measEq.syntax: Syntax for measurement equivalence
In semTools: Useful Tools for Structural Equation Modeling
measEq.syntax R Documentation
Syntax for measurement equivalence
Description
Automatically generates lavaan model syntax to specify a confirmatory
factor analysis (CFA) model with equality constraints imposed on
user-specified measurement (or structural) parameters. Optionally returns
the fitted model (if data are provided) representing some chosen level of
measurement equivalence/invariance across groups and/or repeated measures.
Usage
measEq.syntax(configural.model, ..., ID.fac = "std.lv",
ID.cat = "Wu.Estabrook.2016", ID.thr = c(1L, 2L), group = NULL,
group.equal = "", group.partial = "", longFacNames = list(),
longIndNames = list(), long.equal = "", long.partial = "",
auto = "all", warn = TRUE, debug = FALSE, return.fit = FALSE)
Arguments
configural.model
A model with no measurement-invariance constraints
(i.e., representing only configural invariance), unless required for model
identification. configural.model can be either:
lavaan model.syntax or a parameter table
(as returned by parTable) specifying the
configural model. Using this option, the user can also provide
either raw data or summary statistics via sample.cov
and (optionally) sample.mean. See argument descriptions in
lavaan. In order to include thresholds in
the generated syntax, either users must provide raw data,
or the configural.model syntax must specify all thresholds
(see first example). If raw data are not provided, the
number of blocks (groups, levels, or combination) must be
indicated using an arbitrary sample.nobs argument (e.g.,
3 groups could be specified using sample.nobs=rep(1, 3)).
a fitted lavaan model (e.g., as returned by
cfa) estimating the configural model
Note that the specified or fitted model must not contain any latent
structural parameters (i.e., it must be a CFA model), unless they are
higher-order constructs with latent indicators (i.e., a second-order CFA).
...
Additional arguments (e.g., data, ordered, or
parameterization) passed to the lavaan
function. See also lavOptions.
ID.fac
character. The method for identifying common-factor
variances and (if meanstructure = TRUE) means. Three methods are
available, which go by different names in the literature:
Standardize the common factor (mean = 0, SD = 1) by
specifying any of: "std.lv", "unit.variance",
"UV", "fixed.factor",
"fixed-factor"
Choose a reference indicator by specifying any of:
"auto.fix.first", "unit.loading", "UL",
"marker", "ref", "ref.indicator",
"reference.indicator", "reference-indicator",
"marker.variable", "marker-variable"
Apply effects-code constraints to loadings and intercepts by
specifying any of: "FX", "EC", "effects",
"effects.coding", "effects-coding",
"effects.code", "effects-code"
See Kloessner & Klopp (2019) for details about all three methods.
ID.cat
character. The method for identifying (residual)
variances and intercepts of latent item-responses underlying any
ordered indicators. Four methods are available:
To follow Wu & Estabrook's (2016) guidelines (default), specify
any of: "Wu.Estabrook.2016", "Wu.2016",
"Wu.Estabrook", "Wu", "Wu2016". For
consistency, specify ID.fac = "std.lv".
To use the default settings of Mplus and lavaan,
specify any of: "default", "Mplus", "Muthen".
Details provided in Millsap & Tein (2004).
To use the constraints recommended by Millsap & Tein (2004; see
also Liu et al., 2017, for the longitudinal case)
specify any of: "millsap", "millsap.2004",
"millsap.tein.2004". For consistency, specify
ID.fac = "marker" and parameterization = "theta".
To use the default settings of LISREL, specify "LISREL"
or "Joreskog". Details provided in Millsap & Tein (2004).
For consistency, specify parameterization = "theta".
See Details and References for more information.
ID.thr
integer. Only relevant when
ID.cat = "Millsap.Tein.2004". Used to indicate which thresholds
should be constrained for identification. The first integer indicates the
threshold used for all indicators, the second integer indicates the
additional threshold constrained for a reference indicator (ignored if
binary).
group
optional character indicating the name of a grouping
variable. See cfa.
group.equal
optional character vector indicating type(s) of
parameter to equate across groups. Ignored if is.null(group).
