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. The group factors of the returned solution are sorted and relabelled, so their column order can differ from the input solution's (see Details).

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).
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, and the standard-error arguments (se, b_boot, ci, seed) are not accepted because the second-order fit is an internal step run against a placeholder sample size and only its loadings are kept.

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()).

Where the first-order factors come from a loading matrix – an efa_fit() or a psych::fa() solution, or a pattern matrix supplied with Phi – they are sorted by the number in their column labels, so that "F10" follows "F2" rather than "F1". The sort needs a number in every column label; columns that carry no labels, or a label without a number, keep the order they arrive in. A second-order lavaan solution is not sorted at all: its first-order factors keep the order the model declares them in.

The columns are then labelled "F1" to "Fk" by position, on every route, and the input solution's own factor names are not carried over. A factor a lavaan model calls "F3" can therefore come back as "F1". Where the sort applies it is independent of how the input orders its factors, so the group factors of the returned sl matrix can also be in a different order from the columns they came from. A psych::fa() solution shows this most readily: it orders its columns by their sums of squared loadings, but keeps each factor's own number in its label, so those numbers arrive out of order. A solution whose columns are "PA2", "PA3", "PA1" comes back with those same three factors sorted as PA1, PA2, PA3 and labelled "F1", "F2", "F3". The first group factor of the result is then the third column of the input. The same holds against psych::schmid(), whose columns keep the input order: the two solutions agree column for column only after one of them is permuted to the other's order. An efa_fit() solution already labels its factors "F1" to "Fk" in that order, so nothing moves for one; the reordering shows itself for a psych::fa() solution, and for a pattern matrix supplied with labels of its own.

Read the group factors from the returned matrix, therefore, rather than from the input. An indicator-to-factor map is matched to the group factors by position, so one built in the input solution's column order lines up only where the columns did not move; where they did, efa_reliability() or OMEGA() scores each composite against the wrong factor. Build such a map from the "F1" to "Fk" columns of the returned sl matrix instead, which is right on every route. The fac_names of efa_reliability() are matched by position in the same way, so names given in the input solution's order label the wrong subscales, and do so without any sign.

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
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")

}
# }