EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools

Description

Provides functions to perform exploratory factor analysis (EFA) procedures and compare their solutions. The goal is to provide state-of-the-art factor retention methods and a high degree of flexibility in the EFA procedures. This way, for example, implementations from R 'psych' and 'SPSS' can be compared. Moreover, functions for Schmid-Leiman transformation and the computation of omegas are provided. To speed up the analyses, some of the iterative procedures, like principal axis factoring (PAF), are implemented in C++.

Author(s)

Maintainer: Markus Steiner markus.d.steiner@gmail.com

Authors:

• Silvia Grieder silvia.grieder@gmail.com

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

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

Pipe operator

Description

See magrittr::%>% for details.

Usage

lhs %>% rhs

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.

Confidence intervals around mean

Description

Confidence intervals around mean

Usage

.calc_cis(means, sds, p = 0.05, n)

Arguments

Details

The standard error (SE) to use in the creation of the CIs is calculated according to Rubin's (1987) formula (eq. 11 in Hayes & Enders, 2023):

\sqrt{mean(SE^2) + var(\hat{\theta_{m}}) + var(\hat{\theta_{m}}) / M}

According to Hayes & Enders (2023) p. 42:

[T]he first term under the radical represents the average squared standard error (the within imputation sampling variance [...]), the second term depends on the variance of the M parameter estimates around their average (the between imputation variance [...]), and the final term represents the squared standard error of the pooled estimate [...]. Conceptually, the first term estimates the sampling error of a complete-data analysis, and the next two terms are essentially correction factors that inflate the standard error to compensate for uncertainty due to the imputations–that is, additional uncertainty (sampling variability) in the parameter estimates caused by missing data.

Currently, it is not possible to calculate the first term, because EFA does not calculate SEs for the loadings. Only the second and third terms are used in the calculation of SE for the CIs.

The CI is generally calculated as:

CI = Point estimate ± Margin of error,

where

Margin of error = Critical value × SE of point estimate.

To account for situations where the sample size is small, instead of using z-values, the critical value is derived from the t distribution with n - 1 degrees of freedom (Hazra, 2017).

Value

A list with the lower and upper CIs.

Author(s)

Andreas Soteriades

This function is used internally by EFA_POOLED to calculate confidence intervals (CIs) around the pooled loadings and pooled interfactor correlations.

References

Hayes, T. & Enders, C. K. (2023). Maximum likelihood and multiple imputation missing data handling: how they work, and how to make them work in practice. In APA Handbook of Research Methods in Psychology, Second Edition Vol. 3. Data Analysis and Research Publication, H. Cooper (Editor-in-Chief).

Hazra, A. (2017). Using the confidence interval confidently. Journal of Thoracic Disease 9(10), 4125–4130.

Rubin, D. B. (1987). Multiple imputation for nonresponse in surveys. Wiley. https://doi.org/10.1002/9780470316696

Covert a "LOADINGS" table to matrix or a matrix to "LOADINGS"

Description

Covert a "LOADINGS" table to matrix or a matrix to "LOADINGS"

Usage

.change_class(x, cl = "matrix")

Arguments

Value

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

Author(s)

Andreas Soteriades

The loadings tables returned by EFA are of class "LOADINGS", which prevents applying functions on them. This function allows to change their class to "matrix", and to change back to "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 consensus engine

Description

This internal helper performs a single consensus-target run from one starting target. Users should normally call [CONSENSUS_PROCRUSTES()] instead.

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,
  oblique_maxit = 300,
  oblique_eps = 1e-05,
  oblique_max_line_search = 10,
  oblique_step0 = 1,
  oblique_normalize = FALSE,
  oblique_random_starts = 0,
  oblique_random_starts_stage = c("final", "none", "outer", "both"),
  oblique_screen_keep = 2,
  oblique_triage_maxit = 25,
  oblique_triage_improve_tol = 0,
  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 for each permutation run in EFA_POOLED.

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

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.

Get reference values for nest.

Description

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

Usage

.nest_sym(nf, N, M, nreps = 1000L)

Arguments

Format numbers for print method

Description

Helper function used in the print method for class LOADINGS and SLLOADINGS. Strips the 0 in front of the decimal point of a number if number < 1, only keeps the first digits number of digits, and adds an empty space in front of the number if the number is positive. This way all returned strings (except for those > 1, which are exceptions in LOADINGS) have the same number of characters.

Usage

.numformat(x, digits = 2, print_zero = FALSE, pad = TRUE)

Arguments

Value

A formated number

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 'Phi = t(T) matrix 'T' under the oblique normalization constraint 'diag(t(T)

The line search is monotone: a candidate is accepted only if it satisfies the sufficient-decrease condition, or, as a numerical fallback, if it at least decreases the objective after all step halvings are exhausted. 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.

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

Closed-form orthogonal Procrustes rotation

Description

Rotate 'A' to the orthogonal target 'B' by minimizing '||A

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 PARALLEL so usually no call to this is needed by the user. Provides a C++ implementation of the PARALLEL simulation procedure

Usage

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

Arguments

Calculate statistics for a list of matrices

Description

Calculate statistics for a list of matrices

Usage

.stat_over_list(alist, stat)

Arguments

Value

A matrix with the aggregated results

Author(s)

Andreas Soteriades

Given a list of matrices, this function calculates user-supplied statistics (e.g. mean, median) over the matrices. This function is useful for averaging results from EFA (e.g. loadings) over all permutation runs in EFA_POOLED.

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

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

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

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 (KMO) can still be used.

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

The BARTLETT function can also be called together with the (KMO) function and with factor retention criteria in the N_FACTORS 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

KMO for another measure to determine suitability for factor analysis.

N_FACTORS as a wrapper function for this function, KMO and several factor retention criteria.

Examples

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 (2017) available at https://osf.io/x5cz2/?view_only=d03efba1fd0f4c849a87db82e6705668

Usage

CD(
  x,
  n_factors_max = NA,
  N_pop = 10000,
  N_samples = 500,
  alpha = 0.3,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall"),
  max_iter = 50
)

Arguments

Details

"Parallel analysis (PA) is an effective stopping rule that compares the eigenvalues of randomly generated data with those for the actual data. PA takes into account sampling error, and at present it is widely considered the best available method. We introduce a variant of PA that goes even further by reproducing the observed correlation matrix rather than generating random data. Comparison data (CD) with known factorial structure are first generated using 1 factor, and then the number of factors is increased until the reproduction of the observed eigenvalues fails to improve significantly" (Ruscio & Roche, 2012, p. 282).

The CD implementation here is based on the code by Ruscio and Roche (2012), but is slightly adapted to increase speed by performing the principal axis factoring using a C++ based function.

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 CD and then pass the complete data.

The CD function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class CD containing

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

Other factor retention criteria: EKC, HULL, KGC, PARALLEL, SMT

N_FACTORS as a wrapper function for this and all the above-mentioned factor retention criteria.

Examples

# determine n factors of the GRiPS
CD(GRiPS_raw)

# determine n factors of the DOSPERT risk subscale
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 COMPARE containing summary information of the differences of the objects.

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 COMPARE containing summary statistics on the differences of x and y.

Examples

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

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

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

Consensus Procrustes alignment across multiple loading matrices

Description

Align several loading matrices to a common Procrustes consensus target.

Usage

CONSENSUS_PROCRUSTES(
  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,
  oblique_maxit = 500,
  oblique_eps = 1e-05,
  oblique_max_line_search = 10,
  oblique_step0 = 1,
  oblique_normalize = FALSE,
  oblique_random_starts = 0,
  oblique_random_starts_stage = c("final", "none", "outer", "both"),
  oblique_screen_keep = 2,
  oblique_triage_maxit = 25,
  oblique_triage_improve_tol = 0,
  verbose = FALSE
)

Arguments

Details

The function iterates between two steps:

1. each loading matrix is independently aligned to the current target with 'PROCRUSTES()'; and 2. the target is updated to the centroid, i.e., the elementwise average of the aligned matrices.

