings, Phi, the structure coefficients, the uniquenesses and the communalities – is unaffected and still reported. Use those, or se = "np-boot", when the unrotated loadings themselves are the quantity of interest. The detection covers the Pearson and polychoric paths; the two-stage cor_method = "fiml" sandwich carries no such diagnostic, so a weakly determined orientation is not flagged there. A Heywood case (a uniqueness at its lower boundary) is separate: the Wald approximation fails there for every parameter, so no analytic standard error is reported at all.

• "sandwich" returns robust (Godambe sandwich) standard errors from raw data, combining the estimator weight with an asymptotic-distribution-free covariance of the correlations, so it stays valid under non-normality and weight misspecification (Browne, 1984; Satorra & Bentler, 1994). It is available either for ordinal data with cor_method = "poly" or "tetra" and estimator one of "ML", "ULS", or "DWLS" (the meat is the polychoric / tetrachoric asymptotic covariance), or for continuous data with cor_method = "pearson" and estimator = "ML" or "ULS" (the meat is the fourth-moment ADF covariance of the sample correlations, the basis of the MLM / MLR robust statistics). It reports the same coverage as "information", propagated by the same delta method, and additionally fills the model fit's chi-square block with a scaled (Satorra-Bentler / scaled-and-shifted) chi-square (see Fit indices). The statistic reported as chi is always the scaled-and-shifted one (the WLSMV default, flagged by chi_scaled_type), for the continuous Pearson path as well as the ordinal one; the mean-adjusted statistic that lavaan's MLM reports for continuous data is returned alongside it as chi_mean_adjusted. Because the asymptotic covariance must describe the same cases as the correlation matrix, the sandwich (like estimator = "DWLS") is computed on the listwise-complete cases; on data with missing values the reported N, the correlation matrix, and the point estimate therefore reflect the complete cases regardless of use.

• "np-boot" draws a non-parametric (case-resampling) bootstrap and needs raw data. A correlation matrix carries no cases to resample; alone among the unsupported combinations this one does not error but warns (condition class "efa_boot_cormat") and downgrades se to "none", so the fit is returned without an SE slot. It is the most general method – available for any estimator, rotation, and cor_method – and the most robust to non-normality and misfit, at the cost of speed; its intervals are bootstrap percentile intervals. The replicate fits are run across replicates with the future framework. By default they run sequentially; to run them in parallel, register a plan with future::plan() (e.g. future::plan(future::multisession, workers = 2); see examples). With a fixed seed the bootstrap is reproducible and yields the same result regardless of the number of workers. Under cor_method = "fiml" each resample also re-runs the EM moment estimation and is therefore slow, so a smaller b_boot may be advisable.

  The percentile intervals are centred on the point estimate for the loadings, the factor correlations, the structure coefficients and the residuals, but not for the indices derived from the chi-square (RMSEA, AIC, BIC and ECVI). A resample is drawn from the sample, which already carries the model's misfit, and is then refitted and evaluated against itself, so the replicate chi-square is on average about chi + df rather than chi and the whole interval rides upward with it – often far enough that the point estimate falls below its own lower bound. That is the shift, not a miscomputed interval: read those intervals as a spread rather than as a range for the point estimate. Correcting the location needs resampling from a population transformed to fit the model (Bollen & Stine, 1992), which is not what this bootstrap does. CFI and TLI are not affected, being ratios in which the baseline chi-square shifts along with the model one.

The analytic methods ("information" and "sandwich") are not available with the "promax" or "simplimax" rotations, which have no usable analytic rotation Jacobian; use "np-boot" there. Under cor_method = "fiml", "information" and "sandwich" instead return, for estimator = "ML" or "ULS", the corrected two-stage (Yuan & Bentler, 2000; Savalei & Bentler, 2009) sandwich standard errors, built on the saturated FIML asymptotic covariance with the estimator's own Stage-2 weight: the model is fitted to the EM-estimated correlation, so the naive Stage-2 standard errors (treating that correlation as complete data) are inconsistent under missingness and are not reported (estimator = "PAF" carries no Stage-2 weight, so use se = "np-boot" there).