See lavOptions.
group.partial
optional character vector or a parameter table
indicating exceptions to group.equal (see
lavOptions). Any variables not appearing in the
configural.model will be ignored, and any parameter constraints
needed for identification (e.g., two thresholds per indicator when
ID.cat = "Millsap") will be removed.
longFacNames
optional named list of character vectors,
each indicating multiple factors in the model that are actually the same
construct measured repeatedly. See Details and Examples.
longIndNames
optional named list of character vectors,
each indicating multiple indicators in the model that are actually the
same indicator measured repeatedly. See Details and
Examples.
long.equal
optional character vector indicating type(s) of
parameter to equate across repeated measures. Ignored if no factors are
indicated as repeatedly measured in longFacNames.
long.partial
optional character vector or a parameter table
indicating exceptions to long.equal. Any longitudinal variable
names not appearing in names(longFacNames) or
names(longIndNames) will be ignored, and any parameter constraints
needed for identification will be removed.
auto
Used to automatically included autocorrelated measurement errors
among repeatedly measured indicators in longIndNames. Specify a
single integer to set the maximum order (e.g., auto = 1L
indicates that an indicator's unique factors should only be correlated
between adjacently measured occasions). auto = TRUE or "all"
will specify residual covariances among all possible lags per repeatedly
measured indicator in longIndNames.
warn, debug
logical. Passed to lavaan
and lavParseModelString.
See lavOptions.
return.fit
logical indicating whether the generated syntax
should be fitted to the provided data (or summary statistics, if
provided via sample.cov). If configural.model is a fitted
lavaan model, the generated syntax will be fitted using the update
method (see lavaan), and ... will be passed to
lavaan. If neither data nor a fitted lavaan model
were provided, this must be FALSE. If TRUE, the generated
measEq.syntax object will be included in the lavaan object's
@external slot, accessible by fit@external$measEq.syntax.
Details
This function is a pedagogical and analytical tool to generate model syntax
representing some level of measurement equivalence/invariance across any
combination of multiple groups and/or repeated measures. Support is provided
for confirmatory factor analysis (CFA) models with simple or complex
structure (i.e., cross-loadings and correlated residuals are allowed).
For any complexities that exceed the limits of automation, this function is
intended to still be useful by providing a means to generate syntax that
users can easily edit to accommodate their unique situations.
Limited support is provided for bifactor models and higher-order constructs.
Because bifactor models have cross-loadings by definition, the option
ID.fac = "effects.code" is unavailable. ID.fac = "UV" is
recommended for bifactor models, but ID.fac = "UL" is available on
the condition that each factor has a unique first indicator in the
configural.model. In order to maintain generality, higher-order
factors may include a mix of manifest and latent indicators, but they must
therefore require ID.fac = "UL" to avoid complications with
differentiating lower-order vs. higher-order (or mixed-level) factors.
The keyword "loadings" in group.equal or long.equal
constrains factor loadings of all manifest indicators (including loadings on
higher-order factors that also have latent indicators), whereas the keyword
"regressions" constrains factor loadings of latent indicators. Users
can edit the model syntax manually to adjust constraints as necessary, or
clever use of the group.partial or long.partial arguments
could make it possible for users to still automated their model syntax.
The keyword "intercepts" constrains the intercepts of all manifest
indicators, and the keyword "means" constrains intercepts and means
of all latent common factors, regardless of whether they are latent
indicators of higher-order factors. To test equivalence of lower-order and
higher-order intercepts/means in separate steps, the user can either
manually edit their generated syntax or conscientiously exploit the
group.partial or long.partial arguments as necessary.
ID.fac: If the configural.model fixes any (e.g.,
the first) factor loadings, the generated syntax object will retain those
fixed values. This allows the user to retain additional constraints that
might be necessary (e.g., if there are only 1 or 2 indicators). Some methods
must be used in conjunction with other settings:
-
ID.cat = "Millsap" requires ID.fac = "UL" and
parameterization = "theta".
-
ID.cat = "LISREL" requires parameterization = "theta".
-
ID.fac = "effects.code" is unavailable when there are any
cross-loadings.
ID.cat: Wu & Estabrook (2016) recommended constraining
thresholds to equality first, and doing so should allow releasing any
identification constraints no longer needed. For each ordered
indicator, constraining one threshold to equality will allow the item's
intercepts to be estimated in all but the first group or repeated measure.
Constraining a second threshold (if applicable) will allow the item's
(residual) variance to be estimated in all but the first group or repeated
measure. For binary data, there is no independent test of threshold,
intercept, or residual-variance equality. Equivalence of thresholds must
also be assumed for three-category indicators. These guidelines provide the
least restrictive assumptions and tests, and are therefore the default.
The default setting in Mplus is similar to Wu & Estabrook (2016),
except that intercepts are always constrained to zero (so they are assumed
to be invariant without testing them). Millsap & Tein (2004) recommended
parameterization = "theta" and identified an item's residual variance
in all but the first group (or occasion; Liu et al., 2017) by constraining
its intercept to zero and one of its thresholds to equality. A second
threshold for the reference indicator (so ID.fac = "UL") is used to
identify the common-factor means in all but the first group/occasion. The
LISREL software fixes the first threshold to zero and (if applicable) the
second threshold to 1, and assumes any remaining thresholds to be equal
across groups / repeated measures; thus, the intercepts are always
identified, and residual variances (parameterization = "theta") are
identified except for binary data, when they are all fixed to one.
