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In this example, we will show how to use lslx
to conduct multi-group factor analysis.
The example uses data HolzingerSwineford1939
in the package lavaan
.
Hence, lavaan
must be installed.
In the following specification, x1
- x9
is assumed to be measurements of 3 latent factors: visual
, textual
, and speed
.
model_mgfa <- "visual :=> 1 * x1 + x2 + x3 textual :=> 1 * x4 + x5 + x6 speed :=> 1 * x7 + x8 + x9"
The operator :=>
means that the LHS latent factors is defined by the RHS observed variables.
In this model, visual
is mainly measured by x1
- x3
, textual
is mainly measured by x4
- x6
, and speed
is mainly measured by x7
- x9
.
Loadings of x1
, x4
, and x7
are fixed at 1 for scale setting.
The above specification is valid for both groups.
Details of model syntax can be found in the section of Model Syntax via ?lslx
.
lslx
is written as an R6
class.
Everytime we conduct analysis with lslx
, an lslx
object must be initialized.
The following code initializes an lslx
object named lslx_mgfa
.
library(lslx) lslx_mgfa <- lslx$new(model = model_mgfa, data = lavaan::HolzingerSwineford1939, group_variable = "school", reference_group = "Pasteur")
Here, lslx
is the object generator for lslx
object and new
is the build-in method of lslx
to generate a new lslx
object.
The initialization of lslx
requires users to specify a model for model specification (argument model
) and a data set to be fitted (argument sample_data
).
The data set must contain all the observed variables specified in the given model.
Because in this example a multi-group analysis is considered, variable for group labeling (argument group_variable
) must be specified.
In lslx, two types of parameterization can be used in multi-group analysis.
The first type is the same with the traditional multi-group SEM, which treats model parameters in each group separately.
The second type sets one group as reference and treats model parameters in other groups as increments with respect to the reference.
Under the second type of parameterization, the group heterogeneity can be efficiently explored if we treat the increments as penalized parameters.
In this example, Pasteur
is set as reference.
Hence, the parameters in Grant-White
now reflect differences from the reference.
After an lslx
object is initialized, the heterogeneity of a multi-group model can be quickly respecified by $free_heterogeneity()
, $fix_heterogeneity()
, and $penalize_heterogeneity()
methods.
The following code sets x2<-visual
, x3<-visual
, x5<-textual
, x6<-textual
, x8<-speed
, x9<-speed
, and
x2<-1
, x3<-1
, x5<-1
, x6<-1
, x8<-1
, x9<-1
in Grant-White
as penalized parameters.
Note that parameters in Grant-White
now reflect differences since Pasteur
is set as reference.
lslx_mgfa$penalize_heterogeneity(block = c("y<-1", "y<-f"), group = "Grant-White")
Since the homogeneity of latent factor means may not be a reasonable assumption when examining measurement invariance, the following code relaxes this assumption
lslx_mgfa$free_block(block = "f<-1", group = "Grant-White")
To see more methods to modify a specified model, please check the section of Set-Related Method via ?lslx
.
After an lslx
object is initialized, method $fit_mcp()
can be used to fit the specified model into the given data with MCP.
lslx_mgfa$fit_mcp()
All the fitting result will be stored in the fitting
field of lslx_mgfa
.
Unlike traditional SEM analysis, lslx
fits the model into data under all the penalty levels considered.
To summarize the fitting result, a selector to determine an optimal penalty level must be specified.
Available selectors can be found in the section of Penalty Level Selection via ?lslx
.
The following code summarize the fitting result under the penalty level selected by Haughton’s Bayesian information criterion (HBIC).
lslx_mgfa$summarize(selector = "hbic")
In this example, we can see that all of the loadings are invariant across the two groups.
However, the intercepts of x3
and x7
seem to be not invariant.
The $summarize()
method also shows the result of significance tests for the coefficients.
In lslx
, the default standard errors are calculated based on sandwich formula whenever raw data is available.
It is generally valid even when the model is misspecified and the data is not normal.
However, it may not be valid after selecting an optimal penalty level.
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