Fit indices

For ML and ULS, efa_fit() returns the model chi-square (with its p-value and degrees of freedom), the Comparative Fit Index (CFI; Bentler, 1990), the Tucker-Lewis Index (TLI, also called the non-normed fit index; Tucker & Lewis, 1973), the Root Mean Square Error of Approximation (RMSEA) with its 90% confidence interval (Browne & Cudeck, 1992), the Akaike and Bayesian Information Criteria (AIC, BIC), the Expected Cross-Validation Index (ECVI; Browne & Cudeck, 1989), the Root Mean Squared Residual (RMSR), the Standardized Root Mean Squared Residual (SRMR; Bentler, 1995), and the common-part-accounted-for (CAF) index (Lorenzo-Seva, Timmerman, & Kiers, 2011). The print and summary methods show SRMR, not RMSR, because the two residual summaries differ only by the fixed scaling \sqrt{(p - 1) / (p + 1)} for a fixed number of variables; RMSR remains in the returned object. The model chi-square is the Bartlett-corrected discrepancy (matching stats::factanal() for ML); the AIC, BIC, and ECVI are the minimum-fit-function (chi-square-based) forms (\chi^2 - 2\,df and \chi^2 - \log(N)\,df for AIC and BIC, as in psych::fa()) and can therefore be negative. The RMSEA, CFI, and TLI place the model and baseline noncentrality on the uncorrected N - 1 discrepancy scale on which these approximate-fit indices are defined, so the Bartlett small-sample correction enters only the chi-square test, not the approximate-fit indices.

Which indices are reported depends on the estimator:

• ML and ULS compute the full set above.

• PAF returns only the descriptive residual indices (RMSR, SRMR, CAF) and df; the printed model-fit block shows CAF and SRMR. The chi-square-based indices are NA, because PAF minimises no discrepancy.

• DWLS by default returns only RMSR, SRMR, CAF, and df, because the ordinary maximum-likelihood discrepancy is not its fit function. When se = "sandwich", a scaled (Satorra & Bentler, 1994; Asparouhov & Muthén, 2010) chi-square and the CFI, TLI, and RMSEA derived from it are reported (AIC and BIC remain NA). That scaled statistic is a two-stage correction applied to the polychoric-correlation residuals (Browne, 1984), so it is not identical to the full WLSMV test of lavaan or Mplus, which also projects the thresholds.

• cor_method = "fiml" (with ML or ULS) reports Satorra-Bentler-corrected two-stage statistics (Yuan, Marshall, & Bentler, 2002): the normal-theory discrepancy on the EM-estimated correlation, rescaled by the saturated FIML asymptotic covariance, because the plain two-stage likelihood-ratio statistic is not asymptotically \chi^2(df). The CFI, TLI, and RMSEA follow from the scaled statistics; AIC, BIC, and ECVI are left NA, as for any scaled (moment-adjusted) chi-square.

Whenever the chi-square is a scaled one (se = "sandwich", or any cor_method = "fiml" fit), the AIC, BIC, and ECVI are NA and the returned fit_indices additionally carry the scaled-statistic components (see the fit_indices entry in Value). Note that Lorenzo-Seva, Timmerman, and Kiers (2011) introduce the CAF as ranging between 0 and 1, with values close to 1 indicating close fit; this does not match the formula they apply, 1 - KMO(residuals), which only works if the diagonal of the residual matrix is set to 1s and then approximates 0.5 with close fit.

Available combinations

Not every estimator, rotation, standard-error, and correlation method can be combined:

• Estimator and correlation method. estimator = "DWLS" requires ordinal data with cor_method = "poly" or "tetra". cor_method = "fiml" works with PAF, ML, and ULS (not DWLS) and needs raw data with missing values.

• Standard errors. se = "information" requires estimator = "ML" and cor_method = "pearson" or "fiml", and can be computed from a correlation matrix when N is supplied. se = "sandwich" requires raw data, with either a polychoric/tetrachoric cor_method (ML, ULS, or DWLS) or a Pearson cor_method (ML or ULS); it is not available for PAF. Under cor_method = "fiml", "information" and "sandwich" are available for ML and ULS only and both return the corrected two-stage sandwich. se = "np-boot" requires raw data and works with any estimator, rotation, and correlation method. Neither "information" nor "sandwich" is available with the "promax" or "simplimax" rotations.

• Fit indices. The chi-square-based indices are available for ML and ULS (and, as scaled statistics, for cor_method = "fiml" and for DWLS with se = "sandwich"); PAF and DWLS otherwise report only the descriptive residual indices.