Repeated Measures: If each repeatedly measured factor is measured
by the same indicators (specified in the same order in the
configural.model) on each occasion, without any cross-loadings, the
user can let longIndNames be automatically generated. Generic names
for the repeatedly measured indicators are created using the name of the
repeatedly measured factors (i.e., names(longFacNames)) and the
number of indicators. So the repeatedly measured first indicator
("ind") of a longitudinal construct called "factor" would be
generated as "._factor_ind.1".
The same types of parameter can be specified for long.equal as for
group.equal (see lavOptions for a list), except
for "residual.covariances" or "lv.covariances". Instead, users
can constrain autocovariances using keywords "resid.autocov"
or "lv.autocov". Note that group.equal = "lv.covariances" or
group.equal = "residual.covariances" will constrain any
autocovariances across groups, along with any other covariances the user
specified in the configural.model. Note also that autocovariances
cannot be specified as exceptions in long.partial, so anything more
complex than the auto argument automatically provides should instead
be manually specified in the configural.model.
When users set orthogonal=TRUE in the configural.model (e.g.,
in bifactor models of repeatedly measured constructs), autocovariances of
each repeatedly measured factor will still be freely estimated in the
generated syntax.
Missing Data: If users wish to utilize the auxiliary
function to automatically include auxiliary variables in conjunction with
missing = "FIML", they should first generate the hypothesized-model
syntax, then submit that syntax as the model to auxiliary().
If users utilized runMI to fit their configural.model
to multiply imputed data, that model can also be passed to the
configural.model argument, and if return.fit = TRUE, the
generated model will be fitted to the multiple imputations.
Value
By default, an object of class measEq.syntax.
If return.fit = TRUE, a fitted lavaan
model, with the measEq.syntax object stored in the
@external slot, accessible by fit@external$measEq.syntax.
Author(s)
Terrence D. Jorgensen (University of Amsterdam;
TJorgensen314@gmail.com)
References
Kloessner, S., & Klopp, E. (2019). Explaining constraint interaction: How
to interpret estimated model parameters under alternative scaling methods.
Structural Equation Modeling, 26(1), 143–155.
doi: 10.1080/10705511.2018.1517356
Liu, Y., Millsap, R. E., West, S. G., Tein, J.-Y., Tanaka, R., & Grimm,
K. J. (2017). Testing measurement invariance in longitudinal data with
ordered-categorical measures. Psychological Methods, 22(3),
486–506. doi: 10.1037/met0000075
Millsap, R. E., & Tein, J.-Y. (2004). Assessing factorial invariance in
ordered-categorical measures. Multivariate Behavioral Research, 39(3),
479–515. doi: 10.1207/S15327906MBR3903_4
Wu, H., & Estabrook, R. (2016). Identification of confirmatory factor
analysis models of different levels of invariance for ordered categorical
outcomes. Psychometrika, 81(4), 1014–1045.
doi: 10.1007/s11336-016-9506-0
See Also
compareFit
Examples
mod.cat <- ' FU1 =~ u1 + u2 + u3 + u4
FU2 =~ u5 + u6 + u7 + u8 '
## the 2 factors are actually the same factor (FU) measured twice
longFacNames <- list(FU = c("FU1","FU2"))
## CONFIGURAL model: no constraints across groups or repeated measures
syntax.config <- measEq.syntax(configural.model = mod.cat,
# NOTE: data provides info about numbers of
# groups and thresholds
data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", longFacNames = longFacNames)
## print lavaan syntax to the Console
cat(as.character(syntax.config))
## print a summary of model features
summary(syntax.config)
## THRESHOLD invariance:
## only necessary to specify thresholds if you have no data
mod.th <- '
u1 | t1 + t2 + t3 + t4
u2 | t1 + t2 + t3 + t4
u3 | t1 + t2 + t3 + t4
u4 | t1 + t2 + t3 + t4
u5 | t1 + t2 + t3 + t4
u6 | t1 + t2 + t3 + t4
u7 | t1 + t2 + t3 + t4
u8 | t1 + t2 + t3 + t4
'
syntax.thresh <- measEq.syntax(configural.model = c(mod.cat, mod.th),
# NOTE: data not provided, so syntax must
# include thresholds, and number of
# groups == 2 is indicated by:
sample.nobs = c(1, 1),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", group.equal = "thresholds",
longFacNames = longFacNames,
long.equal = "thresholds")
## notice that constraining 4 thresholds allows intercepts and residual
## variances to be freely estimated in all but the first group & occasion
cat(as.character(syntax.thresh))
## print a summary of model features
summary(syntax.thresh)
## Fit a model to the data either in a subsequent step (recommended):
mod.config <- as.character(syntax.config)
fit.config <- cfa(mod.config, data = datCat, group = "g",
ordered = paste0("u", 1:8), parameterization = "theta")
## or in a single step (not generally recommended):
fit.thresh <- measEq.syntax(configural.model = mod.cat, data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", group.equal = "thresholds",
longFacNames = longFacNames,
long.equal = "thresholds", return.fit = TRUE)
## compare their fit to test threshold invariance
anova(fit.config, fit.thresh)