This makes the target symmetric across imputations or samples: no single solution is permanently privileged as the reference. The outer loop can stop when the target stabilizes, when the consensus loss stabilizes, or when both criteria are satisfied.

If 'multi_start = FALSE', one consensus run is performed. If 'multi_start = TRUE', the same single-start 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.

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

An object of class list of length 2.

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 six different domains: social, recreational, gambling, health/ safety, investment, and ethical.

Usage

DOSPERT_raw

Format

An object of class data.frame with 3123 rows and 30 columns.

Source

Blais, A.-R., & Weber, E. U. (2002). A domain-specific risk-taking (DOSPERT) scale for adult populations. Judgment and Decision Making, 15(4), 263–290. doi: 10.1002/bdm.414

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

This function does an EFA with either PAF, ML, or ULS 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). For all rotations except varimax and promax, the GPArotation package is needed.

Usage

EFA(
  x,
  n_factors,
  N = NA,
  method = c("PAF", "ML", "ULS"),
  rotation = c("none", "varimax", "equamax", "quartimax", "geominT", "bentlerT",
    "bifactorT", "promax", "oblimin", "quartimin", "simplimax", "bentlerQ", "geominQ",
    "bifactorQ"),
  se = c("none", "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"),
  b_boot = 1000,
  ci = 0.95,
  ...
)

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.

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, and abs_eigen depend on the type argument.

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

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

type = "SPSS" will use the following argument specification: init_comm = "smc", criterion = .001, criterion_type = "max_individual", 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 (2020), 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. "SPSS" 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 (2020).

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 23 Algorithms manual and as described by Kaiser (1958). However, there is a slight alteration in computing the varimax criterion, which we found to better align with the results obtain from SPSS. Specifically, the original varimax criterion as described in the SPSS 23 Algorithms manual is sum(n*colSums(lambda ^ 4) - colSums(lambda ^ 2) ^ 2) / n ^ 2, where n is the number of indicators, and lambda is the rotated loadings matrix. However, we found the following to produce results more similar to those of SPSS: sum(n*colSums(abs(lambda)) - colSums(lambda ^ 4) ^ 2) / n^2.

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. For these other rotations, the GPArotation package is needed. Additional arguments can also be specified and will be passed to the respective GPArotation function (e.g., maxit to change the maximum number of iterations for the rotation procedure).

The type argument has no effect on ULS and ML. 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 "factanal" which takes the starting values specified in the stats::factanal function.

Value

A list of class EFA containing (a subset of) the following:

Source

Grieder, S., & Steiner, M.D. (2020). Algorithmic Jingle Jungle: A Comparison of Implementations of Principal Axis Factoring and Promax Rotation in R and SPSS. Manuscript in Preparation.

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

Examples

# A type EFAtools (as presented in Steiner and Grieder, 2020) EFA
EFAtools_PAF <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
                    type = "EFAtools", method = "PAF", rotation = "none")

# A type SPSS EFA to mimick the SPSS implementation (this will throw a warning,
# see below)
SPSS_PAF <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
                type = "SPSS", method = "PAF", rotation = "none")

# A type psych EFA to mimick the psych::fa() implementation
psych_PAF <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
                 type = "psych", method = "PAF", rotation = "none")

# Use ML instead of PAF with type EFAtools
EFAtools_ML <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
                   type = "EFAtools", method = "ML", rotation = "none")

# Use oblimin rotation instead of no rotation with type EFAtools
EFAtools_oblim <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
                      type = "EFAtools", method = "PAF", rotation = "oblimin")

# Do a PAF without rotation without specifying a type, so the arguments
# can be flexibly specified (this is only recommended if you know what you're
# doing)
PAF_none <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
                type = "none", method = "PAF", rotation = "none",
                max_iter = 500, init_comm = "mac", criterion = 1e-4,
                criterion_type = "sum", abs_eigen = FALSE)

# Add a promax rotation
PAF_pro <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
               type = "none", method = "PAF", rotation = "promax",
               max_iter = 500, init_comm = "mac", criterion = 1e-4,
               criterion_type = "sum", abs_eigen = FALSE, k = 3,
               P_type = "unnorm", precision= 1e-5, order_type = "eigen",
               varimax_type = "svd")

Model averaging across different EFA methods and types

Description

Not all EFA procedures always arrive at the same solution. This function allows you perform a number of EFAs from different methods (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 function) and provide a summary across the different solutions.

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"),
  show_progress = TRUE
)

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 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.

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. That is, if, for example, the type argument was set to c("none", "SPSS") and one combination of the specific settings entered was identical to the SPSS combination, this combination would be included in the grid and run only once. We include here a list of arguments that are only evaluated under specific conditions:

The arguments init_comm, criterion, criterion_type, abs_eigen are only evaluated if "PAF" is included in method 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 method.

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 case (defined as a communality of >= 1 + .Machine$double.eps). 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, only the common part accounted for (CAF) index (Lorenzo-Seva, Timmerman, & Kiers, 2011) can be computed. As a consequence, if only "PAF" is included in the method argument, averaging can only be performed for the CAF, and the other fit indices are NA. If a combination of "PAF" and "ML" and/or "ULS" are included in the method argument, the CAF is averaged across all non- problematic factor solutions, while all other fit 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 compared to all other fit indices.

Value

A list of class EFA_AVERAGE containing

Source

Grieder, S., & Steiner, M.D. (2020). Algorithmic Jingle Jungle: A Comparison of Implementations of Principal Axis Factoring and Promax Rotation in R and SPSS. Manuscript in Preparation.

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

Examples

## Not run: 
# 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,
                             method = c("PAF", "ULS", "ML"))

# 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,
                         method = c("PAF", "ULS", "ML"), type = "EFAtools",
                         start_method = "psych")

# 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,
                         method = "ML", rotation = "oblique", type = "EFAtools",
                         start_method = "psych")

## End(Not run)

Exploratory factor analysis on multiple data imputations

Description

Fits EFA to each of several imputed datasets, aligns the factor solutions to a common factor space, and pools the resulting estimates and selected fit quantities across imputations.

Usage

EFA_POOLED(
  data_list,
  p = 0.05,
  target_method = c("consensus", "first_target"),
  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

Details

The function first fits the same EFA model to each imputed dataset. Unrotated loading matrices can optionally be put into a common signed/permuted factor order before averaging. Rotated loading matrices are aligned with either a consensus Procrustes target or with the first imputation's rotated solution as a fixed target. For oblique solutions, factor intercorrelations are transformed/aligned together with the loading matrices so that the factor model remains internally consistent.

Point estimates are pooled by arithmetic averaging after alignment. For oblique rotations, the returned structure matrix is computed from the pooled aligned pattern matrix and the pooled factor correlation matrix, Structure = \Lambda \Phi. Communalities are always computed as the diagonal of the common-factor reproduced correlation matrix, diag(\Lambda \Phi \Lambda') for oblique rotations and diag(\Lambda \Lambda') otherwise.

Residuals are not averaged from the per-imputation residual matrices. Instead, the observed correlation matrices are averaged across imputations and residuals are calculated from the pooled observed correlation matrix minus the model-implied correlation matrix of the pooled solution. Consequently, residual-based fit indices such as RMSR/SRMR are based on pooled residuals.

Fit indices based on the model chi-square are not arithmetic means of the per-imputation fit indices. If possible, chi-square-type fit is pooled with the D2 rule. For CFI, the null-model chi-square is D2-pooled as well when complete-data null-model chi-squares are available. The asymptotic chi-square approximation to D2 is then used for RMSEA and CFI. AIC and BIC, if returned, are chi-square-derived descriptive quantities based on this D2 approximation and should not be interpreted as likelihood-based MI information criteria. D3/D4 pooling is not implemented here because the current EFA objects do not expose the likelihood quantities needed for those methods.