Using the type presets

The type argument is evaluated for PAF and for all rotations (mainly important for the varimax and promax rotations). The type-specific settings for these functions are detailed below.

For PAF, the values of init_comm, criterion, criterion_type, max_iter, and abs_eigen depend on the type argument.

type = "EFAtools" will use the following argument specification: ⁠init_comm = "smc", criterion = .001, criterion_type = "sum", max_iter = 300, abs_eigen = TRUE⁠.

type = "psych" will use the following argument specification: ⁠init_comm = "smc", criterion = .001, criterion_type = "sum", max_iter = 50, abs_eigen = FALSE⁠.

type = "SPSS" will use the following argument specification: ⁠init_comm = "smc", criterion = .001, criterion_type = "max_individual", max_iter = 25, abs_eigen = TRUE⁠.

If SMCs fail, SPSS takes "mac". However, as SPSS takes absolute eigenvalues, this is hardly ever the case. Psych, on the other hand, takes "unity" if SMCs fail, but uses the Moore-Penrose Psudo Inverse of a matrix, thus, taking "unity" is only necessary if negative eigenvalues occur afterwards in the iterative PAF procedure. The EFAtools type setting combination was the best in terms of accuracy and number of Heywood cases compared to all the other setting combinations tested in simulation studies in Grieder & Steiner (2022), which is why this type is used as a default here.

For varimax, the values of varimax_type and order_type depend on the type argument.

type = "EFAtools" will use the following argument specification: ⁠varimax_type = "kaiser", order_type = "eigen"⁠.

type = "psych" will use the following argument specification: ⁠varimax_type = "svd", order_type = "eigen"⁠.

type = "SPSS" will use the following argument specification: ⁠varimax_type = "kaiser", order_type = "ss_factors"⁠.

For promax, the values of p_type, order_type, and k depend on the type argument.

type = "EFAtools" will use the following argument specification: ⁠p_type = "norm", order_type = "eigen", k = 4⁠.

type = "psych" will use the following argument specification: ⁠p_type = "unnorm", order_type = "eigen", k = 4⁠.

type = "SPSS" will use the following argument specification: ⁠p_type = "norm", order_type = "ss_factors", k = 4⁠.

The p_type argument can take two values, "unnorm" and "norm". It controls which formula is used to compute the target matrix P in the promax rotation. "unnorm" uses the formula from Hendrickson and White (1964), specifically: P = abs(A^(k + 1)) / A, where A is the unnormalized matrix containing varimax rotated loadings. "norm" uses the normalized varimax rotated loadings. Specifically it used the following formula, which can be found in the SPSS 23 and SPSS 27 Algorithms manuals: P = abs(A / sqrt(rowSums(A^2))) ^(k + 1) * (sqrt(rowSums(A^2)) / A). As for PAF, the EFAtools type setting combination for promax was the best compared to the other setting combinations tested in simulation studies in Grieder & Steiner (2022). Note that all type presets keep the EFAtools default Kaiser normalization (normalize = TRUE), whereas psych::fa() does not normalize before its promax target rotation; set normalize = FALSE to reproduce the psych::fa() promax result to within the varimax convergence tolerance (the residual difference is the convergence noise of the underlying varimax base at precision = 1e-5, not an algorithmic difference).

The varimax_type argument can take two values, "svd", and "kaiser". "svd" uses singular value decomposition, by calling stats::varimax(). "kaiser" performs the varimax procedure as described in the SPSS Algorithms manual and by Kaiser (1958). The varimax simplicity criterion monitored for convergence is sum(n*colSums(lambda ^ 4) - colSums(lambda ^ 2) ^ 2) / n ^ 2, where n is the number of indicators, and lambda is the Kaiser-normalized rotated loadings matrix.

For all other rotations except varimax and promax, the type argument only controls the order_type argument with the same values as stated above for the varimax and promax rotations. Additional arguments can also be specified and will be passed to the rotation procedure (e.g., maxit to change the maximum number of iterations). As for promax, every preset keeps the EFAtools default Kaiser normalization (normalize = TRUE), whereas psych::fa() and GPArotation do not normalize before these criterion rotations; set normalize = FALSE to reproduce them.

