Various factor retention criteria

Choosing the number of factors to retain is one of the most important decisions in an exploratory factor analysis (EFA). Many criteria exist to help with this choice. This function runs several of them together, and can also check whether the data are suitable for factor analysis.

Usage

efa_retain(
  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", "poly", "tetra"),
  n_factors_max = NA,
  N_pop = 10000,
  N_samples = 500,
  alpha = 0.3,
  ...,
  max_iter_CD = 50,
  n_fac_theor = NA,
  estimator = 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 = lifecycle::deprecated(),
  n_datasets_nest = 1000,
  alpha_nest = 0.05,
  show_progress = FALSE,
  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

Arguments

x: data.frame or matrix. Raw data, or a correlation matrix. If "CD" is included as a criterion, x must be raw data.
criteria: character. Which factor retention methods to run: one or more of "CD", "EKC", "HULL", "KGC", "MAP", "NEST", "PARALLEL", "SCREE", and "SMT" (see details). The default runs a subset of commonly used, well-performing methods, listed in the details.
suitability: logical. Whether the data should be checked for suitability for factor analysis using Bartlett's test of sphericity and the Kaiser-Meyer-Olkin criterion (see details). Default is TRUE.
N: numeric. The number of observations. Only needed if x is a correlation matrix.
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 (a two-step estimator). CD, PARALLEL, NEST, HULL, and SMT do not support "poly" / "tetra" and are skipped automatically if you request them together. Default is "pearson".
n_factors_max: numeric. Passed to efa_cd(). The maximum number of factors to test against. Larger numbers will increase the duration the procedure takes, but test more possible solutions. If left NA (default), the maximum number of factors for which the model is still over-identified (df > 0) is used.
N_pop: numeric. Passed to efa_cd(). Size of finite populations of comparison data. Default is 10000.
N_samples: numeric. Passed to efa_cd(). Number of samples drawn from each population. Default is 500.
alpha: numeric. Passed to efa_cd(). The alpha level used to test the significance of the improvement added by an additional factor. Default is .30.
...: Further arguments passed to efa_fit() in efa_parallel(), efa_kgc(), efa_scree(), efa_nest(), and (through its parallel analysis and its own candidate fits) efa_hull(). An argument that efa_fit() does not recognize causes an error. The estimation tuning knobs are not passed here; they live in estimate_control. The standard-error arguments (se, b_boot, ci, seed) are not accepted, because the criterion fits are internal steps whose standard errors are not reported. Arguments listed after ... must be given by their full name (R matches an abbreviated name only against the arguments before ...), so a tuning knob such as max_iter cannot be mistaken for max_iter_CD.
max_iter_CD: numeric. Passed to efa_cd(). The maximum number of iterations to perform after which the iterative PAF procedure is halted. Default is 50.
n_fac_theor: numeric. Passed to efa_hull(). Theoretical number of factors to retain. The Hull method uses one plus the larger of this number and the number of factors suggested by efa_parallel() as its upper bound.
estimator: character. Passed to efa_fit() in efa_hull(), efa_kgc(), efa_scree(), efa_parallel(), and efa_nest(). The estimator to use. One of "PAF", "ULS", or "ML", for principal axis factoring, unweighted least squares, and maximum likelihood, respectively. The default here is "ML". Some criteria default to something else when called on their own (for example, efa_hull() defaults to "PAF"), so results from efa_retain() can differ from calling that criterion directly unless you set estimator to match. In efa_kgc(), efa_scree(), and efa_parallel() it only takes effect when the respective eigen_type includes "EFA".
gof: character. Passed to efa_hull(). The goodness of fit index to use. Either "CAF", "CFI", or "RMSEA", or any combination of them. With the "PAF" estimator, only the CAF can be used as goodness of fit index. For details on the CAF, see Lorenzo-Seva, Timmerman, and Kiers (2011).
eigen_type_HULL: character. Passed to efa_parallel() in efa_hull(). What the eigenvalues in the parallel analysis are based on. One of "SMC", "PCA", or "EFA" – different ways of estimating how much variance each indicator shares with the others before the eigenvalues are computed. "SMC" (default) uses each indicator's squared multiple correlation with the others (its diagonal value in the correlation matrix). "PCA" leaves the diagonal at 1, so each indicator's total variance – not just the shared part – feeds into the eigenvalues. "EFA" uses the communalities from a fitted EFA solution instead.
eigen_type_other: character. Passed to efa_kgc(), efa_scree(), and efa_parallel(). The same as eigen_type_HULL, but multiple inputs are possible here (any combination of "PCA", "SMC", and "EFA"). Default is "SMC".
n_factors: numeric. Passed to efa_parallel() (also within efa_hull()), efa_kgc(), and efa_scree(). Number of factors to extract if "EFA" is included in eigen_type_HULL or eigen_type_other. Default is 1.
n_datasets: numeric. Passed to efa_parallel() (also within efa_hull()). The number of datasets to simulate. Default is 1000.
percent: numeric. Passed to efa_parallel() (also within efa_hull()). The percentile to take from the simulated eigenvalues. Default is 95.
decision_rule: character. Passed to efa_parallel() (also within efa_hull()). Which rule to use to determine the number of factors to retain. Default is "means", which uses 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).
ekc_type: Accepted and ignored. It used to select between two ways to compute the efa_ekc() reference values. The "AM2019" reference values do not depend on the observed eigenvalues. They therefore skip the empirical correction that defines the criterion, so they are no longer computed.
n_datasets_nest: numeric. Passed to efa_nest(). The number of datasets to simulate. Default is 1000.
alpha_nest: numeric. Passed to efa_nest(). The alpha level to use. The reference values are the eigenvalues at the (1 - alpha_nest) percentile. Default is .05.
show_progress: logical. Whether a progress bar should be shown in the console. Default is FALSE.
estimate_control: an estimate_control() object with the estimation settings for the efa_fit() fits run by the criteria that fit a model (efa_hull(), efa_kgc(), efa_scree(), efa_parallel(), efa_nest(), and efa_smt()). NULL (default) uses the efa_fit() defaults. It only applies to criteria that fit a model, and only to the parts of that fit each criterion actually runs. efa_cd(), efa_ekc(), and efa_map() fit no model. efa_kgc(), efa_scree(), and efa_parallel() only fit one when their eigen_type includes "EFA". efa_smt() fits with maximum likelihood by definition, so only start_method takes effect there. All fits are unrotated, so no rotation settings apply.

