SL: Schmid-Leiman Transformation

In EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools

Schmid-Leiman Transformation

Description

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 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

SL(
  x,
  Phi = NULL,
  type = c("EFAtools", "psych", "SPSS", "none"),
  method = c("PAF", "ML", "ULS"),
  g_name = "g",
  ...
)

Arguments

x	object of class EFA, class psych::fa, class lavaan or matrix. If class EFA or class psych::fa, pattern coefficients and factor intercorrelations are taken from this object. If class 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) or a matrix of first-order factor loadings from a higher-order confirmatory factor analysis (see L2).
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.
type	character. One of "EFAtools" (default), "psych", "SPSS", or "none". This is used to control the procedure of the second-order factor analysis. See EFA for details.
method	character. One of "PAF", "ML", or "ULS" to use principal axis factoring, maximum likelihood, or unweighted least squares (also called minres), respectively, used in EFA to find the second-order loadings.
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".
...	Arguments to be passed to EFA.

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 again factor analyzed to find second-order factor loadings. If there is only one higher-order factor, this step of the procedure stops there, resulting in a second-order factor 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 OMEGA).

Value

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

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

Examples

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


## Use with an output from the psych::fa function with type psych in SL
fa_mod <- psych::fa(test_models$baseline$cormat, nfactors = 3, n.obs = 500,
                    fm = "pa", rotate = "Promax")
SL_psych <- SL(fa_mod, type = "psych", method = "PAF")


## 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(test_models$baseline$cormat, N = 500, n_factors = 3,
               type = "EFAtools", method = "PAF", rotation = "promax")
SL_flex <- SL(EFA_mod$rot_loadings, Phi = EFA_mod$Phi, type = "EFAtools",
              method = "PAF")


## Use with a lavaan second-order CFA output

# 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 <- SL(fit, g_name = "g")

EFAtools documentation built on Jan. 6, 2023, 5:16 p.m.