Multi-Group Factor Analysis In lslx: Semi-Confirmatory Structural Equation Modeling via Penalized Likelihood or Least Squares

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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.

Model Specification

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.

Object Initialization

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.

Model Respecification

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.

Model Fitting

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.

Model Summarizing

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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lslx documentation built on April 28, 2020, 1:09 a.m.