Empirical Kaiser criterion

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

Usage

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

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. https://doi.org/10.1037/met0000074

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

Arguments

x: data.frame or matrix. Dataframe or matrix of raw data or matrix with correlations.
N: numeric. The number of observations. Only needed if x is a correlation matrix. Must be larger than the number of variables.
use: character. Passed to stats::cor() if raw data is given as input. Default is "pairwise.complete.obs".
cor_method: character. Correlation computed from raw data: "pearson", "spearman", or "kendall" (passed to stats::cor()), or "poly" / "tetra" for polychoric / tetrachoric correlations of ordinal / binary data (a two-step estimator). Default is "pearson". Note that the EKC reference values rest on the Marchenko-Pastur law for the eigenvalues of a sample correlation matrix of independent variables, which assumes the sampling behaviour of product-moment correlations; with "poly" / "tetra" (and, to a lesser degree, the rank-based methods) the reference series is therefore an approximation.
type: Accepted and ignored. It selected between two ways to compute the reference values. The "AM2019" reference values do not depend on the observed eigenvalues, so they do not apply the empirical correction that defines the criterion, and they are no longer computed.

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 numeric vector of length one, named "BvA2017", with the suggested number of factors. The factors up to the first observed eigenvalue that fails to exceed its reference value are retained. The "all-exceed" convention of parallel analysis (efa_parallel()), which retains all J factors when no such crossing is found, cannot be reached here: the reference values are never below 1, while the eigenvalues of a correlation matrix sum to J and are sorted downwards, so the last of them is never above 1.
results: A list with one record, holding the eigenvalues, the reference eigenvalues, and the retained solution used for printing and plotting.
settings: A list with the settings used.

Details

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

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

Examples

efa_ekc(test_models$baseline$cormat, N = 500)
#> ── Empirical Kaiser Criterion ──────────────────────────────────────────────────
#> 
#> • Braeken & van Assen (2017): 3