1. Confirmatory Factor Analysis Using OpenMx & R
(CFA) Practical Session Timothy C. Bates

2. Confirmatory Factor Analysis to test theory
Are there 5 independent domains of personality?
Do domains have facets?
Is Locus of control the same as primary control?
Is prosociality one thing , or three?
Is Grit independent of Conscientiousness?

3. Prosocial Obligations
Prosociality covers several domains, e.g.
Workplace: voluntary overtime
Civic life: Giving evidence in court
Welfare of others: paying for other’s healthcare
CFA can test the fit of theories
Which hypothesised model fits best?
3 indicators of one factor?
3 correlated factors?
3 independent factors?

4. Factor structure of Prosociality
Distinct domains?
Work
Civic
Welfare
Work1
Work 2
Work 3
Civic 1
Civic 2
Civic 3
Welfare 1
Welfare 2
Welfare 3

5. Factor structure of Prosociality
One underlying prosociality factor?
Prosociality
Work1
Work 2
Work 3
Civic 1
Civic 2
Civic 3
Welfare 1
Welfare 2
Welfare 3

6. Factor structure of Prosociality
Hierarchical structure: Common and distinct factors?
Prosociality
Work
Civic
Welfare
Work1
Work 2
Work 3
Civic 1
Civic 2
Civic 3
Welfare 1
Welfare 2
Welfare 3

7. Note: These last two models are equivalent!
Work
Civic
Welfare
Work1
Work 2
Work 3
Civic 1
Civic 2
Civic 3
Welfare 1
Welfare 2
Welfare 3

8. CFA with R & OpenMx library(OpenMx) What is OpenMx? # library(sem)
“OpenMx is free and open source software for use with R that allows estimation of a wide variety of advanced multivariate statistical models.”
Homepage:
# library(sem)

9. Our data Dataset: Midlife Study of Well-Being in the US (Midus)
Prosocial obligations - c individuals
How much obligation do you feel…
to testify in court about an accident you witnessed?’ [civic]
to do more than most people would do on your kind of job?’ [work]
to pay more for your healthcare so that everyone had access to healthcare? [welfare]

10. Preparatory code First we need to load OpenMx
require(OpenMx)
Now lets get the data (R can read data off the web)
myData= read.csv("
? What are the names in the data?
summary(myData) # We have some NAs
myData= na.omit(myData) # imputation is another solution
summary(myData)

11. Core OpenMx functions mxModel() mxRun()
this will contain all the objects in our model
mxPath(): each path in our model
mxData(): the data for the model
mxRun()
Generate parameter estimates by sending the model to an optimizer
Returns a fitted model

12. CFA on our competing models
Three competing models to test today:
One general factor
Three uncorrelated factors
Three correlated factors

13. Model 1: One general factor
Prosociality
Work1
Work 2
Work 3
Civic 1
Civic 2
Civic 3
Welfare 1
Welfare 2
Welfare 3

14. Quite a bit more code than you have used before:
We’ll go through it step-by-step

15. Model 1 – Step 1 latents = "F1” manifests = names(myData) # Get set up
observedCov = cov(myData)
numSubjects = nrow(myData)
# Create a model using the mxModel function
cfa1<- mxModel("Common Factor Model”, type="RAM”,

16. Model 1 – Step 2
# Here we set the measured and latent variables manifestVars = manifests, latentVars = latents,

17. Model 1 – Step 3
# Create residual variance for manifest variables using mxPath mxPath( from=manifests, arrows=2, free=T, values=1, labels=paste("error”, 1:9, sep=“”) ), # using mxPath, fix the latent factor variance to 1 mxPath( from="F1", arrows=2, free=F, values=1, labels ="varF1”), # Using mxPath, specify the factor loadings mxPath(from="F1”, to=manifests, arrows=1, free=T, values=1, labels =paste(”l”, 1:9, sep=“”)),

18. Model 1 – Step 4
# Give mxData the covariance matrix of ‘data2’ for analysis mxData(observed=observedCov , type="cov", numObs=numSubjects ) # make sure your last statement DOESN’T have a comma after it ) # Close model

19. Model 1 – Run the model
# The mxRun function fits the model cfa1<- mxRun(cfa1) # Lets see the summary output summary(cfa1) # What is in a model?

21. Summary Output
# Ideally, chi-square should be non-significant chi-square: ; p: < .001 # Lower is better for AIC and BIC AIC (Mx): BIC (Mx): # < .06 is good fit: This model is a bad fit RMSEA: 0.15

22. Story so far…
Model 1 (one common factor) is a poor fit to the data)

23. Model 2 Work Civic Welfare Work1 Work 2 Work 3 Civic 1 Civic 2 Civic 3

24. Model 2 – Step 1
# Create a model using the mxModel function
cfa3<-mxModel("Three Distinct Factors Model Path Specification",
type="RAM",

25. Model 2 – Step 2
# Here we name the measured variables manifestVars=manifests, # Here we name the latent variable latentVars=c("F1","F2", "F3"),

26. Model 2 – Step 3
# Create residual variance for manifest variables mxPath(from=manifests, arrows=2, free=T,values=1, labels=c("error”, 1:9, sep=“”)),

27. Model 2 – Step 4
# Specify latent factor variances mxPath(from=c("F1","F2", "F3"), arrows=2,free=F, values=1, labels=c("varF1","varF2", "varF3") ),

28. Model 2 – Step 5
# Fix the covariance between latent factors to zero (i.e. specify the factors as uncorrelated) mxPath( from="F1", to="F2", arrows=2, free=FALSE, values=0, labels="cov1” ),

29. Model 2 – Step 6
mxPath(from="F1”,to="F3", arrows=2, free=F, values=0, labels="cov2"), mxPath(from="F2",to="F3", arrows=2, free=F, values=0, labels="cov3"),

30. Model 2 – Step 7
# Specify the factor loading i.e. here we specify that F1 loads on work1, 2, and 3 mxPath( from="F1", to=c("work1","work2","work3"), arrows=1, free=T, values=1, labels=c("l1","l2","l3") ), from="F2", to=c("civic1","civic2", "civic3"), labels=c("l4","l5", "l6")

31. Model 2 – Step 8
mxPath( from="F3", to=c("welfare1", "welfare2", "welfare3"), arrows=1, free=T, values=1, labels=c("l7", "l8", "l9") ), # Use the covariance matrix of ‘myData’ for analysis mxData(observed=cov(myData), type="cov", numObs=nrow(myData)) ) # Close the model

32. Model 2 – Run the model
# The mxRun function fits the model threeDistinctFactorFit <- mxRun(threeDistinctfactorModel) # The following line will deliver a lot of output # Lets see the summary output summary(threeDistinctFactorFit)

34. Summary Output
# Ideally, chi-square should be non-significant chi-square: ; p: <.0001 # Lower is better for AIC and BIC AIC (Mx): BIC (Mx): # <.08 is a reasonable fit: This model is a poor fit RMSEA: 0.16

35. Now your turn…
Install OpenMx repos <- c(' If you have your laptop: install.packages('OpenMx', repos=repos) lib <- 'C:/workspace’ install.packages('OpenMx', repos=repos, lib=lib) library(OpenMx, lib=lib)