If each component EFA call was run with se = "np-boot" and returned boot.arrays, pooled bootstrap SEs and Wald-type MI confidence intervals are computed for loadings, communalities, residuals, and, when applicable, factor correlations and structure coefficients. Importantly, the rotated bootstrap loading matrices stored by the component EFA calls are not reused directly, because they were aligned to imputation-specific targets. Instead, the unrotated bootstrap loading matrices are re-aligned to the final MI target before estimating within-imputation bootstrap covariance matrices. Rubin-type MI pooling is then applied with T = Ubar + (1 + 1 / m) B. Confidence intervals for loadings, Phi, communalities, residuals, and structure coefficients are Wald-type MI intervals. The confidence level of these pooled intervals is controlled by p; the ci argument passed through ... to the component EFA calls is not used for the pooled intervals. boot.CI$fit_indices_pooled_algorithm, when available, is a percentile-style summary obtained by re-running the pooled-fit algorithm over matched bootstrap replicate indices.

Value

A list of class "EFA_POOLED" containing pooled estimates, residuals, fit indices, the individual fits, and MI diagnostics.

Author(s)

Andreas Soteriades, Markus Steiner

Examples

# create a list of three datasets, mimicking a list you would obtain from
# e.g. mice.
dat_list <- lapply(1:3, function(x) GRiPS_raw[sample(1:nrow(GRiPS_raw), replace = TRUE),])
mod <- EFA_POOLED(dat_list, n_factors = 1, method = "ML")
mod


# add computation of standard errors and CIs
mod <- EFA_POOLED(dat_list, n_factors = 1, method = "ML", se = "np-boot")
mod

Empirical Kaiser Criterion

Description

The empirical Kaiser criterion incorporates random sampling variations of the eigenvalues from the Kaiser-Guttman criterion (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

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

The EKC function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class EKC containing

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

Other factor retention criteria: CD, HULL, KGC, PARALLEL, SMT

N_FACTORS as a wrapper function for this and all the above-mentioned factor retention criteria.

Examples

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

Estimate factor scores for an EFA model

Description

This is a wrapper function for psych::factor.scores to be used directly with an output from EFA or by manually specifying the factor loadings and intercorrelations. Calculates factor scores according to the specified methods if raw data are provided, and only factor weights if a correlation matrix is provided.

Usage

FACTOR_SCORES(
  x,
  f,
  Phi = NULL,
  method = c("Thurstone", "tenBerge", "Anderson", "Bartlett", "Harman", "components"),
  impute = c("none", "means", "median")
)

Arguments

Value

A list of class FACTOR_SCORES containing the following:

Examples

# Example with raw data with method "Bartlett" and no imputation
EFA_raw <- EFA(DOSPERT_raw, n_factors = 10, type = "EFAtools", method = "PAF",
               rotation = "oblimin")
fac_scores_raw <- FACTOR_SCORES(DOSPERT_raw, f = EFA_raw, method = "Bartlett",
                                impute = "none")

# Example with a correlation matrix (does not return factor scores)
EFA_cor <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
               type = "EFAtools", method = "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

An object of class data.frame with 810 rows and 8 columns.

Source

Zhang, D. C., Highhouse, S., & Nye, C. D. (2018). 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 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

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"),
  n_datasets = 1000,
  percent = 95,
  decision_rule = c("means", "percentile", "crawford"),
  n_factors = 1,
  ...
)

Arguments

Details

The Hull method aims to find a model with an optimal balance between model fit and number of parameters. That is, it aims to retrieve only major factors (Lorenzo-Seva, Timmerman, & Kiers, 2011). To this end, it performs the following steps (Lorenzo-Seva, Timmerman, & Kiers, 2011, p.351):

1. It performs parallel analysis and adds one to the identified number of factors (this number is denoted J). J is taken as an upper bound of the number of factors to retain in the hull method. Alternatively, a theoretical number of factors can be entered. In this case J will be set to whichever of these two numbers (from parallel analysis or based on theory) is higher.

2. For all 0 to J factors, the goodness-of-fit (one of CAF, RMSEA, or CFI) and the degrees of freedom (df) are computed.

3. The solutions are ordered according to their df.

4. Solutions that are not on the boundary of the convex hull are eliminated (see Lorenzo-Seva, Timmerman, & Kiers, 2011, for details).

5. All the triplets of adjacent solutions are considered consecutively. The middle solution is excluded if its point is below or on the line connecting its neighbors in a plot of the goodness-of-fit versus the degrees of freedom.

6. Step 5 is repeated until no solution can be excluded.

7. The st values of the “hull” solutions are determined.

8. The solution with the highest st value is selected.

The 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.

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 stats::factanal starting values. See also the EFA documentation.

The HULL function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class HULL containing the following objects

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

Other factor retention criteria: CD, EKC, KGC, PARALLEL, SMT

N_FACTORS as a wrapper function for this and all the above-mentioned factor retention criteria.

Examples

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

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

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


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

## End(Not run)

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:10.1177/10731911198450

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

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

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"),
  n_factors = 1,
  ...
)

Arguments

Details

Originally, the Kaiser-Guttman criterion was intended for the use with prinicpal 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 (PARALLEL), the Hull method HULL, or sequential chi^2 model tests (SMT; see Auerswald & Moshagen, 2019).

The KGC function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class KGC containing

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

Other factor retention criteria: CD, EKC, HULL, PARALLEL, SMT

N_FACTORS as a wrapper function for this and all the above-mentioned factor retention criteria.

Examples

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

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

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\limits_{i<j}\sum r_{ij}^2}{\sum\limits_{i<j}\sum r_{ij}^2 + \sum\limits_{i<j}\sum q_{ij}^2}

with Q = SR^{-1}S and S = (diag R^{-1})^{-1/2} where \sum\limits_{i<j}\sum r_{ij}^2 is the sum of squares of the upper off-diagonal elements of R and \sum\limits_{i<j}\sum q_{ij}^2 is the sum of squares of the upper off-diagonal elements of Q (see also Cureton & D'Augustino, 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 BARTLETT for another test of suitability for factor analysis.

The KMO function can also be called together with the BARTLETT function and with factor retention criteria in the N_FACTORS 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'Augustino, R. B. (1983). Factor analysis: An applied approach. Hillsdale, N.J.: Lawrence Erlbaum Associates, Inc.

See Also

BARTLETT for another measure to determine suitability for factor analysis.

N_FACTORS as a wrapper function for this function, BARTLETT and several factor retention criteria.

Examples

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

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

Arguments

Details

#' MAP is based on the idea that systematic common variance is increasingly removed from a correlation matrix R as principal components are partialled out. After removing the first m components, a residual (partial) covariance matrix is obtained as

C_m = R - A_m A_m',

where A_m contains the first m principal component loading vectors (PCA loadings). This residual matrix is then standardized to a partial correlation matrix

R^*_m = D_m^{-1/2} \, C_m \, D_m^{-1/2},

with D_m = \mathrm{diag}(C_m). The MAP criteria summarize the off-diagonal association remaining in R^*_m. The recommended number of factors/components is the m that minimizes the chosen criterion.

This function returns two MAP criteria:

• TR2 (original MAP):

  \mathrm{MAP}_m = \frac{\mathrm{Trace}(R^{*2}_m) - p}{p(p-1)}

  which is algebraically equivalent to the mean squared off-diagonal partial correlations and corresponds to Velicer's original MAP procedure.

• TR4 (revised MAP):

  \mathrm{MAP4}_m = \frac{\mathrm{Trace}(R^{*4}_m) - p}{p(p-1)}

  a higher-order variant that places more weight on dominant residual association structure.