## --------------------------------------------------------
## RECOMMENDED PRACTICE: fit one invariance model at a time
## --------------------------------------------------------
## - A downside of setting return.fit=TRUE is that if the model has trouble
## converging, you don't have the opportunity to investigate the syntax,
## or even to know whether an error resulted from the syntax-generator or
## from lavaan itself.
## - A downside of automatically fitting an entire set of invariance models
## (like the old measurementInvariance() function did) is that you might
## end up testing models that shouldn't even be fitted because less
## restrictive models already fail (e.g., don't test full scalar
## invariance if metric invariance fails! Establish partial metric
## invariance first, then test equivalent of intercepts ONLY among the
## indicators that have invariate loadings.)
## The recommended sequence is to (1) generate and save each syntax object,
## (2) print it to the screen to verify you are fitting the model you expect
## to (and potentially learn which identification constraints should be
## released when equality constraints are imposed), and (3) fit that model
## to the data, as you would if you had written the syntax yourself.
## Continuing from the examples above, after establishing invariance of
## thresholds, we proceed to test equivalence of loadings and intercepts
## (metric and scalar invariance, respectively)
## simultaneously across groups and repeated measures.
## Not run:
## metric invariance
syntax.metric <- measEq.syntax(configural.model = mod.cat, data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", longFacNames = longFacNames,
group.equal = c("thresholds","loadings"),
long.equal = c("thresholds","loadings"))
summary(syntax.metric) # summarize model features
mod.metric <- as.character(syntax.metric) # save as text
cat(mod.metric) # print/view lavaan syntax
## fit model to data
fit.metric <- cfa(mod.metric, data = datCat, group = "g",
ordered = paste0("u", 1:8), parameterization = "theta")
## test equivalence of loadings, given equivalence of thresholds
anova(fit.thresh, fit.metric)
## scalar invariance
syntax.scalar <- measEq.syntax(configural.model = mod.cat, data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", longFacNames = longFacNames,
group.equal = c("thresholds","loadings",
"intercepts"),
long.equal = c("thresholds","loadings",
"intercepts"))
summary(syntax.scalar) # summarize model features
mod.scalar <- as.character(syntax.scalar) # save as text
cat(mod.scalar) # print/view lavaan syntax
## fit model to data
fit.scalar <- cfa(mod.scalar, data = datCat, group = "g",
ordered = paste0("u", 1:8), parameterization = "theta")
## test equivalence of intercepts, given equal thresholds & loadings
anova(fit.metric, fit.scalar)
## For a single table with all results, you can pass the models to
## summarize to the compareFit() function
compareFit(fit.config, fit.thresh, fit.metric, fit.scalar)
## ------------------------------------------------------
## NOT RECOMMENDED: fit several invariance models at once
## ------------------------------------------------------
test.seq <- c("thresholds","loadings","intercepts","means","residuals")
meq.list <- list()
for (i in 0:length(test.seq)) {
if (i == 0L) {
meq.label <- "configural"
group.equal <- ""
long.equal <- ""
} else {
meq.label <- test.seq[i]
group.equal <- test.seq[1:i]
long.equal <- test.seq[1:i]
}
meq.list[[meq.label]] <- measEq.syntax(configural.model = mod.cat,
data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv",
ID.cat = "Wu.Estabrook.2016",
group = "g",
group.equal = group.equal,
longFacNames = longFacNames,
long.equal = long.equal,
return.fit = TRUE)
}
compareFit(meq.list)
## -----------------
## Binary indicators
## -----------------
## borrow example data from Mplus user guide
myData <- read.table("http://www.statmodel.com/usersguide/chap5/ex5.16.dat")
names(myData) <- c("u1","u2","u3","u4","u5","u6","x1","x2","x3","g")
bin.mod <- '
FU1 =~ u1 + u2 + u3
FU2 =~ u4 + u5 + u6
'
## Must SIMULTANEOUSLY constrain thresholds, loadings, and intercepts
test.seq <- list(strong = c("thresholds","loadings","intercepts"),
means = "means",
strict = "residuals")
meq.list <- list()
for (i in 0:length(test.seq)) {
if (i == 0L) {
meq.label <- "configural"
group.equal <- ""
long.equal <- ""
} else {
meq.label <- names(test.seq)[i]
group.equal <- unlist(test.seq[1:i])
# long.equal <- unlist(test.seq[1:i])
}
meq.list[[meq.label]] <- measEq.syntax(configural.model = bin.mod,
data = myData,
ordered = paste0("u", 1:6),
parameterization = "theta",
ID.fac = "std.lv",
ID.cat = "Wu.Estabrook.2016",
group = "g",
group.equal = group.equal,
#longFacNames = longFacNames,
#long.equal = long.equal,
return.fit = TRUE)
}
compareFit(meq.list)