The type tuning arguments have no effect on ULS and ML; type itself still governs the checks applied to the correlation matrix. For ULS, no additional arguments are needed. For ML, an additional argument start_method is needed to determine the starting values for the optimization procedure. Default for this argument is "psych" which takes the starting values specified in psych::fa().

Value

A list of class c("efa", "EFA") containing (a subset of) the following:

Source

Bollen, K. A., & Stine, R. A. (1992). Bootstrapping goodness-of-fit measures in structural equation models. Sociological Methods & Research, 21, 205–229. doi: 10.1177/0049124192021002004

Grieder, S., & Steiner, M. D. (2022). Algorithmic jingle jungle: A comparison of implementations of principal axis factoring and promax rotation in R and SPSS. Behavior Research Methods, 54, 54–74. doi: 10.3758/s13428-021-01581-x

Hendrickson, A. E., & White, P. O. (1964). Promax: A quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 17 , 65–70. doi: 10.1111/j.2044-8317.1964.tb00244.x

Lorenzo-Seva, U., Timmerman, M. E., & Kiers, H. A. L. (2011). The Hull Method for Selecting the Number of Common Factors, Multivariate Behavioral Research, 46, 340-364, doi: 10.1080/00273171.2011.564527

Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200. doi: 10.1007/BF02289233

Lawley, D. N., & Maxwell, A. E. (1971). Factor analysis as a statistical method (2nd ed.). Butterworths.

Cudeck, R. (1989). Analysis of correlation matrices using covariance structure models. Psychological Bulletin, 105, 317–327. doi: 10.1037/0033-2909.105.2.317

Olkin, I., & Siotani, M. (1976). Asymptotic distribution of functions of a correlation matrix. In S. Ikeda (Ed.), Essays in probability and statistics (pp. 235–251). Shinko Tsusho.

Jennrich, R. I. (1973). Standard errors for obliquely rotated factor loadings. Psychometrika, 38, 593–604. doi: 10.1007/BF02291497

Zhang, G., & Preacher, K. J. (2015). Factor rotation and standard errors in exploratory factor analysis. Journal of Educational and Behavioral Statistics, 40, 579–603. doi: 10.3102/1076998615606098

Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83. doi: 10.1111/j.2044-8317.1984.tb00789.x

Satorra, A., & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C. C. Clogg (Eds.), Latent variables analysis (pp. 399–419). Sage.

Asparouhov, T., & Muthén, B. (2010). Simple second order chi-square correction. Mplus Technical Appendix.

Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115–132. doi: 10.1007/BF02294210

Yuan, K.-H., & Bentler, P. M. (2000). Three likelihood-based methods for mean and covariance structure analysis with nonnormal missing data. Sociological Methodology, 30, 165–200. doi: 10.1111/0081-1750.00078

Yuan, K.-H., Marshall, L. L., & Bentler, P. M. (2002). A unified approach to exploratory factor analysis with missing data, nonnormal data, and in the presence of outliers. Psychometrika, 67, 95–121. doi: 10.1007/BF02294711

Savalei, V., & Bentler, P. M. (2009). A two-stage approach to missing data: Theory and application to auxiliary variables. Structural Equation Modeling, 16, 477–497. doi: 10.1080/10705510903008238

Little, R. J. A., & Rubin, D. B. (2002). Statistical analysis with missing data (2nd ed.). Wiley.

Bartlett, M. S. (1951). The effect of standardization on approximation in factor analysis. Biometrika, 38, 337–344.

Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238–246. doi: 10.1037/0033-2909.107.2.238

Tucker, L. R., & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38, 1–10. doi: 10.1007/BF02291170

Browne, M. W., & Cudeck, R. (1989). Single sample cross-validation indices for covariance structures. Multivariate Behavioral Research, 24, 445–455. doi: 10.1207/s15327906mbr2404_4

Browne, M. W., & Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods & Research, 21, 230–258. doi: 10.1177/0049124192021002005

Bentler, P. M. (1995). EQS structural equations program manual. Multivariate Software.

See Also

estimate_control() and rotate_control() for the estimation and rotation tuning knobs. efa_retain() for choosing n_factors, and efa_scores(), efa_reliability(), efa_schmid_leiman(), and efa_compare() for working with the fitted solution.

