Schmid-Leiman transformation

This function implements the Schmid-Leiman (SL) transformation (Schmid & Leiman, 1957). It takes the pattern coefficients and factor intercorrelations from an oblique factor solution as input and can reproduce the results from psych::schmid() and from the SPSS implementation from Wolff & Preising (2005). Other arguments from efa_fit() can be used to control the procedure to find the second-order loadings more flexibly. The function can also be used on a second-order confirmatory factor analysis (CFA) solution from lavaan.

Usage

efa_schmid_leiman(
  x,
  Phi = NULL,
  estimator = c("PAF", "ML", "ULS", "MINRES"),
  g_name = "g",
  estimate_control = NULL,
  ...
)

Source

Schmid, J. & Leiman, J. M. (1957). The development of hierarchical factor solutions. Psychometrika, 22(1), 53–61. doi:10.1007/BF02289209

Wolff, H.-G., & Preising, K. (2005). Exploring item and higher order factor structure with the Schmid-Leiman solution: Syntax codes for SPSS and SAS. Behavior Research Methods, 37 , 48–58. doi:10.3758/BF03206397

Arguments

x

object of class efa_fit(), class psych::fa(), class lavaan::lavaan(), a matrix, or an efa_loadings/loadings object. If class efa_fit() or class psych::fa(), pattern coefficients and factor intercorrelations are taken from this object. If class lavaan::lavaan(), it must be a second-order CFA solution. In this case first-order and second-order factor loadings are taken from this object and the g_name argument has to be specified. x can also be a pattern matrix from an oblique factor solution (see Phi).

Phi

matrix. A matrix of factor intercorrelations from an oblique factor solution. Only needs to be specified if a pattern matrix is entered directly into x.

estimator

character. One of "PAF", "ML", or "ULS" to use principal axis factoring, maximum likelihood, or unweighted least squares, respectively, used in efa_fit() to find the second-order loadings. "MINRES" is accepted as a synonym for "ULS" (the same estimator). The value is matched case-insensitively.

g_name

character. The name of the general factor. This needs only be specified if x is a lavaan second-order solution. Default is "g".

estimate_control

an estimate_control() object with the estimation settings for the second-order efa_fit() fit, including the type preset. NULL (default) uses the efa_fit() defaults. The second-order fit is unrotated, so no rotation settings apply.

...

Arguments to be passed to efa_fit(). The estimation tuning knobs are not passed here; they live in estimate_control.

Value

A list of class c("efa_schmid_leiman", "SL") containing the following

orig_R

Original correlation matrix.

sl

A matrix with general factor loadings, group factor loadings, communalities, and uniquenesses.

L2

Second-order factor loadings.

vars_accounted

A matrix of explained variances and sums of squared loadings.

iter

The number of iterations needed for convergence in EFA.

convergence

Integer convergence code of the second-order EFA (0 = converged); NA for a lavaan input. See efa_fit().

settings

list. The settings (arguments) used in EFA to get the second-order loadings.

Details

The SL transformation (also called SL orthogonalization) is a procedure with which an oblique factor solution is transformed into a hierarchical, orthogonalized solution. As a first step, the factor intercorrelations are factor analyzed to extract a single second-order (general) factor, yielding a two-level hierarchical structure. The first-order factor and the second-order factor are then orthogonalized, resulting in an orthogonalized factor solution with proportionality constraints. The procedure thus makes a suggested hierarchical data structure based on factor intercorrelations explicit. One major advantage of SL transformation is that it enables variance partitioning between higher-order and first-order factors, including the calculation of McDonald's omegas (see efa_reliability()).

See also

Other factor rotation: efa_procrustes()

Examples

## Use with an output from the EFAtools::efa_fit function, both with type EFAtools
EFA_mod <- efa_fit(test_models$baseline$cormat, N = 500, n_factors = 3,
                   estimator = "PAF", rotation = "promax")
SL_EFAtools <- efa_schmid_leiman(EFA_mod, estimator = "PAF",
                                 estimate_control = estimate_control(type = "EFAtools"))

# \donttest{
## Use with an output from the psych::fa function with type psych
fa_mod <- psych::fa(test_models$baseline$cormat, nfactors = 3, n.obs = 500,
                    fm = "pa", rotate = "Promax")
SL_psych <- efa_schmid_leiman(fa_mod, estimator = "PAF",
                              estimate_control = estimate_control(type = "psych"))
# }

## Use more flexibly by entering a pattern matrix and phi directly (useful if
## a factor solution found with another program should be subjected to SL
## transformation)

## For demonstration, take pattern matrix and phi from an EFA output
## This gives the same solution as the first example
EFA_mod <- efa_fit(test_models$baseline$cormat, N = 500, n_factors = 3,
                   estimator = "PAF", rotation = "promax")
SL_flex <- efa_schmid_leiman(EFA_mod$rot_loadings, Phi = EFA_mod$Phi, estimator = "PAF",
                             estimate_control = estimate_control(type = "EFAtools"))

# \donttest{
## Use with a lavaan second-order CFA output
if (requireNamespace("lavaan", quietly = TRUE)) {

# Create and fit model in lavaan (assume all variables have SDs of 1)
mod <- 'F1 =~ V1 + V2 + V3 + V4 + V5 + V6
        F2 =~ V7 + V8 + V9 + V10 + V11 + V12
        F3 =~ V13 + V14 + V15 + V16 + V17 + V18
        g =~ F1 + F2 + F3'
fit <- lavaan::cfa(mod, sample.cov = test_models$baseline$cormat,
                   sample.nobs = 500, estimator = "ml")

SL_lav <- efa_schmid_leiman(fit, g_name = "g")

}
# }