In doomlab/learnSEM: Learning Tutorials for Structural Equation Modeling

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

knitr::opts_chunk$set(echo = TRUE)
library(lavaan)
library(semPlot)

Second Order Latent Models

• A second order model is one that is structured into an order or rank.

• Two types we will consider:

  • Higher-order: Describes when latent variables are structured so that latents influence other latents into levels.
  • Bi-factor: Generally used to describe a CFA with two sets of latent variables (not hierarchical).

Second Order Models

• The idea of a higher order model is:

  • You have some latent variables that are measured by the observed variables.
  • A portion of the variance in the latent variables can be explained by a second (set) of latent variables.
  • Therefore, we are switching out the covariances between factors and using another latent to explain them.

Second Order Models

• When are these used?
• When there are multiple latent variables that covary with each other (and a lot).
• A second set of latents explains that covariance.

Second Order Models

knitr::include_graphics("pictures/second_order.png")

Higher Order Models

• The covariance of the first order is accounted for by the second order plus a specific factor.
• Specific factors are error that is not explained by the second order latents.
• The higher order is thought to indirectly influence the manifest variables through the first order.

Identification

• Remember that each portion of the model has to be identified.

  • The section with each latent variable has to be identified.
  • The section with the latents has to be identified.

Identification

• You can achieve identification in a couple of ways:

  • Set some of the loadings in the upper portion of the model to be equal by giving them the same name.
  • You can set the variance in the upper latent to be one.
  • You can set some of the error variances of the latents in the lower portion to be equal.

Bi-Factor Models

• Special type of model with two sets of latents, but they are not hierarchically structured.

• Best used when:

  • General factor that accounts for variance in the manifest variables.
  • Domain specific areas that are thought to influence the manifest variables.

Bi-Factor Models

knitr::include_graphics("pictures/bi_factor.png")

Bi-Factor Models

• One thing to note is that the latent variables are left uncorrelated in this type of model.
• This structure represents the domain specific part of the interpretation.

Model Differences

• Differences between bi-factor and hierarchical:

  • In hierarchical models, the second order influences the first order, while the two sets of latents in bi-factors are uncorrelated.
  • What does that allow you to test differently?

Model Differences

• Advantages:

  • Allows you to see how the first order latent variables influence the manifest variables separately from the other latent variables.
  • After accounting for the general latent variable, are the domain specific items still accounting for variance?
  • You can compare models with and without the domain specific areas

Examples: WISC

• Going back to the WISC data, this time with more of the subscales that are available.

library(lavaan)
library(semPlot)

##import the data
wisc4.cov <- lav_matrix_lower2full(c(8.29,
                                    5.37,9.06,
                                    2.83,4.44,8.35,
                                    2.83,3.32,3.36,8.88,
                                    5.50,6.66,4.20,3.43,9.18,
                                    6.18,6.73,4.01,3.33,6.77,9.12,
                                    3.52,3.77,3.19,2.75,3.88,4.05,8.88,
                                    3.79,4.50,3.72,3.39,4.53,4.70,4.54,8.94,
                                    2.30,2.67,2.40,2.38,2.06,2.59,2.65,2.83,8.76,
                                    3.06,4.04,3.70,2.79,3.59,3.67,3.44,4.20,4.53,9.73))

wisc4.sd <- c(2.88,3.01,2.89,2.98,3.03,3.02,2.98,2.99,2.96,3.12) 

names(wisc4.sd) <- 
  colnames(wisc4.cov) <- 
  rownames(wisc4.cov) <- c("Comprehension", "Information", 
                         "Matrix.Reasoning", "Picture.Concepts", 
                         "Similarities", "Vocabulary",  "Digit.Span", 
                         "Letter.Number",  "Coding", "Symbol.Search") 

Examples: WISC

• Let's start with a first order model for the WISC.
• By starting with the measurement model, we can ensure the measurement section is identified and works correctly.
• You should also check out the correlations/covariances between factors to make sure that they are even related.

