In doomlab/learnSEM: Learning Tutorials for Structural Equation Modeling

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

options(scipen = 999)
knitr::opts_chunk$set(echo = TRUE)

Terminology

• Measured variables – the real scores from the experiment
• Squares on a diagram
• Latent variables – the construct the measured variables are supposed to represent
• Not measured directly
• Circles on a diagram

Terminology

library(lavaan)
library(semPlot)
HS.model <- ' visual  =~ x1 + x2 + x3
              textual =~ x4 + x5 + x6
              speed   =~ x7 + x8 + x9 '

fit <- cfa(HS.model, data = HolzingerSwineford1939)
semPaths(fit,
         whatLabels = "std",
         edge.label.cex = 1)

What is EFA?

• Factor analysis attempts to achieve parsimony (data reduction) by:
• Explaining the maximum amount of common variance in a correlation matrix
• Using the smallest number of explanatory constructs (factors)
• Common variance is the overlapping variance between items
• Unique variance is the variance only related to that item (error variance)

Directionality

• Factors are thought to cause the measured variables
• Therefore, we are saying that the measured variable is predicted by the factor

library(lavaan)
library(semPlot)
HS.model <- ' visual  =~ x1 + x2 + x3
              textual =~ x4 + x5 + x6
              speed   =~ x7 + x8 + x9 '

fit <- cfa(HS.model, data = HolzingerSwineford1939)
semPaths(fit,
         whatLabels = "std",
         edge.label.cex = 1)

Why use EFA?

• Understand structure of set of variables
• Construct a scale to measure the latent variable
• Reduce data set to smaller size that still measures original information

The Example Data

• We are going to build a scale that measures the anxiety that statistics provokes in students

• Dataset is from the Field textbook

• Statistics makes me cry

• My friends will think I'm stupid for not being able to cope with R
• Standard deviations excite me
• I dream that Pearson is attacking me with correlation coefficients
• I don't understand statistics
• I have little experience of computers
• All computers hate me
• I have never been good at mathematics
• My friends are better at statistics than me
• Computers are useful only for playing games
• I did badly at mathematics at school
• People try to tell you that R makes statistics easier to understand but it doesn't
• I worry that I will cause irreparable damage because of my incompetence with computers
• Computers have minds of their own and deliberately go wrong whenever I use them
• Computers are out to get me
• I weep openly at the mention of central tendency
• I slip into a coma whenever I see an equation
• R always crashes when I try to use it
• Everybody looks at me when I use R
• I can't sleep for thoughts of eigenvectors
• I wake up under my duvet thinking that I am trapped under a normal distribution
• My friends are better at R than I am
• If I'm good at statistics my friends will think I'm a nerd

The Example Data

library(rio)
library(psych)
master <- import("data/lecture_efa.csv")
head(master)

Steps to Analysis

• Data screening (not shown, use last week's notes!)
• Additionally, you need at least interval measurement for the analyses shown here
• Enough items to group into factors (recommended 3-4 per potential factor)
• How many factors should you use?
• Simple structure
• Adequate solutions

How many factors do I use?

• Theory
• Kaiser criterion
• Scree plots
• Parallel analysis

Kaiser criterion

• Old rule: extract the number of eigenvalues over 1
• New rule: extract the number of eigenvalues over .7
• What the heck is an eigenvalue?
• A mathematical representation of the variance accounted for by that grouping of items

Scree plots

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

Parallel analysis

• A statistical test to tell you how many eigenvalues are greater than chance
• Calculates the eigenvalues for your data
• Randomizes your data and recalculates the eigenvalues
• Then compares them to determine if they are equal

Finding factors/components

number_items <- fa.parallel(master, #data frame
                            fm="ml", #math
                            fa="fa") #only efa

Eigenvalues

sum(number_items$fa.values > 1)
sum(number_items$fa.values > .7)

Simple structure

• Simple structure covers two pieces:
• The math used to achieve the solution: maximum likelihood
• Rotation to increase communality between items and aid in interpretation

Rotation

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

Rotation

• Orthogonal assume uncorrelated factors: varimax, quartermax, equamax
• Oblique allows factors to be correlated: oblimin, promax
• Why would we even use orthogonal?

Simple structure/solution

• Looking at the loadings: the relationship between the item and the factor/component
• Want these to be related at least .3
• Remember that r = .3 is a medium effect size that is ~10% variance
• Can eliminate items that load poorly
• Difference here in scale development versus exploratory clustering

Run an EFA

EFA_fit <- fa(master, #data
              nfactors = 2, #number of factors
              rotate = "oblimin", #rotation
              fm = "ml") #math

Look at the results

EFA_fit

Item 23

EFA_fit2 <- fa(master[ , -23], #data
              nfactors = 2, #number of factors
              rotate = "oblimin", #rotation
              fm = "ml") #math

EFA_fit2

Plots of the results

• The built in plot functions from psych are fantastic!

 fa.plot(EFA_fit2, 
     labels = colnames(master[ , -23]))

Plots of the results

fa.diagram(EFA_fit2)

Adequate solution

• Fit indices: a measure of how well the model matches the data
• Goodness of fit statistics: measure the overlap between the reproduced correlation matrix and the original, want high numbers close to 1
• Badness of fit statistics (residual): measure the mismatch, want low numbers close to zero
• Theory/interpretability
• Reliability

Fit statistics

EFA_fit2$rms #Root mean square of the residuals
EFA_fit2$RMSEA #root mean squared error of approximation
EFA_fit2$TLI #tucker lewis index
1 - ((EFA_fit2$STATISTIC-EFA_fit2$dof)/
       (EFA_fit2$null.chisq-EFA_fit2$null.dof)) #CFI 

Reliability

factor1 = c(1:7, 9:10, 12:16, 18:22)
factor2 = c(8, 11, 17)
##we use the psych::alpha to make sure that R knows we want the alpha function from the psych package.
##ggplot2 has an alpha function and if we have them both open at the same time
##you will sometimes get a color error without this :: information. 
psych::alpha(master[, factor1], check.keys = T)
psych::alpha(master[, factor2], check.keys = T)

Interpretation

Factor 1:

1. Statistics makes me cry
2. My friends will think I'm stupid for not being able to cope with R
3. Standard deviations excite me
4. I dream that Pearson is attacking me with correlation coefficients
5. I don't understand statistics
6. I have little experience of computers
7. All computers hate me
8. My friends are better at statistics than me
9. Computers are useful only for playing games
10. People try to tell you that R makes statistics easier to understand but it doesn't
11. I worry that I will cause irreparable damage because of my incompetence with computers
12. Computers have minds of their own and deliberately go wrong whenever I use them
13. Computers are out to get me
14. I weep openly at the mention of central tendency
15. R always crashes when I try to use it
16. Everybody looks at me when I use R
17. I can't sleep for thoughts of eigenvectors
18. I wake up under my duvet thinking that I am trapped under a normal distribution
19. My friends are better at R than I am

Factor 2:

1. I have never been good at mathematics
2. I did badly at mathematics at school
3. I slip into a coma whenever I see an equation

Bad:

1. If I'm good at statistics my friends will think I'm a nerd

Wrapping Up

• You've learned about exploratory factor analysis, which we will revisit when we cover confirmatory factor analysis
• You learned how to examine for the number of possible latent variables
• You learned how to determine simple structure
• You learned how to determine if that simple structure was an adequate model

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