Parallel analysis

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

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

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.

Glorfeld, L. W. (1995). An improvement on Horn's parallel analysis methodology for selecting the correct number of factors to retain. Educational and Psychological Measurement, 55(3), 377-393.

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

Arguments

x: matrix or data.frame. The real data to compare the simulated eigenvalues against. Must not contain variables of classes other than numeric. Can be a correlation matrix or raw data.
N: numeric. The number of cases / observations to simulate. Only has to be specified if x is either a correlation matrix or NULL. If x contains raw data, N is found from the dimensions of x.
n_vars: numeric. The number of variables / indicators to simulate. Only has to be specified if x is left as NULL as otherwise the dimensions are taken from x.
n_datasets: numeric. The number of datasets to simulate. Default is 1000.
percent: numeric. The percentile to take from the simulated eigenvalues. Default is 95.
eigen_type: character. On what the eigenvalues should be found. Can be either "SMC", "PCA", or "EFA". If using "SMC", the diagonal of the correlation matrix is replaced by the squared multiple correlations (SMCs) of the indicators. If using "PCA", the diagonal values of the correlation matrices are left to be 1. If using "EFA", eigenvalues are found on the correlation matrices with the final communalities of an EFA solution as diagonal.
use: character. Passed to stats::cor() if raw data is given as input. Default is "pairwise.complete.obs".
cor_method: character. One of "pearson", "spearman", or "kendall", passed to stats::cor(). "poly" and "tetra" are not supported because PARALLEL compares the data against simulated continuous reference data. Default is "pearson".
decision_rule: character. Which rule to use to determine the number of factors to retain. Default is "means", which will use the average simulated eigenvalues. "percentile", uses the percentiles specified in percent. "crawford" uses the 95th percentile for the first factor and the mean afterwards (based on Crawford et al, 2010). The "means" rule retains a factor whenever its real eigenvalue exceeds the average simulated one and thus tends to retain more factors than the more conservative "percentile" rule (Glorfeld, 1995).
n_factors: numeric. Number of factors to extract if "EFA" is included in eigen_type. Default is 1.
estimate_control: an estimate_control() object with the estimation settings for the efa_fit() fits (of both the real and the simulated data) when "EFA" is included in eigen_type. NULL (default) uses the efa_fit() defaults. The fits are unrotated, so no rotation settings apply.
...: Additional arguments passed to efa_fit(). For example, estimator, to change the estimator (default is "PAF"). PAF is more robust, but it will take longer compared to the other estimators available ("ML" and "ULS"). The estimation tuning knobs are not passed here; they live in estimate_control.

Value

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

n_factors: A named numeric vector with the suggested number of factors for each requested eigenvalue type ("PCA", "SMC", and/or "EFA"). These are NA when no real data are supplied (i.e. only N and n_vars are given). When every real eigenvalue exceeds its reference (no crossing is found), all n_vars components are retained and a warning is issued.
results: A list with one record per eigenvalue type, each holding the real eigenvalues (when real data were supplied) and the simulated reference eigenvalues (means and percentiles) used for printing and plotting.
settings: A list of the settings used.

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 (efa_kgc()).

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 reference eigenvalues are obtained from simulated data, so the suggested number of factors varies slightly from run to run. Call set.seed() beforehand to make a run reproducible; the result is then also independent of the parallel plan set via future::plan(), so it can be reproduced on a machine with a different number of cores.

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

Examples

# \donttest{
# example without real data
pa_unreal <- efa_parallel(N = 500, n_vars = 10)

# example with correlation matrix with all eigen_types and PAF estimation
pa_paf <- efa_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 <- efa_parallel(test_models$case_11b$cormat, N = 500, estimator = "ML")
# }

if (FALSE) { # \dontrun{
# for parallel computation
future::plan(future::multisession)
pa_faster <- efa_parallel(test_models$case_11b$cormat, N = 500)
} # }