Other factor analysis: efa_average(), efa_group(), efa_mi(), plot.efa_group(), print.efa_group()

Examples

# Principal axis factoring with oblimin rotation
mod_oblimin <- efa_fit(test_models$baseline$cormat, n_factors = 3, N = 500,
                       rotation = "oblimin")
mod_oblimin
summary(mod_oblimin)

# ML estimation with oblimin rotation
mod_oblimin <- efa_fit(test_models$baseline$cormat, n_factors = 3, N = 500,
                       estimator = "ML", rotation = "oblimin")
mod_oblimin
summary(mod_oblimin)

# Tuning knobs are supplied through the control objects. Here the SPSS preset is
# used for the estimation and rotation, with the maximum PAF iterations raised.
mod_spss <- efa_fit(test_models$baseline$cormat, n_factors = 3, N = 500,
                    rotation = "promax",
                    estimate_control = estimate_control(type = "SPSS", max_iter = 500),
                    rotate_control = rotate_control(type = "SPSS"))
mod_spss

# Analytic (expected-information) standard errors for the above
ML_info <- efa_fit(test_models$baseline$cormat, n_factors = 3, N = 500,
                   estimator = "ML", rotation = "oblimin", se = "information")
ML_info
summary(ML_info)


# Robust (sandwich) standard errors and a scaled chi-square for ordinal raw data.
# These need a polychoric/tetrachoric correlation method and estimator ML, ULS, or DWLS.
DWLS_rob <- efa_fit(DOSPERT_raw, n_factors = 6, cor_method = "poly",
                    estimator = "DWLS", rotation = "oblimin", se = "sandwich")
DWLS_rob
summary(DWLS_rob)

# The same robust SEs and scaled chi-square for continuous data: a Pearson
# correlation with estimator ML or ULS (the fourth-moment ADF covariance).
ML_rob <- efa_fit(GRiPS_raw, n_factors = 1, cor_method = "pearson",
                  estimator = "ML", rotation = "none", se = "sandwich")
ML_rob
summary(ML_rob)



# Two-stage FIML correlations from raw data with missing values: the saturated
# multivariate-normal moments are EM-estimated (assuming the data are missing at
# random) and the standardized covariance is analysed.
x_miss <- GRiPS_raw
x_miss[cbind(1:20, 1)] <- NA
efa_fiml <- efa_fit(x_miss, n_factors = 1, estimator = "ML", cor_method = "fiml")
efa_fiml


## Not run: 
# Bootstrap standard errors from raw data, reproducible via a fixed seed and run
# in parallel across replicates.
future::plan(future::multisession, workers = 2)
efa_boot <- efa_fit(GRiPS_raw, n_factors = 1, estimator = "PAF", rotation = "none",
                    se = "np-boot", b_boot = 1000, seed = 42)
future::plan(future::sequential)

## End(Not run)

Multigroup exploratory factor analysis

Description

Fit an exploratory factor analysis in each of several groups at a common number of factors and bring the per-group solutions into one shared orientation so their loadings can be compared. Each group is fitted with efa_fit(); the solutions are then aligned either to a symmetric consensus target or to a chosen reference group (see Alignment).

Usage

efa_group(
  x,
  groups = NULL,
  n_factors,
  N = NA,
  reference_group = NULL,
  b_boot = 0L,
  ci = 0.95,
  seed = NULL,
  delta = 0.1,
  invariance = FALSE,
  ...
)

Arguments

Details

Input

Groups can be supplied in two ways: raw data together with a grouping vector (x a data frame or matrix, groups one value per row), or a named list of per-group data sets in x (with groups left NULL). The list may hold raw data frames or correlation matrices (supply N), but not a mix of the two. All groups must contain the same items in the same order; a different item set or order is an error rather than being silently reordered.

Every group is fitted at the same n_factors (a common-k multigroup model). Extra arguments in ... (for example estimator, rotation, cor_method, or an estimate_control() / rotate_control() carrying the tuning knobs) are forwarded unchanged to each efa_fit() call, so the estimator and rotation are common to all groups.

The k-factor model must be identified for the shared item set: its degrees of freedom ((p - k)^2 - (p + k)) / 2 must be non-negative. Unlike a single efa_fit() fit – which only warns on an under-identified model – a multigroup fit aborts, because a shared alignment target across an under-identified group is not interpretable.

