efa_schmid_leiman: Schmid-Leiman transformation
In EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools
View source: R/efa_schmid_leiman.R
efa_schmid_leiman R Documentation
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_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,
...
)
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.
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.
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.
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
See Also
Other factor rotation:
efa_procrustes()
Other reliability coefficients:
efa_reliability(),
print.efa_reliability()
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"))
## 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"))
## 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")
}
EFAtools documentation built on Aug. 21, 2026, 5:16 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
View source: R/efa_schmid_leiman.R
| efa_schmid_leiman | R Documentation |
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_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,
...
)
Arguments
x |
object of class |
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 |
estimator |
character. One of "PAF", "ML", or "ULS" to use
principal axis factoring, maximum likelihood, or unweighted least squares,
respectively, used in |
g_name |
character. The name of the general factor. This needs only be
specified if |
estimate_control |
an |
... |
Arguments to be passed to |
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.
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); |
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
See Also
Other factor rotation:
efa_procrustes()
Other reliability coefficients:
efa_reliability(),
print.efa_reliability()
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"))
## 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"))
## 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")
}
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.