Input handling. x can be a correlation matrix or raw data. If x is not a correlation matrix, correlations are computed using cor with the requested missing-data handling (use) and association measure (cor_method). If a correlation matrix is supplied, N must be provided.

Matrix conditioning. The function stops if the correlation matrix is singular (non-invertible), because subsequent computations rely on stable matrix operations. If the correlation matrix is not positive definite (e.g., due to sampling error), it is smoothed using cor.smooth.

PCA-based partialing. The PCA loading matrix A is obtained from the eigen-decomposition of R as A = V \Lambda^{1/2}. For each m = 0, \dots, p-1, the first m columns of A are used to compute C_m = R - A_m A_m'. The residual is re-standardized to the partial correlation matrix R^*_m using D_m^{-1/2} (i.e., dividing by the square roots of residual variances).

Termination. If residual variances (the diagonal of C_m) become non-positive or numerically unstable, the loop terminates early because R^*_m cannot be formed reliably.

Value

An object of class "MAP" with the following elements:

• eigenvalues: Eigenvalues of the (possibly smoothed) correlation matrix.

• n_factors_TR2: Index m that minimizes the TR2 (original MAP) criterion.

• n_factors_TR4: Index m that minimizes the TR4 (revised MAP) criterion.

• criteria: A matrix with columns m, TR2 (orig. MAP), and TR4 (revised MAP).

• settings: A list containing use, cor_method, and N.

References

Velicer, W. F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321–327.

Velicer, W. F., Eaton, C. A., & Fava, J. L. (2000). Construct explication through factor or component analysis: A review and evaluation of alternative procedures for determining the number of factors or components. In Goffin, R. D. & Helmes, E. (Eds.), Problems and Solutions in Human Assessment: Honoring Douglas N. Jackson at Seventy (pp. 41–71). Boston: Kluwer.

Examples

## Example with raw data
res <- MAP(GRiPS_raw)
res

## Example with a correlation matrix
res2 <- MAP(test_models$baseline$cormat)
res2

Next eigenvalue sufficiency test (NEST)

Description

NEST uses many synthetic datasets to generate reference eigenvalues against which to compare the empirical eigenvalues. This is similar to parallel analysis, but other than parallel analysis, NEST does not just rely on synthetic eigenvalues based on an identity matrix as null model. It was introduced by Achim (2017), see also Brandenburg and Papenberg (2024) and Caron (2025) for further simulation studies including NEST.

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"),
  n_datasets = 1000,
  ...
)

Arguments

Details

NEST compares the first empirical eigenvalue against the first eigenvalues of n_dataset synthetic datasets based on a null model (i.e., with uncorrelated variables; same as in parallel analysis, see PARALLEL). The following eigenvalues are compared against synthetic datasets based on an EFA-model with one fewer factors than the position of the respective empirical eigenvalue. E.g, the second empirical eigenvalue is compared against synthetic data based on a one-factor model. The alpha-level defines against which percentile of the synthetic eigenvalue distribution to compare the empirical eigenvalues against, i.e., an alpha of .05 (the default) uses the 95th percentile as reference value.

For details on the method, including simulation studies, see Achim (2017), Brandenburg and Papenberg (2024), and Caron (2025).

The NEST function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class NEST containing the following objects

Source

Achim, A. (2017). Testing the number of required dimensions in exploratory factor analysis. The Quantitative Methods for Psychology, 13(1), 64–74. https://doi.org/10.20982/tqmp.13.1.p064

Brandenburg, N., & Papenberg, M. (2024). Reassessment of innovative methods to determine the number of factors: A simulation-based comparison of exploratory graph analysis and Next Eigenvalue Sufficiency Test. Psychological Methods, 29(1), 21–47. https://doi.org/10.1037/met0000527

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

Examples

# with correlation matrix
NEST(test_models$baseline$cormat, N = 500)

# with raw data
NEST(GRiPS_raw)

Various Factor Retention Criteria

Description

Among the most important decisions for an exploratory factor analysis (EFA) is the choice of the number of factors to retain. Several factor retention criteria have been developed for this. With this function, various factor retention criteria can be performed simultaneously. Additionally, the data can be checked for their suitability for factor analysis.

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"),
  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

Details

By default, the entered data are checked for suitability for factor analysis using the following methods (see respective documentations for details):

• Bartlett's test of sphericity (see BARTLETT)

• Kaiser-Meyer-Olkin criterion (see KMO)

The available factor retention criteria are the following (see respective documentations for details):

• Comparison data (see CD)

• Empirical Kaiser criterion (see EKC)

• Hull method (see HULL)

• Kaiser-Guttman criterion (see KGC)

• Parallel analysis (see PARALLEL)

• Next Eigenvalue Sufficiency Test, NEST (see NEST)

• Scree plot (see SCREE)

• Sequential chi-square model tests, RMSEA lower bound, and AIC (see SMT)

Value

A list of class N_FACTORS containing

Examples

# Default criteria, with correlation matrix and fit method "ML" (where needed)
# This will throw a warning for CD, as no raw data were specified
nfac_all <- N_FACTORS(test_models$baseline$cormat, N = 500, method = "ML")

# The same as above, but without "CD"
nfac_wo_CD <- N_FACTORS(test_models$baseline$cormat, criteria = c("EKC",
                        "HULL", "PARALLEL", "NEST"), N = 500,
                        method = "ML")

# Use PAF instead of ML (this will take longer). For this, gof has
# to be set to "CAF" for the Hull method.
nfac_PAF <- N_FACTORS(test_models$baseline$cormat, criteria = c("EKC",
                        "HULL", "PARALLEL", "NEST"), N = 500,
                      gof = "CAF")

# Do KGC and PARALLEL with only "PCA" type of eigenvalues
nfac_PCA <- N_FACTORS(test_models$baseline$cormat, criteria = c("EKC",
                      "HULL", "PARALLEL", "NEST"), N = 500,
                      method = "ML", eigen_type_other = "PCA")

# Use raw data, such that CD can also be performed
nfac_raw <- N_FACTORS(GRiPS_raw, method = "ML")

McDonald's omega

Description

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, SL, 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; Sijtsma, 2009), 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).

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 / grouup factors. Low values indicate that a latent variable is not well defined by its indicators.

The ECV (Sijtsma, 2009, 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 coefficents are obtained from the transformed (orthogonalized) solution (see SL 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. The type argument is not evaluated if model is of class lavaan.

If model is of class SL 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 calculation of the total variance (for the whole scale as well as the subscale composites) can also be controlled in this function using the variance argument. For both types—"EFAtools" and "psych" —variance is set to "correlation" by default, which means that total variances are found using the correlation matrix. If variance = "sums_load" the total variance is calculated using the squared sums of general loadings and group factor loadings and the sum of the uniquenesses. This will only get comparable results to variance = "correlation" if no cross-loadings are present and simple structure is well-achieved in general with the SL solution (i.e., the uniquenesses should capture almost all of the variance not explained by the general factor and the variable's allocated group factor).

Value

If found for an SL or lavaan second-order of 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.

Sijtsma, K. (2009). On the use, the misuse, and the very limited usefulness of Cronbach’s alpha. Psychometrika, 74, 107–120.

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.

Examples

## Use with lavaan outputs

# 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(test_models$baseline$cormat, N = 500, n_factors = 3,
               type = "EFAtools", method = "PAF", rotation = "promax")
sl_mod <- SL(efa_mod, type = "EFAtools", method = "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(test_models$baseline$cormat, N = 500, n_factors = 3,
               type = "EFAtools", method = "PAF", rotation = "promax")
sl_mod <- SL(efa_mod, type = "EFAtools", method = "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

Various methods for performing parallel analysis. This function uses future_lapply for which a parallel processing plan can be selected. To do so, call library(future) and, for example, plan(multisession); see examples.