Examples: WISC Build the Model

##first order model
wisc4.fourFactor.model <- '
gc =~ Comprehension + Information +  Similarities + Vocabulary 
gf =~ Matrix.Reasoning + Picture.Concepts
gsm =~  Digit.Span + Letter.Number
gs =~ Coding + Symbol.Search
'   

Examples: WISC Analyze the Model

wisc4.fourFactor.fit <- cfa(model = wisc4.fourFactor.model, 
                            sample.cov = wisc4.cov, 
                            sample.nobs = 550)

Examples: WISC Summarize the Model

• Logical solution:

  • Positive variances
  • SMCs + Correlations < 1
  • No error messages
  • SEs are not "huge"

• Estimates:

  • Do our questions load appropriately?

• Model fit:

  • What do the fit indices indicate?
  • Can we improve model fit without overfitting?

summary(wisc4.fourFactor.fit, 
        fit.measure = TRUE, 
        standardized = TRUE,
        rsquare = TRUE)

Examples: WISC Diagram the Model

semPaths(wisc4.fourFactor.fit, 
         whatLabels="std", 
         edge.label.cex = 1,
         edge.color = "black",
         what = "std",
         layout="tree")

Examples: WISC Interpretation

• In the first order model, we find that the correlations are pretty high.
• That’s a good sign that maybe a second order model is appropriate.
• Or that the factors should be collapsed together, as they are not distinct.

Examples: Second-Order WISC

• Build the model:

wisc4.higherOrder.model <- '
gc =~ Comprehension + Information + Similarities + Vocabulary 
gf =~ Matrix.Reasoning + Picture.Concepts
gsm =~  Digit.Span + Letter.Number
gs =~ Coding + Symbol.Search

g =~ gf + gc  + gsm + gs 
'

Examples: Second-Order WISC

• Analyze the model:

wisc4.higherOrder.fit <- cfa(model = wisc4.higherOrder.model, 
                             sample.cov = wisc4.cov, 
                             sample.nobs = 550)

Examples: Second-Order WISC

• Summarize the model:

  • Note: the fit indices should not change.
  • Let's look at loadings, where the shift should have happened.

summary(wisc4.higherOrder.fit, 
        fit.measure=TRUE, 
        standardized=TRUE, 
        rsquare = TRUE)

Examples: Second-Order WISC

semPaths(wisc4.higherOrder.fit, 
         whatLabels="std", 
         edge.label.cex = 1,
         edge.color = "black",
         what = "std",
         layout="tree")

Examples: Bi-factor WISC

• Build the model:

  • In this particular model, because we have only two indicators, we have to set them equal for identification for a bifactor to run properly?
  • Why not the second order model?
  • After you define the measurement model, you would define a second latent that every variable is related to.
  • Note: these are the variables not the latents like a hierarchical model.

wisc4.bifactor.model <- '
gc =~ Comprehension + Information +  Similarities + Vocabulary 
gf =~ a*Matrix.Reasoning + a*Picture.Concepts  
gsm =~  b*Digit.Span + b*Letter.Number
gs =~ c*Coding + c*Symbol.Search 
g =~ Information + Comprehension + Matrix.Reasoning + Picture.Concepts + Similarities + Vocabulary +  Digit.Span + Letter.Number + Coding + Symbol.Search
'

Examples: Bi-factor WISC

• Analyze the model:

  • Last, we want to turn off the automatic exogenous only correlations, which we do with the orthogonal = TRUE model.
  • The argument is that all the correlations normally seen between the domain specific latents are due to the general factor.

wisc4.bifactor.fit <- cfa(model = wisc4.bifactor.model, 
                          sample.cov = wisc4.cov,
                          sample.nobs = 550, 
                          orthogonal = TRUE)

Examples: Bi-factor WISC

• Summarize the model:

  • Notice the covariances are zero.
  • Notice the equal loading estimates for the variables set to the same.

summary(wisc4.bifactor.fit, 
        fit.measure = TRUE, 
        rsquare = TRUE, 
        standardized = TRUE)

Examples: Bi-factor WISC

• Diagram the model:

semPaths(wisc4.bifactor.fit, 
         whatLabels="std", 
         edge.label.cex = 1,
         edge.color = "black",
         what = "std",
         layout="tree")

Summary

• In this lecture you've learned:

  • How to account for strong correlations between latent variables
  • How to program and run a higher-order model (hierarchical)
  • How to program and run a bi-factor model (not hierarchical)

doomlab/learnSEM documentation built on Jan. 25, 2024, 2 p.m.