Semi-Confirmatory Structural Equation Modeling 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 semi-confirmatory structural equation modeling. The example uses data PoliticalDemocracy in the package lavaan. Hence, lavaan must be installed.

Model Specification

In the following specification, x1 - x3 and y1 - y8 is assumed to be measurements of 3 latent factors: ind60, dem60, and dem65.

model_sem <- "fix(1) * x1 + x2 + x3      <=: ind60
fix(1) * y1 + y2 + y3 + y4 <=: dem60
fix(1) * y5 + y6 + y7 + y8 <=: dem65
dem60 <= ind60
dem65 <= ind60 + dem60"

The operator <=: means that the RHS latent factors is defined by the LHS observed variables. In particular, the loadings are freely estimated. In this model, ind60 is measured by x1 - x3, dem60 is mainly measured by y1 - y4, and dem65 is mainly measured by y5 - y8. The operator <= means that the regression coefficients from the RHS variables to the LHS variables are freely estimated. In this model, dem60 is influenced by ind60 and dem65 is influenced by dem60 and ind60. 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_sem.

library(lslx)
lslx_sem <- lslx\$new(model = model_sem,
sample_cov = cov(lavaan::PoliticalDemocracy),
sample_size = nrow(lavaan::PoliticalDemocracy))

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 sample moments to be fitted (argument sample_cov and sample_size). The sample moment must contain all the observed variables specified in the given model.

Model Respecification

After an lslx object is initialized, model can be respecified by \$free_coefficient(), \$fix_coefficient(), and \$penalize_coefficient() methods. The following code sets y1<->y5, y2<->y4, y2<->y6, y3<->y7, y4<->y8, and y6<->y8 as penalized parameters.

lslx_sem\$penalize_coefficient(name = c("y1<->y5",
"y2<->y4",
"y2<->y6",
"y3<->y7",
"y4<->y8",
"y6<->y8"))

To see more methods for respecifying model, please check the section of Set-Related Method via ?lslx.

Model Fitting

After an lslx object is initialized, method \$fit_lasso() can be used to fit the specified model into the given data with LASSO penalty function.

lslx_sem\$fit_lasso()

The \$fit_lasso() requires users to specify the considered penalty levels (argument lambda_grid). In this example, the lambda grid is automatically initialized by default. Note that MCP with delta = Inf is equivalent to the LASSO penalty. All the fitting result will be stored in the fitting field of lslx_sem.

Model Summarizing

Unlike traditional SEM analysis, lslx fit 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 adjusted Bayesian information criterion (ABIC).

lslx_sem\$summarize(selector = "abic")

In this example, we can see that the PL estimate under the selected penalty level doesn't contain any zero value, which indicates that all of the covariance of measurements are relevant. 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. In this example, because raw data is not used for lslx object initialization, standard error is calculated by using observed Fisher information matrix. It may not be valid when the model is misspecified and the data are not normal. Also, it is generally invalid after choosing a penalty level.

Objects Extraction

In lslx, many quantities related to SEM can be extracted by extract-related method. For example, the coefficient estimate and its asymptotic variance can be obtained by

lslx_sem\$extract_coefficient(selector = "abic", type = "effective")
diag(lslx_sem\$extract_coefficient_acov(selector = "abic", type = "effective"))

Here, the type argument is used to specify which types of parameters are used to calculate related quantities. type = "effective" indicates that only freely estimated and penalized non-zero parameters are used. By default, type = "all"

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