HULL {EFAtools}

Hull method for determining the number of factors to retain

Description

Implementation of the Hull method suggested by Lorenzo-Seva, Timmerman, and Kiers (2011), with an extension to principal axis factoring. See details for parallelization.

Usage

HULL(
  x,
  N = NA,
  n_fac_theor = NA,
  method = c("PAF", "ULS", "ML"),
  gof = c("CAF", "CFI", "RMSEA"),
  eigen_type = c("SMC", "PCA", "EFA"),
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall"),
  n_datasets = 1000,
  percent = 95,
  decision_rule = c("means", "percentile", "crawford"),
  n_factors = 1,
  ...
)

Arguments

x

matrix or data.frame. Dataframe or matrix of raw data or matrix with correlations.

N

numeric. Number of cases in the data. This is passed to PARALLEL. Only has to be specified if x is a correlation matrix, otherwise it is determined based on the dimensions of x.

n_fac_theor

numeric. Theoretical number of factors to retain. The maximum of this number and the number of factors suggested by PARALLEL plus one will be used in the Hull method.

method

character. The estimation method to use. One of "PAF", "ULS", or "ML", for principal axis factoring, unweighted least squares, and maximum likelihood, respectively.

gof

character. The goodness of fit index to use. Either "CAF", "CFI", or "RMSEA", or any combination of them. If method = "PAF" is used, 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

character. On what the eigenvalues should be found in the parallel analysis. Can be one of "SMC", "PCA", or "EFA". If using "SMC" (default), the diagonal of the correlation matrices 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. This is passed to PARALLEL.

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".

n_datasets

numeric. The number of datasets to simulate. Default is 1000. This is passed to PARALLEL.

percent

numeric. A vector of percentiles to take the simulated eigenvalues from. Default is 95. This is passed to PARALLEL.

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). This is passed to PARALLEL.

n_factors

numeric. Number of factors to extract if "EFA" is included in eigen_type. Default is 1. This is passed to PARALLEL.

...

Further arguments passed to EFA, also in PARALLEL.

Details

The Hull method aims to find a model with an optimal balance between model fit and number of parameters. That is, it aims to retrieve only major factors (Lorenzo-Seva, Timmerman, & Kiers, 2011). To this end, it performs the following steps (Lorenzo-Seva, Timmerman, & Kiers, 2011, p.351):

1. It performs parallel analysis and adds one to the identified number of factors (this number is denoted J). J is taken as an upper bound of the number of factors to retain in the hull method. Alternatively, a theoretical number of factors can be entered. In this case J will be set to whichever of these two numbers (from parallel analysis or based on theory) is higher.

2. For all 0 to J factors, the goodness-of-fit (one of CAF, RMSEA, or CFI) and the degrees of freedom (df) are computed.

3. The solutions are ordered according to their df.

4. Solutions that are not on the boundary of the convex hull are eliminated (see Lorenzo-Seva, Timmerman, & Kiers, 2011, for details).

5. All the triplets of adjacent solutions are considered consecutively. The middle solution is excluded if its point is below or on the line connecting its neighbors in a plot of the goodness-of-fit versus the degrees of freedom.

6. Step 5 is repeated until no solution can be excluded.

7. The st values of the “hull” solutions are determined.

8. The solution with the highest st value is selected.

The PARALLEL function and the principal axis factoring of the different number of factors can be parallelized using the future framework, by calling the future::plan function. The examples provide example code on how to enable parallel processing.

Note that if gof = "RMSEA" is used, 1 - RMSEA is actually used to compare the different solutions. Thus, the threshold of .05 is then .95. This is necessary due to how the heuristic to locate the elbow of the hull works.

The ML estimation method uses the stats::factanal starting values. See also the EFA documentation.

The HULL function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class HULL containing the following objects

n_fac_CAF

The number of factors to retain according to the Hull method with the CAF.

n_fac_CFI

The number of factors to retain according to the Hull method with the CFI.

n_fac_RMSEA

The number of factors to retain according to the Hull method with the RMSEA.

solutions_CAF

A matrix containing the CAFs, degrees of freedom, and for the factors lying on the hull, the st values of the hull solution (see Lorenzo-Seva, Timmerman, and Kiers 2011 for details).

solutions_CFI

A matrix containing the CFIs, degrees of freedom, and for the factors lying on the hull, the st values of the hull solution (see Lorenzo-Seva, Timmerman, and Kiers 2011 for details).

solutions_RMSEA

A matrix containing the RMSEAs, degrees of freedom, and for the factors lying on the hull, the st values of the hull solution (see Lorenzo-Seva, Timmerman, and Kiers 2011 for details).

n_fac_max

The upper bound J of the number of factors to extract (see details).

settings

A list of the settings used.

Source

Lorenzo-Seva, U., Timmerman, M. E., & Kiers, H. A. (2011). The Hull method for selecting the number of common factors. Multivariate Behavioral Research, 46(2), 340-364.

See Also

Other factor retention criteria: CD, EKC, KGC, PARALLEL, SMT

N_FACTORS as a wrapper function for this and all the above-mentioned factor retention criteria.

Examples

# using PAF (this will throw a warning if gof is not specified manually
# and CAF will be used automatically)
HULL(test_models$baseline$cormat, N = 500, gof = "CAF")

# using ML with all available fit indices (CAF, CFI, and RMSEA)
HULL(test_models$baseline$cormat, N = 500, method = "ML")

# using ULS with only RMSEA
HULL(test_models$baseline$cormat, N = 500, method = "ULS", gof = "RMSEA")


## Not run: 
# using parallel processing (Note: plans can be adapted, see the future
# package for details)
future::plan(future::multisession)
HULL(test_models$baseline$cormat, N = 500, gof = "CAF")

## End(Not run)