Kaiser-Guttman criterion

Retains as many factors as there are eigenvalues above a cutoff. For the correlation matrix ("PCA"), the cutoff is 1: the classic eigenvalues-greater-than-one rule (Kaiser, 1960). For the reduced correlation matrix, with communality estimates in the diagonal ("SMC" and "EFA"), the cutoff is 0 (Guttman, 1954).

Usage

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

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. https://doi.org/10.1007/BF02289162

Kaiser, H. F. (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20, 141–151. https://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. 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.

eigen_type

character. On what the eigenvalues should be found. Can be either "PCA", "SMC", or "EFA", or some combination of them. If using "PCA", the diagonal values of the correlation matrices are left to be 1. If using "SMC", the diagonal of the correlation matrices is replaced by the squared multiple correlations (SMCs) of the indicators. If using "EFA", eigenvalues are found on the correlation matrices with the final communalities of an exploratory factor analysis solution (default is principal axis factoring extracting 1 factor) as diagonal. Default is c("PCA", "SMC", "EFA"), i.e. all three; "EFA" is the only one that fits a model.

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".

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() fit that provides the communalities when "EFA" is included in eigen_type. NULL (default) uses the efa_fit() defaults. The fit is unrotated, so no rotation settings apply.

...

Additional arguments passed to efa_fit(). For example, estimator, to change the estimator (PAF is default). The estimation tuning knobs are not passed here; they live in estimate_control, and the standard-error arguments (se, b_boot, ci, seed) are not accepted because the fit is an internal step that keeps only its communalities.

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").

results

A list with one record per eigenvalue type, each holding the eigenvalues and the retained solution used for printing and plotting.

settings

A list of the settings used.

Details

The two cutoffs apply to different matrices. The eigenvalues of the correlation matrix sum to the number of variables, and an eigenvalue greater than 1 identifies a component that explains more variance than one standardized variable. The eigenvalues of the reduced correlation matrix sum to the sum of the communalities, so the cutoff 1 does not apply to them. With SMCs in the diagonal, the number of positive eigenvalues is a lower bound to the number of common factors in the population. This bound is the strongest of the three lower bounds of Guttman (1954). "EFA" applies the same cutoff to the final communalities of an EFA solution.

Both rules are often inaccurate in samples. Because of sampling error, the reduced correlation matrix of a sample usually has many small positive eigenvalues, so the cutoff 0 tends to suggest too many factors. The classic rule also tends to suggest too many factors (e.g., Zwick & Velicer, 1986; see Auerswald & Moshagen, 2019, for a comparison with other criteria). Which criteria are informative depends on the data at hand, so rather than substituting one for another, run several of them together and compare their suggestions.

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

See also

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

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

Examples

efa_kgc(test_models$baseline$cormat, eigen_type = c("PCA", "SMC"))
#> ── Kaiser-Guttman criterion ────────────────────────────────────────────────────
#> Eigenvalues found using PCA and SMC; cutoff 1 (PCA), 0 (SMC).
#> 
#> • PCA: 3
#> • SMC: 8