## ---------------------
## Multilevel Invariance
## ---------------------
## To test invariance across levels in a MLSEM, specify syntax as though
## you are fitting to 2 groups instead of 2 levels.
mlsem <- ' f1 =~ y1 + y2 + y3
f2 =~ y4 + y5 + y6 '
## metric invariance
syntax.metric <- measEq.syntax(configural.model = mlsem, meanstructure = TRUE,
ID.fac = "std.lv", sample.nobs = c(1, 1),
group = "cluster", group.equal = "loadings")
## by definition, Level-1 means must be zero, so fix them
syntax.metric <- update(syntax.metric,
change.syntax = paste0("y", 1:6, " ~ c(0, NA)*1"))
## save as a character string
mod.metric <- as.character(syntax.metric, groups.as.blocks = TRUE)
## convert from multigroup to multilevel
mod.metric <- gsub(pattern = "group:", replacement = "level:",
x = mod.metric, fixed = TRUE)
## fit model to data
fit.metric <- lavaan(mod.metric, data = Demo.twolevel, cluster = "cluster")
summary(fit.metric)
## End(Not run)
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| measEq.syntax | R Documentation |
Syntax for measurement equivalence
Description
Automatically generates lavaan model syntax to specify a confirmatory
factor analysis (CFA) model with equality constraints imposed on
user-specified measurement (or structural) parameters. Optionally returns
the fitted model (if data are provided) representing some chosen level of
measurement equivalence/invariance across groups and/or repeated measures.
Usage
measEq.syntax(configural.model, ..., ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016", ID.thr = c(1L, 2L), group = NULL, group.equal = "", group.partial = "", longFacNames = list(), longIndNames = list(), long.equal = "", long.partial = "", auto = "all", warn = TRUE, debug = FALSE, return.fit = FALSE)
Arguments
configural.model |
A model with no measurement-invariance constraints
(i.e., representing only configural invariance), unless required for model
identification.
Note that the specified or fitted model must not contain any latent structural parameters (i.e., it must be a CFA model), unless they are higher-order constructs with latent indicators (i.e., a second-order CFA). |
... |
Additional arguments (e.g., |
ID.fac |
See Kloessner & Klopp (2019) for details about all three methods. |
ID.cat |
See Details and References for more information. |
ID.thr |
|
group |
optional |
group.equal |
optional |
group.partial |
optional |
longFacNames |
optional named |
longIndNames |
optional named |
long.equal |
optional |
long.partial |
optional |
auto |
Used to automatically included autocorrelated measurement errors
among repeatedly measured indicators in |
warn, debug |
|
return.fit |
|
Details
This function is a pedagogical and analytical tool to generate model syntax representing some level of measurement equivalence/invariance across any combination of multiple groups and/or repeated measures. Support is provided for confirmatory factor analysis (CFA) models with simple or complex structure (i.e., cross-loadings and correlated residuals are allowed). For any complexities that exceed the limits of automation, this function is intended to still be useful by providing a means to generate syntax that users can easily edit to accommodate their unique situations.
Limited support is provided for bifactor models and higher-order constructs.
Because bifactor models have cross-loadings by definition, the option
ID.fac = "effects.code" is unavailable. ID.fac = "UV" is
recommended for bifactor models, but ID.fac = "UL" is available on
the condition that each factor has a unique first indicator in the
configural.model. In order to maintain generality, higher-order
factors may include a mix of manifest and latent indicators, but they must
therefore require ID.fac = "UL" to avoid complications with
differentiating lower-order vs. higher-order (or mixed-level) factors.