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"),
  decision_rule = c("means", "percentile", "crawford"),
  n_factors = 1,
  ...
)

Arguments

Details

Parallel analysis (Horn, 1965) compares the eigenvalues obtained from the sample correlation matrix against those of null model correlation matrices (i.e., with uncorrelated variables) of the same sample size. This way, it accounts for the variation in eigenvalues introduced by sampling error and thus eliminates the main problem inherent in the Kaiser-Guttman criterion (KGC).

Three different ways of finding the eigenvalues under the factor model are implemented, namely "SMC", "PCA", and "EFA". PCA leaves the diagonal elements of the correlation matrix as they are and is thus equivalent to what is done in PCA. SMC uses squared multiple correlations as communality estimates with which the diagonal of the correlation matrix is replaced. Finally, EFA performs an EFA with one factor (can be adapted to more factors) to estimate the communalities and based on the correlation matrix with these as diagonal elements, finds the eigenvalues.

Parallel analysis is often argued to be one of the most accurate factor retention criteria. However, for highly correlated factor structures it has been shown to underestimate the correct number of factors. The reason for this is that a null model (uncorrelated variables) is used as reference. However, when factors are highly correlated, the first eigenvalue will be much larger compared to the following ones, as later eigenvalues are conditional on the earlier ones in the sequence and thus the shared variance is already accounted in the first eigenvalue (e.g., Braeken & van Assen, 2017).

The PARALLEL function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class PARALLEL containing the following objects

Source

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

Crawford, A. V., Green, S. B., Levy, R., Lo, W. J., Scott, L., Svetina, D., & Thompson, M. S. (2010). Evaluation of parallel analysis methods for determining the number of factors. Educational and Psychological Measurement, 70(6), 885-901.

Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2), 179–185. doi: 10.1007/BF02289447

See Also

Other factor retention criteria: CD, EKC, HULL, KGC, SMT

N_FACTORS as a wrapper function for this and all the above-mentioned factor retention criteria.

Examples

# example without real data
pa_unreal <- PARALLEL(N = 500, n_vars = 10)

# example with correlation matrix with all eigen_types and PAF estimation
pa_paf <- PARALLEL(test_models$case_11b$cormat, N = 500)

# example with correlation matrix with all eigen_types and ML estimation
# this will be faster than the above with PAF)
pa_ml <- PARALLEL(test_models$case_11b$cormat, N = 500, method = "ML")


## Not run: 
# for parallel computation
future::plan(future::multisession)
pa_faster <- PARALLEL(test_models$case_11b$cormat, N = 500)

## End(Not run)

Rotate a loading matrix to a target using Procrustes alignment

Description

'PROCRUSTES()' aligns one loading matrix to a target loading matrix with the same dimensions. It is used internally by 'EFA_POOLED()', but can also be used directly when factor columns must be brought into a common orientation before averaging or comparing solutions.

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

Details

For 'rotation = "orthogonal"', the function solves the closed-form orthogonal Procrustes problem

\min_T \frac{1}{2}\|A T - B\|_F^2 \quad \textrm{subject to}\quad T'T = I,

where 'A' is the loading matrix and 'B' is 'Target'.

For 'rotation = "oblique"', the function calls the compiled '.oblique_procrustes()' optimizer. The oblique convention is the same as in 'GPArotation::targetQ()':

L = A T^{-T}, \qquad \Phi = T'T, \qquad diag(\Phi) = 1.

Random starts are only used for oblique alignment. For one-factor models, oblique and orthogonal alignment are equivalent, so the function uses the stable one-factor orthogonal solution instead of calling the oblique optimizer.

Value

A list containing aligned 'loadings', transformation matrix 'T', factor intercorrelation matrix 'Phi', target criterion 'value', convergence diagnostics, line-search diagnostics, and multi-start summaries. Row and column names are preserved where possible.

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

An object of class list of length 2.

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

The scree plot was originally introduced by Cattell (1966) to perform the scree test. In a scree plot, the eigenvalues of the factors / components are plotted against the index of the factors / components, ordered from 1 to N factors components, hence from largest to smallest eigenvalue. According to the scree test, the number of factors / components to retain is the number of factors / components to the left of the "elbow" (where the curve starts to level off) in the scree plot.

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"),
  n_factors = 1,
  ...
)

Arguments

Details

As the scree test requires visual examination, the test has been especially criticized for its subjectivity and with this low inter-rater reliability. Moreover, a scree plot can be ambiguous if there are either no clear "elbow" or multiple "elbows", making it difficult to judge just where the eigenvalues do level off. Finally, the scree test has also been found to be less accurate than other factor retention criteria. For all these reasons, the scree test has been recommended against, at least for exclusive use as a factor retention criterion (Zwick & Velicer, 1986)

The SCREE function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class SCREE containing

Source

Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1(2), 245–276. https://doi.org/10.1207/s15327906mbr0102_10

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

Other factor retention criteria: CD, EKC, HULL, PARALLEL, SMT

N_FACTORS as a wrapper function for this and all the above-mentioned factor retention criteria.

Examples

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

Schmid-Leiman Transformation

Description

This function implements the Schmid-Leiman (SL) transformation (Schmid & Leiman, 1957). It takes the pattern coefficients and factor intercorrelations from an oblique factor solution as input and can reproduce the results from psych::schmid and from the SPSS implementation from Wolff & Preising (2005). Other arguments from EFA can be used to control the procedure to find the second-order loadings more flexibly. The function can also be used on a second-order confirmatory factor analysis (CFA) solution from lavaan.

Usage

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

Arguments

Details

The SL transformation (also called SL orthogonalization) is a procedure with which an oblique factor solution is transformed into a hierarchical, orthogonalized solution. As a first step, the factor intercorrelations are again factor analyzed to find second-order factor loadings. If there is only one higher-order factor, this step of the procedure stops there, resulting in a second-order factor structure. The first-order factor and the second-order factor are then orthogonalized, resulting in an orthogonalized factor solution with proportionality constraints. The procedure thus makes a suggested hierarchical data structure based on factor intercorrelations explicit. One major advantage of SL transformation is that it enables variance partitioning between higher-order and first-order factors, including the calculation of McDonald's omegas (see OMEGA).

Value

A list of class SL containing the following

Source

Schmid, J. & Leiman, J. M. (1957). The development of hierarchical factor solutions. Psychometrika, 22(1), 53–61. doi:10.1007/BF02289209

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

Examples

## Use with an output from the EFAtools::EFA function, both with type EFAtools
EFA_mod <- EFA(test_models$baseline$cormat, N = 500, n_factors = 3,
               type = "EFAtools", method = "PAF", rotation = "promax")
SL_EFAtools <- SL(EFA_mod, type = "EFAtools", method = "PAF")


## Use with an output from the psych::fa function with type psych in SL
fa_mod <- psych::fa(test_models$baseline$cormat, nfactors = 3, n.obs = 500,
                    fm = "pa", rotate = "Promax")
SL_psych <- SL(fa_mod, type = "psych", method = "PAF")


## Use more flexibly by entering a pattern matrix and phi directly (useful if
## a factor solution found with another program should be subjected to SL
## transformation)

## For demonstration, take pattern matrix and phi from an EFA output
## This gives the same solution as the first example
EFA_mod <- EFA(test_models$baseline$cormat, N = 500, n_factors = 3,
               type = "EFAtools", method = "PAF", rotation = "promax")
SL_flex <- SL(EFA_mod$rot_loadings, Phi = EFA_mod$Phi, type = "EFAtools",
              method = "PAF")


## Use with a lavaan second-order CFA output

# Create and fit 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 <- lavaan::cfa(mod, sample.cov = test_models$baseline$cormat,
                   sample.nobs = 500, estimator = "ml")

SL_lav <- SL(fit, g_name = "g")

Sequential Chi Square Model Tests, RMSEA lower bound, and AIC

Description

Sequential Chi Square Model Tests (SMT) are a factor retention method where multiple EFAs with increasing numbers of factors are fitted and the number of factors for which the Chi Square value first becomes non-significant is taken as the suggested number of factors. Preacher, Zhang, Kim, & Mels (2013) suggested a similar approach with the lower bound of the 90% confidence interval of the Root Mean Square Error of Approximation (RMSEA; Browne & Cudeck, 1992; Steiger & Lind, 1980), and with the Akaike Information Criterion (AIC). For the RMSEA, the number of factors for which this lower bound first falls below .05 is the suggested number of factors to retain. For the AIC, it is the number of factors where the AIC is lowest.

