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)

Confirmatory Factor Analysis

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

Relation to EFA

• You have a bunch of questions
• You have an idea (or sometimes not!) of how many factors to expect
• You let the questions go where they want
• You remove the bad questions until you get a good fit

CFA models

• You set up the model with specific questions onto specific factors
• Forcing the cross loadings be zero
• You test to see if that model fits
• So, you may think about how confirmatory factor analysis is step two to exploring (exploratory factor analysis)

CFA Models

• Reflective – the latent variable causes the manifest variables scores
• Purpose is to understand the relationships between the measured variables
• Same theoretical concept as EFA

CFA Models Reflective Example

# a famous example, build the model
HS.model <- ' visual  =~ x1 + x2 + x3
              textual =~ x4 + x5 + x6
              speed   =~ x7 + x8 + x9 '

# fit the model 
HS.fit <- cfa(HS.model, data = HolzingerSwineford1939)

# diagram the model
semPaths(HS.fit, 
         whatLabels = "std", 
         layout = "tree",
         edge.label.cex = 1)

CFA Models

• Formative – latent variables are the result of manifest variables
• Similar to principal components analysis theoretical concept
• Potentially a use for demographics?

CFA Models Formative Example

# a famous example, build the model
HS.model <- ' visual  <~ x1 + x2 + x3'

# fit the model 
HS.fit <- cfa(HS.model, data = HolzingerSwineford1939)

# diagram the model
semPaths(HS.fit, 
         whatLabels = "std", 
         layout = "tree",
         edge.label.cex = 1)

CFA Models

• The manifest variables in a CFA are sometimes called indicator variables
• Because they indicate what the latent variable should be since we do not directly measure the latent variable

General Set Up

• The latents will be correlated (because they are exogenous only)
• Similar to an oblique rotation
• Each factor section has to be identified
• You should have three measured variables per latent
• If you only have two, you need to set their coefficients to equal (estimated but equal)
• Arrows go from latent to measured (reflexive)
• We think that latent caused the measured answers
• Error terms on the measured variables, variance on the latent variables
• If you are counting for degrees of freedom for identification

Correlated Error

• Generally, you leave the error terms uncorrelated, as you think they are separate items
• However:
• These questions all measure the same factor right?
• Often they are pretty similar
• Some answers will be related to other items
• So it's not too big of a idea to say that item's errors are related
• We can use modification indices to see if they should be correlated
• Make sure these make sense!

Interpretation

• The latent variable section includes the factor loadings or coefficients
• These are the same idea as EFA - you want the relationship between the latent variable and the manifest variable to be strong
• We used a rule of .300 before but for this rule, you should examine the standardized loading
• Otherwise, why would we think this item measures the latent variable?

Interpretation

• These coefficients are often called:
• Pattern coefficients (unstandardized): for every one unit in the latent variable, the manifest variable increases b units
• Structure coefficients (standardized): the correlation between the latent variable and the manifest variable

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

Scaling

• Remember that scaling is the way we "set the scale" for the latent variable
• We usually do this by setting one of the pattern coefficients to 1 - the marker variable approach
• Another option is to to set the variance of the latent variable to 1 std.lv in the the standardized output
• What does that do?
• Sets the scale to z-score
• Makes double headed arrow between latents correlation
• Make sure you are using unstandardized data!

Scaling

• So what is the std.all as part of the "completely standardized solution?
• Both the latent variable variance and the manifest variable variance is set to 1
• If you are going to report the standardized solution, this version is the most common, as it matches EFA and regression
• All of these options give you different loadings, but should not change model fit

Examples

• Reminder:
• When you use a correlation matrix as your input, the solution is already standardized!
• When you use a covariance matrix as your input, both the unstandardized and standardized solution can be viewed

One-Factor CFA Example

• IQ is often thought of as "g" or this overall cognitive ability
• Let's look at an example of the WISC, which is an IQ test for children
• We have five of the subtest scores including information, similarities, word reasoning, matrix reasoning, and picture concepts

Convert Correlations to Covariance

wisc4.cor <- lav_matrix_lower2full(c(1,
                                     0.72,1,
                                     0.64,0.63,1,
                                     0.51,0.48,0.37,1,
                                     0.37,0.38,0.38,0.38,1))
# enter the SDs
wisc4.sd <- c(3.01 , 3.03 , 2.99 , 2.89 , 2.98)

# give everything names
colnames(wisc4.cor) <- 
  rownames(wisc4.cor) <-
  names(wisc4.sd) <- 
  c("Information", "Similarities", 
    "Word.Reasoning", "Matrix.Reasoning", "Picture.Concepts")

# convert
wisc4.cov <- cor2cov(wisc4.cor, wisc4.sd)

WISC One-Factor Model

• The =~ is used to define a reflexive latent variable
• ~ can be interpreted as Y is predicted by X
• =~ can be interpreted as X is indicated by Ys

wisc4.model <- '
g =~ Information + Similarities + Word.Reasoning + Matrix.Reasoning + Picture.Concepts
'

Analyze the Model

• Notice we changed to the cfa() function
• It has the same basic arguments
• The std.lv option can be used to only see the standardized solution on the latent variable, usually you want to set this to FALSE

wisc4.fit <- cfa(model = wisc4.model, 
                sample.cov = wisc4.cov, 
                sample.nobs = 550,  
                std.lv = FALSE)

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?

Summarize the Model

summary(wisc4.fit,
        standardized=TRUE, 
        rsquare = TRUE,
        fit.measures=TRUE)

New Functions

• std.nox: the standardized estimates are based on both the variances of both (continuous) observed and latent variables, but not the variances of exogenous covariates
• This output is the best way to get the confidence intervals for each parameter

parameterestimates(wisc4.fit,
                   standardized=TRUE)

New Functions

fitted(wisc4.fit) ## estimated covariances
wisc4.cov ## actual covariances

All Fit Indices

fitmeasures(wisc4.fit)

Modification Indices

modificationindices(wisc4.fit, sort = T)

Diagram the Model

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

WISC Two-Factor Model

• Note that we only have two items on the latent variable
• If we see an identification error, we can set these to equal (homework hint!)

wisc4.model2 <- '
V =~ Information + Similarities + Word.Reasoning 
F =~ Matrix.Reasoning + Picture.Concepts
'

# wisc4.model2 <- '
# V =~ Information + Similarities + Word.Reasoning 
# F =~ a*Matrix.Reasoning + a*Picture.Concepts
# '

Analyze the Model

wisc4.fit2 <- cfa(wisc4.model2, 
                  sample.cov=wisc4.cov, 
                  sample.nobs=550,
                  std.lv = F)

Summarize the Model

summary(wisc4.fit2,
        standardized=TRUE, 
        rsquare = TRUE,
        fit.measures=TRUE)

Diagram the Model

semPaths(wisc4.fit2, 
         whatLabels="std", 
         what = "std",
         edge.color = "pink",
         edge.label.cex = 1,
         layout="tree")

Compare the Models

anova(wisc4.fit, wisc4.fit2)
fitmeasures(wisc4.fit, c("aic", "ecvi"))
fitmeasures(wisc4.fit2, c("aic", "ecvi"))

How to Tidy lavaan Output

• https://easystats.github.io/parameters/articles/efa_cfa.html

#install.packages("parameters")
library(parameters)
model_parameters(wisc4.fit, standardize = TRUE)

How to Tidy lavaan Output

library(broom)
tidy(wisc4.fit)
glance(wisc4.fit)

Summary

• In this lecture you've learned:

• How to create a simple confirmatory factor analysis (measurement model)

• How to view parameter estimates, modification indices, and more
• Practiced building these models and comparing them

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