The keyword "loadings" in group.equal or long.equal
constrains factor loadings of all manifest indicators (including loadings on
higher-order factors that also have latent indicators), whereas the keyword
"regressions" constrains factor loadings of latent indicators. Users
can edit the model syntax manually to adjust constraints as necessary, or
clever use of the group.partial or long.partial arguments
could make it possible for users to still automated their model syntax.
The keyword "intercepts" constrains the intercepts of all manifest
indicators, and the keyword "means" constrains intercepts and means
of all latent common factors, regardless of whether they are latent
indicators of higher-order factors. To test equivalence of lower-order and
higher-order intercepts/means in separate steps, the user can either
manually edit their generated syntax or conscientiously exploit the
group.partial or long.partial arguments as necessary.
ID.fac: If the configural.model fixes any (e.g.,
the first) factor loadings, the generated syntax object will retain those
fixed values. This allows the user to retain additional constraints that
might be necessary (e.g., if there are only 1 or 2 indicators). Some methods
must be used in conjunction with other settings:
-
ID.cat = "Millsap"requiresID.fac = "UL"andparameterization = "theta". -
ID.cat = "LISREL"requiresparameterization = "theta". -
ID.fac = "effects.code"is unavailable when there are any cross-loadings.
ID.cat: Wu & Estabrook (2016) recommended constraining
thresholds to equality first, and doing so should allow releasing any
identification constraints no longer needed. For each ordered
indicator, constraining one threshold to equality will allow the item's
intercepts to be estimated in all but the first group or repeated measure.
Constraining a second threshold (if applicable) will allow the item's
(residual) variance to be estimated in all but the first group or repeated
measure. For binary data, there is no independent test of threshold,
intercept, or residual-variance equality. Equivalence of thresholds must
also be assumed for three-category indicators. These guidelines provide the
least restrictive assumptions and tests, and are therefore the default.
The default setting in Mplus is similar to Wu & Estabrook (2016),
except that intercepts are always constrained to zero (so they are assumed
to be invariant without testing them). Millsap & Tein (2004) recommended
parameterization = "theta" and identified an item's residual variance
in all but the first group (or occasion; Liu et al., 2017) by constraining
its intercept to zero and one of its thresholds to equality. A second
threshold for the reference indicator (so ID.fac = "UL") is used to
identify the common-factor means in all but the first group/occasion. The
LISREL software fixes the first threshold to zero and (if applicable) the
second threshold to 1, and assumes any remaining thresholds to be equal
across groups / repeated measures; thus, the intercepts are always
identified, and residual variances (parameterization = "theta") are
identified except for binary data, when they are all fixed to one.
Repeated Measures: If each repeatedly measured factor is measured
by the same indicators (specified in the same order in the
configural.model) on each occasion, without any cross-loadings, the
user can let longIndNames be automatically generated. Generic names
for the repeatedly measured indicators are created using the name of the
repeatedly measured factors (i.e., names(longFacNames)) and the
number of indicators. So the repeatedly measured first indicator
("ind") of a longitudinal construct called "factor" would be
generated as "._factor_ind.1".
The same types of parameter can be specified for long.equal as for
group.equal (see lavOptions for a list), except
for "residual.covariances" or "lv.covariances". Instead, users
can constrain autocovariances using keywords "resid.autocov"
or "lv.autocov". Note that group.equal = "lv.covariances" or
group.equal = "residual.covariances" will constrain any
autocovariances across groups, along with any other covariances the user
specified in the configural.model. Note also that autocovariances
cannot be specified as exceptions in long.partial, so anything more
complex than the auto argument automatically provides should instead
be manually specified in the configural.model.
When users set orthogonal=TRUE in the configural.model (e.g.,
in bifactor models of repeatedly measured constructs), autocovariances of
each repeatedly measured factor will still be freely estimated in the
generated syntax.
Missing Data: If users wish to utilize the auxiliary
function to automatically include auxiliary variables in conjunction with
missing = "FIML", they should first generate the hypothesized-model
syntax, then submit that syntax as the model to auxiliary().
If users utilized runMI to fit their configural.model
to multiply imputed data, that model can also be passed to the
configural.model argument, and if return.fit = TRUE, the
generated model will be fitted to the multiple imputations.
Value
By default, an object of class measEq.syntax.
If return.fit = TRUE, a fitted lavaan
model, with the measEq.syntax object stored in the
@external slot, accessible by fit@external$measEq.syntax.