Alignment

A factor solution is identified only up to a rotation of its factors, so the per-group solutions must be brought into a common orientation before their loadings can be compared. Two strategies are available and are chosen automatically:

• Consensus (the default for orthogonal rotations and for unrotated solutions): a symmetric Generalized Procrustes Analysis target is built across all groups (Gower, 1975), and every group's loadings are rotated to it. The consensus frame is only identified up to a global rotation of its factors, so it is then rotated into a fixed gauge and the same transform is applied to every group. The gauge is the simple structure of the rotation that was requested, evaluated on the consensus target itself, so the shared loadings are in the same kind of frame as the per-group solutions they summarise. Where no criterion identifies a frame – an unrotated solution, and a "bifactorT" request with two factors, whose criterion is identically zero with a single group factor – the principal-axes orientation of the target is used instead. Either way the columns are ordered by decreasing sum of squares and signed by their column sums, and the shared orientation, and hence every reported congruence, difference, and flag, is independent of the order in which the groups are supplied.

• Reference: every group's loadings are aligned by Procrustes rotation to one reference group's loadings, which are kept fixed. This path is used when reference_group is given, and is used automatically for oblique rotations because the consensus iteration is not defined for oblique transforms with more than one factor. When an oblique rotation triggers the reference path without an explicit reference_group, the first group is used and a message reports this; the requested rotation is never silently changed.

In both cases the returned per-group loadings share the column order and sign of the returned target.

Comparing the aligned loadings

Because the per-group loadings share one orientation, they can be compared cell by cell. efa_group() reports a per-pair summary of their differences (diffs) and a per-item, per-factor flag table (flags) marking cells whose absolute difference reaches delta. delta is a descriptive salience heuristic, not a significance test; a bootstrap (b_boot > 0) additionally reports whether each difference's confidence interval excludes zero. With invariance = TRUE, a per-factor verdict grades the matched Tucker congruence by the Lorenzo-Seva and ten Berge (2006) similarity bands – "equal" (⁠>= .95⁠) and "fair" (⁠[.85, .95)⁠) – labelling weaker congruences "incongruent"; when a bootstrap is available the verdict uses the congruence CI lower bound, so a factor is judged "equal" only if even the lower bound clears the band.

Value

An object of class efa_group, a list containing:

References

Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall.

Gower, J. C. (1975). Generalized Procrustes analysis. Psychometrika, 40, 33-51. doi: 10.1007/BF02291478

Lorenzo-Seva, U., and ten Berge, J. M. F. (2006). Tucker's congruence coefficient as a meaningful index of factor similarity. Methodology, 2, 57-64. doi: 10.1027/1614-2241.2.2.57

See Also

Other factor analysis: efa_average(), efa_fit(), efa_mi(), plot.efa_group(), print.efa_group()

Examples

# Raw data split by a grouping vector (unrotated, consensus alignment)
g <- rep(c("g1", "g2"), length.out = nrow(GRiPS_raw))
mg <- efa_group(GRiPS_raw, groups = g, n_factors = 1)
mg$loadings

# Per-pair difference summary and the per-item salience-flag table
mg$diffs
mg$flags


# Percentile bootstrap confidence intervals for the between-group congruences, with an
# approximate-invariance verdict read conservatively off the congruence CI lower bound
mg_ci <- efa_group(GRiPS_raw, groups = g, n_factors = 1, b_boot = 100, seed = 42,
                   invariance = TRUE)
mg_ci$congruence$matched_ci
mg_ci$invariance

# A named list of correlation matrices sharing the same items, common
# three-factor model, orthogonal rotation -> symmetric consensus target
bands <- list(age_6_8 = WJIV_ages_6_8$cormat, age_14_19 = WJIV_ages_14_19$cormat)
Ns <- c(WJIV_ages_6_8$N, WJIV_ages_14_19$N)
efa_group(bands, n_factors = 3, N = Ns, rotation = "varimax")

# An oblique rotation aligns to a reference group (reported via a message)
efa_group(bands, n_factors = 3, N = Ns, rotation = "promax")

Hull method for determining the number of factors to retain

Description

Implementation of the Hull method suggested by Lorenzo-Seva, Timmerman, and Kiers (2011), with an extension to principal axis factoring. See details for parallelization.