Usage

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

Arguments

Details

As a first step in the procedure, a maximum number of factors to extract is determined for which the model is still over-identified (df > 0).

Then, EFAs with increasing numbers of factors from 1 to the maximum number are fitted with maximum likelihood estimation.

For the SMT, first the significance of the chi square value for a model with 0 factors is determined. If this value is not significant, 0 factors are suggested to retain. If it is significant, a model with 1 factor is estimated and the significance of its chi square value is determined, and so on, until a non-significant result is obtained. The suggested number of factors is the number of factors for the model where the chi square value first becomes non-significant.

Regarding the RMSEA, the suggested number of factors is the number of factors for the model where the lower bound of the 90% confidence interval of the RMSEA first falls below the .05 threshold.

Regarding the AIC, the suggested number of factors is the number of factors for the model with the lowest AIC.

In comparison with other prominent factor retention criteria, SMT performed well at determining the number of factors to extract in EFA (Auerswald & Moshagen, 2019). The RMSEA lower bound also performed well at determining the true number of factors, while the AIC performed well at determining the most generalizable model (Preacher, Zhang, Kim, & Mels, 2013).

The SMT function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class SMT containing

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

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

Preacher, K. J., Zhang G., Kim, C., & Mels, G. (2013). Choosing the Optimal Number of Factors in Exploratory Factor Analysis: A Model Selection Perspective, Multivariate Behavioral Research, 48(1), 28-56, doi:10.108/00273171.2012.710386

Steiger, J. H., & Lind, J. C. (1980, May). Statistically based tests for the number of common factors. Paper presented at the annual meeting of the Psychometric Society, Iowa City, IA.

See Also

Other factor retention criteria: CD, EKC, HULL, KGC, PARALLEL

N_FACTORS as a wrapper function for this and all the above-mentioned factor retention criteria.

Examples

SMT_base <- SMT(test_models$baseline$cormat, N = 500)
SMT_base

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 (2020).

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. (2020). Algorithmic Jingle Jungle: A Comparison of Implementations of Principal Axis Factoring and Promax Rotation in R and SPSS. Manuscript in Preparation.

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:10.1177/10731911198450

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 (2020).

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. (2020). Algorithmic Jingle Jungle: A Comparison of Implementations of Principal Axis Factoring and Promax Rotation in R and SPSS. Manuscript in Preparation.

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:10.1177/10731911198450

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

An object of class data.frame with 645 rows and 45 columns.

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.

Plot CD object

Description

Plot method showing a summarized output of the CD function

Usage

## S3 method for class 'CD'
plot(x, ...)

Arguments

Plot EFA_AVERAGE object

Description

Plot method showing a summarized output of the EFA_AVERAGE function

Usage

## S3 method for class 'EFA_AVERAGE'
plot(x, ...)

Arguments

Examples

## Not run: 
EFA_aver <- EFA_AVERAGE(test_models$baseline$cormat, n_factors = 3, N = 500)
EFA_aver

## End(Not run)

Plot EKC object

Description

Plot method showing a summarized output of the EKC function

Usage

## S3 method for class 'EKC'
plot(x, ...)

Arguments

Examples

EKC_base <- EKC(test_models$baseline$cormat, N = 500)
plot(EKC_base)

Plot HULL object

Description

Plot method showing a summarized output of the HULL function

Usage

## S3 method for class 'HULL'
plot(x, ...)

Arguments

Examples

x <- HULL(test_models$baseline$cormat, N = 500, method = "ML")
plot(x)

Plot KGC object

Description

Plot method showing a summarized output of the KGC function

Usage

## S3 method for class 'KGC'
plot(x, ...)

Arguments

Examples

KGC_base <- KGC(test_models$baseline$cormat)
plot(KGC_base)

Plot PARALLEL object

Description

Plot method showing a summarized output of the PARALLEL function

Usage

## S3 method for class 'PARALLEL'
plot(x, ...)

Arguments

Examples

# example with correlation matrix and "ML" estimation
x <- PARALLEL(test_models$case_11b$cormat, N = 500, method = "ML")
plot(x)

Plot SCREE object

Description

Plot method showing a summarized output of the SCREE function

Usage

## S3 method for class 'SCREE'
plot(x, ...)

Arguments

Examples

SCREE_base <- SCREE(test_models$baseline$cormat)
plot(SCREE_base)

population_models

Description

Population factor models, some of which (baseline to case_11e) used for the simulation analyses reported in Grieder and Steiner (2019). All combinations of the pattern matrices and the factor intercorrelations were used in the simulations. Many models are based on cases used in de Winter and Dodou (2012).

Usage

population_models

Format

A list of 3 lists "loadings", "phis_3", and "phis_6".

loadings contains the following matrices of pattern coefficients:

baseline

(matrix) - The pattern coefficients of the baseline model. Three factors with six indicators each, all with pattern coefficients of .6. Same baseline model as used in de Winter and Dodou (2012).

case_1a

(matrix) - Three factors with 2 indicators per factor.

case_1b

(matrix) - Three factors with 3 indicators per factor. Case 5 in de Winter and Dodou (2012).

case_1c

(matrix) - Three factors with 4 indicators per factor.

case_1d

(matrix) - Three factors with 5 indicators per factor.

case_2

(matrix) - Same as baseline model but with low pattern coefficients of .3.

case_3

(matrix) - Same as baseline model but with high pattern coefficients of .9.

case_4

(matrix) - Three factors with different pattern coefficients between factors (one factor with .9, one with .6, and one with .3, respectively). Case 7 in de Winter and Dodou (2012).

case_5

(matrix) - Three factors with different pattern coefficients within factors (each factor has two pattern coefficients of each .9, .6, and .3). Similar to cases 8/ 9 in de Winter and Dodou (2012).

case_6a

(matrix) - Same as baseline model but with one cross loading of .4. Similar to case 10 in de Winter and Dodou (2012).

case_6b

(matrix) - Same as baseline model but with three cross loading of .4 (One factor with 2 and one with 1 crossloading). Similar to case 10 in de Winter and Dodou (2012).

case_7

(matrix) - Three factors with different number of indicators per factor (2, 4, and 6 respectively). Similar to cases 11/ 12 in de Winter and Dodou (2012).

case_8

(matrix) - Three factors with random variation in pattern coefficients added, drawn from a uniform distribution between [-.2, .2]. Case 13 in de Winter and Dodou (2012).

case_9a

(matrix) - Three factors with 2 indicators per factor, with different pattern coefficients within one of the factors.

case_9b

(matrix) - Three factors with 3 indicators per factor, with different pattern coefficients.

case_9c

(matrix) - Three factors with 4 indicators per factor, with different pattern coefficients.

case_9d

(matrix) - Three factors with 5 indicators per factor, with different pattern coefficients.

case_10a

(matrix) - Six factors with 2 indicators per factor, all with pattern coefficients of .6.

case_10b

(matrix) - Six factors with 3 indicators per factor, all with pattern coefficients of .6.

case_10c

(matrix) - Six factors with 4 indicators per factor, all with pattern coefficients of .6.

case_10d

(matrix) - Six factors with 5 indicators per factor, all with pattern coefficients of .6.

case_10e

(matrix) - Six factors with 6 indicators per factor, all with pattern coefficients of .6.