Author(s)
Terrence D. Jorgensen (University of Amsterdam; TJorgensen314@gmail.com)
References
Kloessner, S., & Klopp, E. (2019). Explaining constraint interaction: How to interpret estimated model parameters under alternative scaling methods. Structural Equation Modeling, 26(1), 143–155. doi: 10.1080/10705511.2018.1517356
Liu, Y., Millsap, R. E., West, S. G., Tein, J.-Y., Tanaka, R., & Grimm, K. J. (2017). Testing measurement invariance in longitudinal data with ordered-categorical measures. Psychological Methods, 22(3), 486–506. doi: 10.1037/met0000075
Millsap, R. E., & Tein, J.-Y. (2004). Assessing factorial invariance in ordered-categorical measures. Multivariate Behavioral Research, 39(3), 479–515. doi: 10.1207/S15327906MBR3903_4
Wu, H., & Estabrook, R. (2016). Identification of confirmatory factor analysis models of different levels of invariance for ordered categorical outcomes. Psychometrika, 81(4), 1014–1045. doi: 10.1007/s11336-016-9506-0
See Also
compareFit
Examples
mod.cat <- ' FU1 =~ u1 + u2 + u3 + u4
FU2 =~ u5 + u6 + u7 + u8 '
## the 2 factors are actually the same factor (FU) measured twice
longFacNames <- list(FU = c("FU1","FU2"))
## CONFIGURAL model: no constraints across groups or repeated measures
syntax.config <- measEq.syntax(configural.model = mod.cat,
# NOTE: data provides info about numbers of
# groups and thresholds
data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", longFacNames = longFacNames)
## print lavaan syntax to the Console
cat(as.character(syntax.config))
## print a summary of model features
summary(syntax.config)
## THRESHOLD invariance:
## only necessary to specify thresholds if you have no data
mod.th <- '
u1 | t1 + t2 + t3 + t4
u2 | t1 + t2 + t3 + t4
u3 | t1 + t2 + t3 + t4
u4 | t1 + t2 + t3 + t4
u5 | t1 + t2 + t3 + t4
u6 | t1 + t2 + t3 + t4
u7 | t1 + t2 + t3 + t4
u8 | t1 + t2 + t3 + t4
'
syntax.thresh <- measEq.syntax(configural.model = c(mod.cat, mod.th),
# NOTE: data not provided, so syntax must
# include thresholds, and number of
# groups == 2 is indicated by:
sample.nobs = c(1, 1),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", group.equal = "thresholds",
longFacNames = longFacNames,
long.equal = "thresholds")
## notice that constraining 4 thresholds allows intercepts and residual
## variances to be freely estimated in all but the first group & occasion
cat(as.character(syntax.thresh))
## print a summary of model features
summary(syntax.thresh)
## Fit a model to the data either in a subsequent step (recommended):
mod.config <- as.character(syntax.config)
fit.config <- cfa(mod.config, data = datCat, group = "g",
ordered = paste0("u", 1:8), parameterization = "theta")
## or in a single step (not generally recommended):
fit.thresh <- measEq.syntax(configural.model = mod.cat, data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", group.equal = "thresholds",
longFacNames = longFacNames,
long.equal = "thresholds", return.fit = TRUE)
## compare their fit to test threshold invariance
anova(fit.config, fit.thresh)