Usage

efa_hull(
  x,
  N = NA,
  n_fac_theor = NA,
  estimator = c("PAF", "ULS", "ML"),
  gof = c("CAF", "CFI", "RMSEA"),
  eigen_type = c("SMC", "PCA", "EFA"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  n_datasets = 1000,
  percent = 95,
  decision_rule = c("means", "percentile", "crawford"),
  n_factors = 1,
  estimate_control = NULL,
  ...
)

Arguments

Details

The Hull method aims to find a model with an optimal balance between model fit and number of parameters, retaining only major factors (Lorenzo-Seva, Timmerman, & Kiers, 2011). It fits 0 to J factors – where J is the number of factors suggested by parallel analysis (or n_fac_theor, if that is larger), plus one – keeps the solutions on the upper boundary of the convex hull of goodness-of-fit against degrees of freedom, and selects the one at the sharpest elbow, i.e. with the highest st value.

The efa_parallel function and the principal axis factoring of the different number of factors can be parallelized using the future framework, by calling the future::plan() function. The examples provide example code on how to enable parallel processing.

The upper bound J comes from efa_parallel(), which compares against simulated data, so the suggested number of factors varies slightly from run to run; a criterion-based rotation passed through ... adds its own random starts. Call set.seed() beforehand to make a run reproducible; the result is then also independent of the parallel plan.

Note that if gof = "RMSEA" is used, 1 - RMSEA is actually used to compare the different solutions. Thus, the threshold of .05 is then .95. This is necessary due to how the heuristic to locate the elbow of the hull works.

The ML estimation method uses the psych::fa() starting values. See also the efa_fit documentation.

Value

An object of class efa_retention (see print.efa_retention() and plot.efa_retention() for the print and plot methods). Its main fields are:

Source

Lorenzo-Seva, U., Timmerman, M. E., & Kiers, H. A. (2011). The Hull method for selecting the number of common factors. Multivariate Behavioral Research, 46(2), 340-364.

See Also

efa_retain() as a wrapper function for this and the other factor retention criteria.

Other factor retention criteria: efa_cd(), efa_ekc(), efa_kgc(), efa_map(), efa_nest(), efa_parallel(), efa_retain(), efa_scree(), efa_smt()

Examples

# using PAF (this will throw a warning if gof is not specified manually
# and CAF will be used automatically)
efa_hull(test_models$baseline$cormat, N = 500, gof = "CAF")

# using ML with all available fit indices (CAF, CFI, and RMSEA)
efa_hull(test_models$baseline$cormat, N = 500, estimator = "ML")

# using ULS with only RMSEA
efa_hull(test_models$baseline$cormat, N = 500, estimator = "ULS", gof = "RMSEA")


## Not run: 
# using parallel processing (Note: plans can be adapted, see the future
# package for details)
future::plan(future::multisession)
efa_hull(test_models$baseline$cormat, N = 500, gof = "CAF")

## End(Not run)

Kaiser-Guttman criterion

Description

Probably the most popular factor retention criterion. Kaiser and Guttman suggested to retain as many factors as there are sample eigenvalues greater than 1. This is why the criterion is also known as eigenvalues-greater-than-one rule.

Usage

efa_kgc(
  x,
  eigen_type = c("PCA", "SMC", "EFA"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  n_factors = 1,
  estimate_control = NULL,
  ...
)

Arguments

Details

Originally, the Kaiser-Guttman criterion was intended for the use with principal components, hence with eigenvalues derived from the original correlation matrix. This can be done here by setting eigen_type to "PCA". However, it is well-known that this criterion is often inaccurate and that it tends to overestimate the number of factors, especially for unidimensional or orthogonal factor structures (e.g., Zwick & Velicer, 1986).

The criterion's inaccuracy in these cases is somewhat addressed if it is applied on the correlation matrix with communalities in the diagonal, either initial communalities estimated from SMCs (done setting eigen_type to "SMC") or final communality estimates from an EFA (done setting eigen_type to "EFA"; see Auerswald & Moshagen, 2019). However, although this variant of the KGC is more accurate in some cases compared to the traditional KGC, it is at the same time less accurate than the PCA-variant in other cases, and it is still often less accurate than other factor retention methods, for example parallel analysis (efa_parallel()), the Hull method efa_hull(), or sequential chi^2 model tests (efa_smt(); see Auerswald & Moshagen, 2019).