case_11a

(matrix) - Six factors with 2 indicators per factor, with different pattern coefficients within and between factors (.3, .6, and .9).

case_11b

(matrix) - Six factors with 3 indicators per factor, with different pattern coefficients within and between factors (.3, .6, and .9).

case_11c

(matrix) - Six factors with 4 indicators per factor, with different pattern coefficients within and between factors (.3, .6, and .9).

case_11d

(matrix) - Six factors with 5 indicators per factor, with different pattern coefficients within and between factors (.3, .6, and .9).

case_11e

(matrix) - Six factors with 6 indicators per factor, with different pattern coefficients within and between factors (.3, .6, and .9).

case_12a

(matrix) - One factor, with 2 equal pattern coefficients (.6).

case_12b

(matrix) - One factor, with 3 equal pattern coefficients (.6).

case_12c

(matrix) - One factor, with 6 equal pattern coefficients (.6).

case_12d

(matrix) - One factor, with 10 equal pattern coefficients (.6).

case_12e

(matrix) - One factor, with 15 equal pattern coefficients (.6).

case_13a

(matrix) - One factor, with 2 different pattern coefficients (.3, and .6).

case_13b

(matrix) - One factor, with 3 different pattern coefficients (.3, .6, and .9).

case_13c

(matrix) - One factor, with 6 different pattern coefficients (.3, .6, and .9).

case_13d

(matrix) - One factor, with 10 different pattern coefficients (.3, .6, and .9).

case_13e

(matrix) - One factor, with 15 different pattern coefficients (.3, .6, and .9).

case_14a

(matrix) - No factor, 2 variables (0).

case_14b

(matrix) - No factor, 3 variables (0).

case_14c

(matrix) - No factor, 6 variables (0).

case_14d

(matrix) - No factor, 10 variables (0).

case_14e

(matrix) - No factor, 15 variables (0).

phis_3 contains the following 3x3 matrices:

zero

(matrix) - Matrix of factor intercorrelations of 0. Same intercorrelations as used in de Winter and Dodou (2012).

moderate

(matrix) - Matrix of moderate factor intercorrelations of .3.

mixed

(matrix) - Matrix of mixed (.3, .5, and .7) factor intercorrelations.

strong

(matrix) - Matrix of strong factor intercorrelations of .7. Same intercorrelations as used in de Winter and Dodou (2012).

phis_6 contains the following 6x6 matrices:

zero

(matrix) - Matrix of factor intercorrelations of 0. Same intercorrelations as used in de Winter and Dodou (2012).

moderate

(matrix) - Matrix of moderate factor intercorrelations of .3.

mixed

(matrix) - Matrix of mixed (around .3, .5, and .7; smoothing was necessary for the matrix to be positive definite) factor intercorrelations.

strong

(matrix) - Matrix of strong factor intercorrelations of .7. Same intercorrelations as used in de Winter and Dodou (2012).

Source

Grieder, S., & Steiner, M.D. (2020). Algorithmic Jingle Jungle: A Comparison of Implementations of Principal Axis Factoring and Promax Rotation in R and SPSS. Manuscript in Preparation.

de Winter, J.C.F., & Dodou, D. (2012). Factor recovery by principal axis factoring and maximum likelihood factor analysis as a function of factor pattern and sample size. Journal of Applied Statistics. 39.

Print BARTLETT object

Description

Print BARTLETT object

Usage

## S3 method for class 'BARTLETT'
print(x, ...)

Arguments

Examples

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

Print function for CD objects

Description

Print function for CD objects

Usage

## S3 method for class 'CD'
print(x, plot = TRUE, ...)

Arguments

Examples

# determine n factors of the GRiPS
CD(GRiPS_raw)

Print COMPARE object

Description

Print Method showing a summarized output of the COMPARE function.

Usage

## S3 method for class 'COMPARE'
print(x, ...)

Arguments

Examples

# A type SPSS EFA to mimick the SPSS implementation
EFA_SPSS_5 <- EFA(IDS2_R, n_factors = 5, type = "SPSS")

# A type psych EFA to mimick the psych::fa() implementation
EFA_psych_5 <- EFA(IDS2_R, n_factors = 5, type = "psych")

# compare the two
COMPARE(EFA_SPSS_5$unrot_loadings, EFA_psych_5$unrot_loadings,
        x_labels = c("SPSS", "psych"))

Print EFA object

Description

Print Method showing a summarized output of the EFA function.

Usage

## S3 method for class 'EFA'
print(
  x,
  cutoff = 0.3,
  digits = 3,
  max_name_length = 10,
  ci = c("auto", "none", "separate"),
  ci_filter = c("salient", "all", "nonzero"),
  details = c("standard", "compact", "full"),
  diagnostics = TRUE,
  diagnostics_top_n = 10,
  residual_cutoff = 0.1,
  residual_top_n = 10,
  show_structure = FALSE,
  sort_loadings = c("none", "primary", "clustered"),
  show_loading_legend = TRUE,
  cross_loading_cutoff = cutoff,
  min_primary_gap = 0.2,
  min_salient_per_factor = 3,
  max_factors_per_block = NULL,
  show_mi_diagnostics = NULL,
  ...
)

## S3 method for class 'EFA_POOLED'
print(
  x,
  cutoff = 0.3,
  digits = 3,
  max_name_length = 10,
  ci = c("auto", "none", "separate"),
  ci_filter = c("salient", "all", "nonzero"),
  details = c("standard", "compact", "full"),
  diagnostics = TRUE,
  diagnostics_top_n = 10,
  residual_cutoff = 0.1,
  residual_top_n = 10,
  show_structure = FALSE,
  sort_loadings = c("none", "primary", "clustered"),
  show_loading_legend = TRUE,
  cross_loading_cutoff = cutoff,
  min_primary_gap = 0.2,
  min_salient_per_factor = 3,
  max_factors_per_block = NULL,
  show_mi_diagnostics = NULL,
  ...
)

## S3 method for class 'EFA'
format(
  x,
  cutoff = 0.3,
  digits = 3,
  max_name_length = 10,
  ci = c("auto", "none", "separate"),
  ci_filter = c("salient", "all", "nonzero"),
  details = c("standard", "compact", "full"),
  diagnostics = TRUE,
  diagnostics_top_n = 10,
  residual_cutoff = 0.1,
  residual_top_n = 10,
  show_structure = FALSE,
  sort_loadings = c("none", "primary", "clustered"),
  show_loading_legend = TRUE,
  cross_loading_cutoff = cutoff,
  min_primary_gap = 0.2,
  min_salient_per_factor = 3,
  max_factors_per_block = NULL,
  show_mi_diagnostics = NULL,
  ...
)

## S3 method for class 'EFA_POOLED'
format(
  x,
  cutoff = 0.3,
  digits = 3,
  max_name_length = 10,
  ci = c("auto", "none", "separate"),
  ci_filter = c("salient", "all", "nonzero"),
  details = c("standard", "compact", "full"),
  diagnostics = TRUE,
  diagnostics_top_n = 10,
  residual_cutoff = 0.1,
  residual_top_n = 10,
  show_structure = FALSE,
  sort_loadings = c("none", "primary", "clustered"),
  show_loading_legend = TRUE,
  cross_loading_cutoff = cutoff,
  min_primary_gap = 0.2,
  min_salient_per_factor = 3,
  max_factors_per_block = NULL,
  show_mi_diagnostics = NULL,
  ...
)

Arguments

Details

The method is shared by single-imputation 'EFA' objects and pooled 'EFA_POOLED' objects. It prints, in order, a compact model header, optional diagnostics, the loading table, optional confidence-interval tables, variance accounted for, model fit, bootstrap notes, optional multiple-imputation uncertainty diagnostics, and residual diagnostics.

