EFA_AVERAGE: Model averaging across different EFA methods and types

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

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

x	data.frame or matrix. Dataframe or matrix of raw data or matrix with correlations. If raw data is entered, the correlation matrix is found from the data.
n_factors	numeric. Number of factors to extract.
N	numeric. The number of observations. Needs only be specified if a correlation matrix is used. If input is a correlation matrix and N = NA (default), not all fit indices can be computed.
method	character vector. Any combination of "PAF", "ML", and "ULS", to use principal axis factoring, maximum likelihood, or unweighted least squares (also called minres), respectively, to fit the EFAs. Default is "PAF".
rotation	character vector. Either perform no rotation ("none"), any combination of orthogonal rotations ("varimax", "equamax", "quartimax", "geominT", "bentlerT", and "bifactorT"; using "orthogonal" runs all of these), or of oblique rotations ("promax", "oblimin", "quartimin", "simplimax", "bentlerQ", "geominQ", and "bifactorQ"; using "oblique" runs all of these). Rotation types (no rotation, orthogonal rotations, and oblique rotations) cannot be mixed. Default is "promax".
type	character vector. Any combination of "none" (default), "EFAtools", "psych", and "SPSS" can be entered. "none" allows the specification of various combinations of the arguments controlling both factor extraction methods and the rotations. The others ("EFAtools", "psych", and "SPSS"), control the execution of the respective factor extraction method and rotation to be in line with how it is executed in this package (i.e., the respective default procedure), in the psych package, and in SPSS. A specific psych implementation exists for PAF, ML, varimax, and promax. The SPSS implementation exists for PAF, varimax, and promax. For details, see EFA.
averaging	character. One of "mean" (default), and "median". Controls whether the different results should be averaged using the (trimmed) mean, or the median.
trim	numeric. If averaging is set to "mean", this argument controls the trimming of extremes (for details see base::mean). By default no trimming is done (i.e., trim = 0).
salience_threshold	numeric. The threshold to use to classify a pattern coefficient or loading as salient (i.e., substantial enough to assign it to a factor). Default is 0.3. Indicator-to-factor correspondences will be inferred based on this threshold. Note that this may not be meaningful if rotation = "none" and n_factors > 1 are used, as no simple structure is present there.
max_iter	numeric. The maximum number of iterations to perform after which the iterative PAF procedure is halted with a warning. Default is 10,000. Note that non-converged procedures are excluded from the averaging procedure.
init_comm	character vector. Any combination of "smc", "mac", and "unity". Controls the methods to estimate the initial communalities in PAF if "none" is among the specified types. "smc" will use squared multiple correlations, "mac" will use maximum absolute correlations, "unity" will use 1s (for details see EFA). Default is c("smc", "mac", "unity").
criterion	numeric vector. The convergence criterion used for PAF if "none" is among the specified types. If the change in communalities from one iteration to the next is smaller than this criterion the solution is accepted and the procedure ends. Default is 0.001.
criterion_type	character vector. Any combination of "max_individual" and "sum". Type of convergence criterion used for PAF if "none" is among the specified types. "max_individual" selects the maximum change in any of the communalities from one iteration to the next and tests it against the specified criterion. "sum" takes the difference of the sum of all communalities in one iteration and the sum of all communalities in the next iteration and tests this against the criterion (for details see EFA). Default is c("sum", "max_individual").
abs_eigen	logical vector. Any combination of TRUE and FALSE. Which algorithm to use in the PAF iterations if "none" is among the specified types. If FALSE, the loadings are computed from the eigenvalues. This is also used by the psych::fa function. If TRUE the loadings are computed with the absolute eigenvalues as done by SPSS (for details see EFA). Default is TRUE.
varimax_type	character vector. Any combination of "svd" and "kaiser". The type of the varimax rotation performed if "none" is among the specified types and "varimax", "promax", "orthogonal", or "oblique" is among the specified rotations. "svd" uses singular value decomposition, as stats::varimax does, and "kaiser" uses the varimax procedure performed in SPSS. This is the original procedure from Kaiser (1958), but with slight alterations in the varimax criterion (for details, see EFA and Grieder & Steiner, 2020). Default is c("svd", "kaiser").
normalize	logical vector. Any combination of TRUE and FALSE. TRUE performs a kaiser normalization before the specified rotation(s). Default is TRUE.
k_promax	numeric vector. The power used for computing the target matrix P in the promax rotation if "none" is among the specified types and "promax" or "oblique" is among the specified rotations. Default is 2:4.
k_simplimax	numeric. The number of 'close to zero loadings' for the simplimax rotation (see GPArotation::GPFoblq) if "simplimax" or "oblique" is among the specified rotations. Default is ncol(x), where x is the entered data.
P_type	character vector. Any combination of "norm" and "unnorm". This specifies how the target matrix P is computed in promax rotation if "none" is among the specified types and "promax" or "oblique" is among the specified rotations. "unnorm" will use the unnormalized target matrix as originally done in Hendrickson and White (1964). "norm" will use a normalized target matrix (for details see EFA). Default is c("norm", "unnorm").
precision	numeric vector. The tolerance for stopping in the rotation procedure(s). Default is 10^-5.
start_method	character vector. Any combination of "psych" and "factanal". How to specify the starting values for the optimization procedure for ML. "psych" takes the starting values specified in psych::fa. "factanal" takes the starting values specified in the stats::factanal function. Default is c("psych", "factanal").
use	character. Passed to stats::cor if raw data is given as input. Default is "pairwise.complete.obs".
cor_method	character. Passed to stats::cor. Default is "pearson".
show_progress	logical. Whether a progress bar should be shown in the console. Default is TRUE.

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 loading or communality of >= .998). 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

orig_R	Original correlation matrix.
h2	A list with the average, standard deviation, minimum, maximum, and range of the final communality estimates across the factor solutions.
loadings	A list with the average, standard deviation, minimum, maximum, and range of the final loadings across the factor solutions. If rotation was "none", the unrotated loadings, otherwise the rotated loadings (pattern coefficients).
Phi	A list with the average, standard deviation, minimum, maximum, and range of the factor intercorrelations across factor solutions obtained with oblique rotations.
ind_fac_corres	A matrix with each cell containing the proportion of the factor solutions in which the respective indicator-to-factor correspondence occurred, i.e., in which the loading exceeded the specified salience threshold. Note: Rowsums can exceed 1 due to cross-loadings.
vars_accounted	A list with the average, standard deviation, minimum, maximum, and range of explained variances and sums of squared loadings across the factor solutions. Based on the unrotated loadings.
fit_indices	A matrix containing the average, standard deviation, minimum, maximum, and range for all applicable fit indices across the respective factor solutions, and the degrees of freedom (df). If the method argument contains ML or ULS: Fit indices derived from the unrotated factor loadings: Chi Square (chisq), including significance level, Comparative Fit Index (CFI), Root Mean Square Error of Approximation (RMSEA), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC)and the common part accounted for (CAF) index as proposed by Lorenzo-Seva, Timmerman, & Kiers (2011). For PAF, only the CAF can be calculated (see details).
implementations_grid	A matrix containing, for each performed EFA, the setting combination, if an error occurred (logical), the error message (character), an integer code for convergence as returned by stats:optim (0 indicates successful completion.), if heywood cases occurred (logical, see details for definition), if the solution was admissible (logical, see details for definition), and the fit indices.
efa_list	A list containing the outputs of all performed EFAs. The names correspond to the rownames from the implementations_grid.
settings	A list of the settings used.

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)

EFAtools documentation built on Jan. 6, 2023, 5:16 p.m.