The efa_kgc function can also be called together with other factor retention criteria in the efa_retain() function.

Value

An object of class efa_retention (see print.efa_retention() and plot.efa_retention() for the print and plot methods). Its main fields are:

Source

Auerswald, M., & Moshagen, M. (2019). How to determine the number of factors to retain in exploratory factor analysis: A comparison of extraction methods under realistic conditions. Psychological Methods, 24(4), 468–491. https://doi.org/10.1037/met0000200

Guttman, L. (1954). Some necessary conditions for common-factor analysis. Psychometrika, 19, 149 –161. http://dx.doi.org/10.1007/BF02289162

Kaiser, H. F. (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20, 141–151. http://dx.doi.org/10.1177/001316446002000116

Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99, 432–442. http://dx.doi.org/10.1037/0033-2909.99.3.432

See Also

efa_retain() as a wrapper function for this and the other factor retention criteria.

Other factor retention criteria: efa_cd(), efa_ekc(), efa_hull(), efa_map(), efa_nest(), efa_parallel(), efa_retain(), efa_scree(), efa_smt()

Examples

efa_kgc(test_models$baseline$cormat, eigen_type = c("PCA", "SMC"))

Kaiser-Meyer-Olkin criterion

Description

This function computes the Kaiser-Meyer-Olkin (KMO) criterion overall and for each variable in a correlation matrix. The KMO represents the degree to which each observed variable is predicted by the other variables in the dataset and with this indicates the suitability for factor analysis.

Usage

efa_kmo(
  x,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra")
)

Arguments

Details

Kaiser (1970) proposed this index, originally called measure of sampling adequacy (MSA), that indicates how near the inverted correlation matrix R^{-1} is to a diagonal matrix S to determine a given correlation matrix's (R) suitability for factor analysis. The index is

KMO = \frac{\sum_{i \neq j} r_{ij}^2}{\sum_{i \neq j} r_{ij}^2 + \sum_{i \neq j} q_{ij}^2}

with Q = SR^{-1}S and S = (diag R^{-1})^{-1/2} where \sum_{i \neq j} r_{ij}^2 is the sum of squares of the off-diagonal elements of R and \sum_{i \neq j} q_{ij}^2 is the sum of squares of the off-diagonal elements of Q (see also Cureton & D'Agostino, 1983).

So KMO varies between 0 and 1, with larger values indicating higher suitability for factor analysis. Kaiser and Rice (1974) suggest that KMO should at least exceed .50 for a correlation matrix to be suitable for factor analysis.

This function was heavily influenced by the psych::KMO() function.

See also efa_bartlett() for another test of suitability for factor analysis.

The efa_kmo function can also be called together with the efa_bartlett() function and with factor retention criteria in the efa_retain() function.

Value

A list containing

Source

Kaiser, H. F. (1970). A second generation little jiffy. Psychometrika, 35, 401-415.

Kaiser, H. F. & Rice, J. (1974). Little jiffy, mark IV. Educational and Psychological Measurement, 34, 111-117.

Cureton, E. E. & D'Agostino, R. B. (1983). Factor analysis: An applied approach. Hillsdale, N.J.: Lawrence Erlbaum Associates, Inc.

See Also

efa_bartlett() for another measure to determine suitability for factor analysis.

efa_retain() as a wrapper function for this function, efa_bartlett() and several factor retention criteria.

Other factor analysis suitability: efa_bartlett(), efa_screen(), print.efa_screen()

Examples

efa_kmo(test_models$baseline$cormat)

Velicer's minimum average partial (MAP) criterion

Description

Computes Velicer's Minimum Average Partial (MAP) criterion for determining the number of factors/components to retain. The function implements the original MAP criterion (Velicer, 1976), expressed via the \mathrm{TR2} representation, and the revised \mathrm{TR4} variant proposed by Velicer, Eaton, and Fava (2000).

Usage

efa_map(
  x,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra")
)

Arguments

Details

MAP partials successive principal components out of the correlation matrix and, after removing m components, summarizes the off-diagonal partial correlations r^*_{ij} that remain in the m-th partial correlation matrix M (which has a unit diagonal); the suggested number of factors is the m that minimizes the criterion. Two criteria are returned, each rescaling the trace of a matrix power of M by the number of off-diagonal cells p(p-1):