Value

A list of class c("efa_retain", "N_FACTORS"), the trailing class keeping inherits(x, "N_FACTORS") working for code written against the superseded name. It contains

suitability: A list with the results from efa_bartlett() and efa_kmo() (bartlett and kmo), or NULL if suitability = FALSE.
outputs: A named list with one efa_retention object per factor retention criterion that was run (see, e.g., efa_ekc()).
n_factors: A named numeric vector with the suggested number of factors per criterion and, where a criterion has several variants, per variant (e.g. EKC_BvA2017 or PARALLEL_SMC). Criteria without a numeric suggestion (the scree plot) are not included. The printed summary's "most common" value is based on each criterion's own most frequent (modal) suggestion among its variants, not a plain tally of this vector, so counting entries here by hand can give a different answer.
not_run: A named character vector with the criteria that were skipped or failed and the reason, or NULL if all requested criteria ran.
settings: A list of the settings used. Its criteria element records the requested criteria, in the order they were given, while outputs and n_factors are in the order in which the criteria were run. gof records the requested Hull goodness-of-fit indices, and gof_used records the ones the Hull method actually computed (it reduces them to "CAF" for the PAF estimator). gof_used is NA when HULL was not requested, was skipped, or failed.

Details

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

Bartlett's test of sphericity (see efa_bartlett())
Kaiser-Meyer-Olkin criterion (see efa_kmo())

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

Comparison data (see efa_cd())
Empirical Kaiser criterion (see efa_ekc())
Hull method (see efa_hull())
Kaiser-Guttman criterion (see efa_kgc())
Velicer's minimum average partial, MAP (see efa_map())
Next Eigenvalue Sufficiency Test, NEST (see efa_nest())
Parallel analysis (see efa_parallel())
Scree plot (see efa_scree())
Sequential chi-square model tests, RMSEA lower bound, and AIC (see efa_smt())

The default criteria are comparison data, the empirical Kaiser criterion, the Hull method, MAP, NEST, and parallel analysis. No single criterion is the most accurate in all conditions. efa_retain() therefore runs several criteria together, and the printed summary gives the range of their suggestions and the most common one. Auerswald and Moshagen (2019) compare the criteria and give guidance on the selection.

The comparison data, parallel analysis, and NEST criteria compare the data against simulated reference data, so their suggested numbers of factors vary slightly from run to run. The Hull method also varies, because it calls efa_parallel() to set its upper bound. Call base::set.seed() before efa_retain() to make the results reproducible.

Examples

# \donttest{
# Default criteria, with correlation matrix and estimator "ML" (where needed)
# This will throw a warning for CD, as no raw data were specified
# The simulation-based criteria are seeded to make the run reproducible
set.seed(42)
nfac_all <- efa_retain(test_models$baseline$cormat, N = 500, estimator = "ML",
                       n_datasets = 100, n_datasets_nest = 100)
#> Warning: `x` is a correlation matrix, but "CD" needs raw data.
#> ℹ Skipping "CD".

# The same as above, but without "CD"
nfac_wo_CD <- efa_retain(test_models$baseline$cormat, criteria = c("EKC",
                         "HULL", "PARALLEL", "NEST"), N = 500,
                         estimator = "ML", n_datasets = 100,
                         n_datasets_nest = 100)

# Use PAF instead of ML (this will take longer). PAF only supports "CAF" as
# gof for the Hull method, so set it explicitly to avoid the automatic message.
nfac_PAF <- efa_retain(test_models$baseline$cormat, criteria = c("EKC",
                       "HULL", "PARALLEL", "NEST"), N = 500,
                       estimator = "PAF", gof = "CAF", n_datasets = 100,
                       n_datasets_nest = 100)

# Back to the default ML estimator (unlike above), with only "PCA" type eigenvalues
nfac_PCA <- efa_retain(test_models$baseline$cormat, criteria = c("EKC",
                       "HULL", "PARALLEL", "NEST"), N = 500,
                       estimator = "ML", eigen_type_other = "PCA",
                       n_datasets = 100, n_datasets_nest = 100)

# Use raw data, such that CD can also be performed
nfac_raw <- efa_retain(GRiPS_raw, estimator = "ML", N_pop = 500,
                        N_samples = 20, n_datasets = 100,
                        n_datasets_nest = 100)
#> Warning: The suggested maximum number of factors was 2, but the Hull method needs at
#> least 3.
#> ℹ Setting it to 3.
# }