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)

Full Structural Models

• A fully latent or full structural model has two parts:

• Measurement model: the latent variable with it's measured variables (CFA)

• Structural model: the relationship between other variables and the measurement model

• The structural model may be a second order latent, like we did last week

• The structural model could include other predictors
• Or you could simply convert from correlations to a specific prediction direction as part of your structural model

Example Model

knitr::include_graphics("pictures/full_sem2.png")

Full Structural Models

• A reminder from earlier this semester:

  • Reflective indicators: we assume the latent variables are the cause, so they are exogenous
  • Formative indicators: we assume the latent variables are the criterion, so they are endogenous

Full Structural Models

• Example of formative indicator

  • Income, education level, and occupation all predict your SES
  • Stress caused by outside factors

• Other names people use for these:

  • Composite causes
  • MIMIC – multiple indicators, multiple causes models

Example Model

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

Identification

• Identification rules of thumb:

  • Latent variables should have four indicators
  • Latent variables have three indicators AND Error variances do not covary
  • Latent variables have two indicators AND Error variances do not covary and Loadings are set to equal each other.

Structural Model Identification

• Scaling is also required to identify the structural part

• 2+ emitted paths rule

  • Composite variable must have direct effects on two other endogenous variables

Things to Consider

• Parceling

  • When you have large structural models, they can be very complex to fit to a Full SEM if each latent variable has lots of indicators (items).
  • Parceling occurs when you create subsets of items to be able to get the model to run and to balance out the number of indicators on each latent.
  • At the moment, this topic is still pretty controversial.

How to Model

• Test each CFA piece separately to make sure they run.

  • CFAs that are bad, do not suddenly make good models when the structural component is added!

• Slowly add structural paths to see if you can get the full model to work.

  • If not, try parceling.
  • Drop non-significant paths.

How to Model

• As you add the structural components, you should not see a big change in the loadings to the indicators

  • If you do, it means the model is not invariant
  • Causes interpretation difficulties

When to Stop?

• We have discussed using modificationindices() and other tricks to improve model fit
• In theory, we could add all paths until the model is "perfect"

• What should the stopping rule be?

  • Based on theory
  • Fit indices do not greatly improve
  • Parsimony

Example Model

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

Example Model: Setup

library(lavaan)
library(semPlot)

family.cor <- lav_matrix_lower2full(c(1.00, 
                                      .74,  1.00,   
                                      .27,  .42,    1.00,   
                                      .31,  .40,    .79,    1.00,   
                                      .32,  .35,    .66,    .59,    1.00))
family.sd <- c(32.94,   22.75, 13.39,   13.68,  14.38)
rownames(family.cor) <- 
  colnames(family.cor) <-
  names(family.sd) <- c("father", "mother", "famo", "problems", "intimacy")

family.cov <- cor2cov(family.cor, family.sd)

Example Model: Build the CFA

• First, we are going to test the measurement model – just the CFAs with a covariance between them.
• Then, we are going to change it to a full SEM, predicting the direction of the relationship between latents.
• We should ensure the measurement model does not change significantly.

Example Model: Build the CFA

family.model <- '
adjust =~ problems + intimacy
family =~ father + mother + famo'

Example Model: Analyze the CFA

family.fit <- cfa(model = family.model,
                  sample.cov = family.cov,
                  sample.nobs = 203)

Example Model: Deal with Error

inspect(family.fit, "cov.lv")
inspect(family.fit, "cor.lv")

Example Model: Analyze the CFA

family.fit <- cfa(model = family.model,
                  sample.cov = family.cor,
                  sample.nobs = 203)

Example Model: Summarize the Model

summary(family.fit, 
        rsquare = TRUE, 
        standardized = TRUE,
        fit.measures = TRUE)

Example Model: Improve the Model?

modificationindices(family.fit, sort = T)

Example Model: Improve the Model?

family.model2 <- '
adjust =~ problems + intimacy
family =~ father + mother + famo
father ~~ mother'

family.fit2 <- cfa(model = family.model2,
                  sample.cov = family.cov,
                  sample.nobs = 203)

inspect(family.fit2, "cor.lv")

Example Model: Diagram the Model

semPaths(family.fit, 
         whatLabels="std", 
         layout="tree", 
         edge.label.cex = 1)

Example Model: Build Full SEM

predict.model <- '
adjust =~ problems + intimacy
family =~ father + mother + famo
adjust~family'

Example Model: Analyze Full SEM

predict.fit <- sem(model = predict.model,
                   sample.cov = family.cor,
                   sample.nobs = 203)

Example Model: Summarize Full SEM

summary(predict.fit, 
        rsquare = TRUE, 
        standardized = TRUE,
        fit.measures = TRUE)

Example Model: Diagram Full SEM

semPaths(predict.fit, 
         whatLabels="std", 
         layout="tree", 
         edge.label.cex = 1)

Example 2: Composite Variables

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

Example 2: Setup

family.cor <- lav_matrix_lower2full(c(1.00, 
                                     .42,   1.00, 
                                    -.43,   -.50,   1.00, 
                                    -.39,   -.43,   .78,    1.00,   
                                    -.24,   -.37,   .69,    .73,    1.00, 
                                    -.31,   -.33,   .63,    .87,    .72,    1.00,   
                                    -.25,   -.25,   .49,    .53,    .60,    .59,    1.00, 
                                     -.25,  -.26,   .42,    .42,    .44,    .45,    .77,    1.00,   
                                     -.16,  -.18,   .23,    .36,    .38,    .38,    .59,    .58, 1.00))

family.sd <- c(13.00,   13.50,  13.10,  12.50,  13.50,  14.20,  9.50,   11.10,  8.70)

rownames(family.cor) <- 
  colnames(family.cor) <-
  names(family.sd) <- c("parent_psych","low_SES","verbal",
                        "reading","math","spelling","motivation","harmony","stable")

family.cov <- cor2cov(family.cor, family.sd)

Example 2: Build the Model

• How to define a composite variable?
• We have been using =~ to define a latent variable that predicts manifest variables.
• Use <~ to create a composite variable that is predicted by the manifest variables.

Example 2: Build the Model

composite.model <- '
risk <~ low_SES + parent_psych + verbal
achieve =~ reading + math + spelling
adjustment =~ motivation + harmony + stable
risk =~ achieve + adjustment
'

Example 2: Analyze the Model

composite.fit <- sem(model = composite.model, 
                      sample.cov = family.cov, 
                      sample.nobs = 158)

Example 2: Summarize the Model

summary(composite.fit, 
        rsquare = TRUE, 
        standardized = TRUE,
        fit.measures = TRUE)

Example 2: Improve the Model?

modificationindices(composite.fit, sort = T)

Example 2: Diagram the Model

semPaths(composite.fit, 
         whatLabels="std", 
         layout="tree",
         edge.label.cex = 1)

Summary

• In this lecture you've learned:

  • How to build on our previous work of measurement models
  • How to build a composite variable and run those models
  • How to examine for invariance when building a full SEM

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