## --------------------------------------------------------
## RECOMMENDED PRACTICE: fit one invariance model at a time
## --------------------------------------------------------
## - A downside of setting return.fit=TRUE is that if the model has trouble
## converging, you don't have the opportunity to investigate the syntax,
## or even to know whether an error resulted from the syntax-generator or
## from lavaan itself.
## - A downside of automatically fitting an entire set of invariance models
## (like the old measurementInvariance() function did) is that you might
## end up testing models that shouldn't even be fitted because less
## restrictive models already fail (e.g., don't test full scalar
## invariance if metric invariance fails! Establish partial metric
## invariance first, then test equivalent of intercepts ONLY among the
## indicators that have invariate loadings.)
## The recommended sequence is to (1) generate and save each syntax object,
## (2) print it to the screen to verify you are fitting the model you expect
## to (and potentially learn which identification constraints should be
## released when equality constraints are imposed), and (3) fit that model
## to the data, as you would if you had written the syntax yourself.
## Continuing from the examples above, after establishing invariance of
## thresholds, we proceed to test equivalence of loadings and intercepts
## (metric and scalar invariance, respectively)
## simultaneously across groups and repeated measures.
## Not run:
## metric invariance
syntax.metric <- measEq.syntax(configural.model = mod.cat, data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", longFacNames = longFacNames,
group.equal = c("thresholds","loadings"),
long.equal = c("thresholds","loadings"))
summary(syntax.metric) # summarize model features
mod.metric <- as.character(syntax.metric) # save as text
cat(mod.metric) # print/view lavaan syntax
## fit model to data
fit.metric <- cfa(mod.metric, data = datCat, group = "g",
ordered = paste0("u", 1:8), parameterization = "theta")
## test equivalence of loadings, given equivalence of thresholds
anova(fit.thresh, fit.metric)
## scalar invariance
syntax.scalar <- measEq.syntax(configural.model = mod.cat, data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv", ID.cat = "Wu.Estabrook.2016",
group = "g", longFacNames = longFacNames,
group.equal = c("thresholds","loadings",
"intercepts"),
long.equal = c("thresholds","loadings",
"intercepts"))
summary(syntax.scalar) # summarize model features
mod.scalar <- as.character(syntax.scalar) # save as text
cat(mod.scalar) # print/view lavaan syntax
## fit model to data
fit.scalar <- cfa(mod.scalar, data = datCat, group = "g",
ordered = paste0("u", 1:8), parameterization = "theta")
## test equivalence of intercepts, given equal thresholds & loadings
anova(fit.metric, fit.scalar)
## For a single table with all results, you can pass the models to
## summarize to the compareFit() function
compareFit(fit.config, fit.thresh, fit.metric, fit.scalar)
## ------------------------------------------------------
## NOT RECOMMENDED: fit several invariance models at once
## ------------------------------------------------------
test.seq <- c("thresholds","loadings","intercepts","means","residuals")
meq.list <- list()
for (i in 0:length(test.seq)) {
if (i == 0L) {
meq.label <- "configural"
group.equal <- ""
long.equal <- ""
} else {
meq.label <- test.seq[i]
group.equal <- test.seq[1:i]
long.equal <- test.seq[1:i]
}
meq.list[[meq.label]] <- measEq.syntax(configural.model = mod.cat,
data = datCat,
ordered = paste0("u", 1:8),
parameterization = "theta",
ID.fac = "std.lv",
ID.cat = "Wu.Estabrook.2016",
group = "g",
group.equal = group.equal,
longFacNames = longFacNames,
long.equal = long.equal,
return.fit = TRUE)
}
compareFit(meq.list)
## -----------------
## Binary indicators
## -----------------
## borrow example data from Mplus user guide
myData <- read.table("http://www.statmodel.com/usersguide/chap5/ex5.16.dat")
names(myData) <- c("u1","u2","u3","u4","u5","u6","x1","x2","x3","g")
bin.mod <- '
FU1 =~ u1 + u2 + u3
FU2 =~ u4 + u5 + u6
'
## Must SIMULTANEOUSLY constrain thresholds, loadings, and intercepts
test.seq <- list(strong = c("thresholds","loadings","intercepts"),
means = "means",
strict = "residuals")
meq.list <- list()
for (i in 0:length(test.seq)) {
if (i == 0L) {
meq.label <- "configural"
group.equal <- ""
long.equal <- ""
} else {
meq.label <- names(test.seq)[i]
group.equal <- unlist(test.seq[1:i])
# long.equal <- unlist(test.seq[1:i])
}
meq.list[[meq.label]] <- measEq.syntax(configural.model = bin.mod,
data = myData,
ordered = paste0("u", 1:6),
parameterization = "theta",
ID.fac = "std.lv",
ID.cat = "Wu.Estabrook.2016",
group = "g",
group.equal = group.equal,
#longFacNames = longFacNames,
#long.equal = long.equal,
return.fit = TRUE)
}
compareFit(meq.list)
## ---------------------
## Multilevel Invariance
## ---------------------
## To test invariance across levels in a MLSEM, specify syntax as though
## you are fitting to 2 groups instead of 2 levels.
mlsem <- ' f1 =~ y1 + y2 + y3
f2 =~ y4 + y5 + y6 '
## metric invariance
syntax.metric <- measEq.syntax(configural.model = mlsem, meanstructure = TRUE,
ID.fac = "std.lv", sample.nobs = c(1, 1),
group = "cluster", group.equal = "loadings")
## by definition, Level-1 means must be zero, so fix them
syntax.metric <- update(syntax.metric,
change.syntax = paste0("y", 1:6, " ~ c(0, NA)*1"))
## save as a character string
mod.metric <- as.character(syntax.metric, groups.as.blocks = TRUE)
## convert from multigroup to multilevel
mod.metric <- gsub(pattern = "group:", replacement = "level:",
x = mod.metric, fixed = TRUE)
## fit model to data
fit.metric <- lavaan(mod.metric, data = Demo.twolevel, cluster = "cluster")
summary(fit.metric)
## End(Not run)
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