For 'EFA_POOLED' objects, the header and diagnostics report the number of imputations and the alignment/pooling settings stored in the object. Loading and factor-correlation confidence intervals are printed only when 'boot.CI' is present. Pooled intervals created by 'EFA_POOLED()' are labelled as bootstrap/MI intervals.

'ci_filter' controls which loading intervals are shown. 'salient' reports intervals for loadings whose absolute point estimate is at least 'cutoff'; 'nonzero' reports intervals excluding zero; and 'all' reports every finite interval.

Examples

mod <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
           method = "PAF", rotation = "promax")
mod

# With SEs and CIs, needs raw-data
mod <- EFA(GRiPS_raw, n_factors = 1, method = "ML", se = "np-boot")
mod
# with additional infos
print(mod, details = "full")
# omit some diagnostic parts
print(mod, details = "compact")

Print EFA_AVERAGE object

Description

Print Method showing a summarized output of the EFA_AVERAGE function

Usage

## S3 method for class 'EFA_AVERAGE'
print(x, stat = c("average", "range"), plot = TRUE, ...)

Arguments

Examples

## Not run: 
EFA_aver <- EFA_AVERAGE(test_models$baseline$cormat, n_factors = 3, N = 500)
EFA_aver

## End(Not run)

Print function for EKC objects

Description

Print function for EKC objects

Usage

## S3 method for class 'EKC'
print(x, plot = TRUE, ...)

Arguments

Examples

EKC_base <- EKC(test_models$baseline$cormat, N = 500)
EKC_base

Print function for HULL objects

Description

Print function for HULL objects

Usage

## S3 method for class 'HULL'
print(x, plot = TRUE, ...)

Arguments

Examples

HULL(test_models$baseline$cormat, N = 500, method = "ML")

Print function for KGC objects

Description

Print function for KGC objects

Usage

## S3 method for class 'KGC'
print(x, plot = TRUE, ...)

Arguments

Examples

KGC_base <- KGC(test_models$baseline$cormat)
KGC_base

Print KMO object

Description

Print KMO object

Usage

## S3 method for class 'KMO'
print(x, ...)

Arguments

Examples

KMO_base <- KMO(test_models$baseline$cormat)
KMO_base

Print LOADINGS object

Description

Print LOADINGS object

Usage

## S3 method for class 'LOADINGS'
print(
  x,
  cutoff = 0.3,
  digits = 3,
  max_name_length = 10,
  h2 = NULL,
  color = TRUE,
  name_style = c("truncate", "abbreviate", "full"),
  max_factor_name_length = NULL,
  max_factors_per_block = NULL,
  sort_loadings = c("none", "primary", "clustered"),
  legend = FALSE,
  ...
)

## S3 method for class 'LOADINGS'
format(
  x,
  cutoff = 0.3,
  digits = 3,
  max_name_length = 10,
  h2 = NULL,
  color = TRUE,
  name_style = c("truncate", "abbreviate", "full"),
  max_factor_name_length = NULL,
  max_factors_per_block = NULL,
  sort_loadings = c("none", "primary", "clustered"),
  legend = FALSE,
  ...
)

Arguments

Details

The method prints a loading matrix in a compact, console-oriented table. Loadings with absolute value greater than or equal to 'cutoff' are emphasized, smaller loadings are de-emphasized, and Heywood-relevant communality/ uniqueness values are marked when 'h2' is supplied. Long variable names can be truncated, abbreviated, or printed in full. If the matrix has many factor columns, the table is split into column blocks so that the output remains readable in narrower consoles.

If 'h2' is named and 'x' has row names, 'h2' is matched to the row names of 'x' before any optional row sorting is applied. If 'x' has no row names, a named 'h2' vector is used in the supplied order.

Examples

EFAtools_PAF <- EFA(test_models$baseline$cormat, n_factors = 3, N = 500,
                    type = "EFAtools", method = "PAF", rotation = "promax")
EFAtools_PAF

Print function for MAP objects

Description

Print function for MAP objects

Usage

## S3 method for class 'MAP'
print(x, plot = TRUE, ...)

Arguments

Examples

MAP_base <- MAP(test_models$baseline$cormat)
MAP_base

Print function for NEST objects

Description

Print function for NEST objects

Usage

## S3 method for class 'NEST'
print(x, ...)

Arguments

Examples

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

Print function for N_FACTORS objects

Description

Print function for N_FACTORS objects

Usage

## S3 method for class 'N_FACTORS'
print(x, ...)

Arguments

Examples

# All criteria except "CD", with correlation matrix and fit method "ML"
# (where needed)
N_FACTORS(test_models$baseline$cormat, criteria = c("EKC", "HULL", "KGC",
          "PARALLEL", "SCREE", "SMT"), N = 500, method = "ML")

Print OMEGA object

Description

Print OMEGA object

Usage

## S3 method for class 'OMEGA'
print(x, digits = 3, ...)

Arguments

Examples

efa_mod <- EFA(test_models$baseline$cormat, N = 500, n_factors = 3,
               type = "EFAtools", method = "PAF", rotation = "promax")
sl_mod <- SL(efa_mod, type = "EFAtools", method = "PAF")

OMEGA(sl_mod, type = "EFAtools",
factor_corres = sl_mod$sl[, c("F1", "F2", "F3")] >= .2)

Print function for PARALLEL objects

Description

Print function for PARALLEL objects

Usage

## S3 method for class 'PARALLEL'
print(x, plot = TRUE, ...)

Arguments

Examples

# example without real data
PARALLEL(N = 500, n_vars = 10)

# example with correlation matrix and "ML" estimation
PARALLEL(test_models$case_11b$cormat, N = 500, method = "ML")

Print function for SCREE objects

Description

Print function for SCREE objects

Usage

## S3 method for class 'SCREE'
print(x, plot = TRUE, ...)

Arguments

Examples

SCREE_base <- SCREE(test_models$baseline$cormat)
SCREE_base

Print SL object

Description

Print Method showing a summarized output of the SL function.

Usage

## S3 method for class 'SL'
print(x, ...)

Arguments

Examples

EFA_mod <- EFA(test_models$baseline$cormat, N = 500, n_factors = 3,
               type = "EFAtools", method = "PAF", rotation = "promax")
SL(EFA_mod, type = "EFAtools", method = "PAF")

Print SLLOADINGS object

Description

Print SLLOADINGS object

Usage

## S3 method for class 'SLLOADINGS'
print(x, cutoff = 0.2, digits = 3, ...)

Arguments

Examples

EFA_mod <- EFA(test_models$baseline$cormat, N = 500, n_factors = 3,
               type = "EFAtools", method = "PAF", rotation = "promax")
SL(EFA_mod, type = "EFAtools", method = "PAF")

Print SMT object

Description

Print SMT object

Usage

## S3 method for class 'SMT'
print(x, ...)

Arguments

Examples

SMT_base <- SMT(test_models$baseline$cormat, N = 500)
SMT_base

Residuals function for EFA objects

Description

Residuals function for EFA objects

Usage

## S3 method for class 'EFA'
residuals(object, print = TRUE, digits = 3, ...)

Arguments

Four test models used in Grieder and Steiner (2020)

Description

Correlation matrices created from simulated data from four of the population_models cases, each with strong factor intercorrelations. These are used in Grieder & Steiner (2020) to compare the psych and SPSS implementations in this package with the actual implementations of the programs. For details on the cases, see population_models.

Usage

test_models

Format

A list of 4 lists "baseline", "case_1a", "case_6b", and"case_11b", each with the following elements.

cormat

(matrix) - The correlation matrix of the simulated data.

n_factors

(numeric) - The true number of factors.

N

(numeric) - The sample size of the generated data.

Source

Grieder, S., & Steiner, M.D. (2020). Algorithmic Jingle Jungle: A Comparison of Implementations of Principal Axis Factoring and Promax Rotation in R and SPSS. Manuscript in Preparation.