Week 1: Factor Analysis
Week One
Overview of the next 5 weeks of MV stats
- Factor analysis
- Structural Equation Modelling
- CFA
- Path Analysis
- Latent variable modeling
Introduces you to the idea of latent variables, the concept of a path diagram, and also how to do a factor analysis in R.
Show how the EFA needs to fix latent variances at 1 and covariances at 0, and fix the upper right triangle of whatever matrix contains the latent loadings.
Show what rotation does (R is great for that - just pull the loadings matrix and throw in at promax() or varimax().
Introduce indeterminacy by talking about how the rotations do not differ in fit.
How many factors?
Horn’s Parallel analysis
Rotation
We now now about loadings
Scores: Why scores are not just data %*% loadings
http://stackoverflow.com/questions/4145400/how-to-create-factors-from-factanal
Can we construct scores on our factors?
An intuition might be: Factors = data matrix * Loadings matrix.
- Thomson Regression method restriction that the scores are uncorrelated, centered around 0 and with variance = 1.
- ‘scale(m1) %% solve(cor(m1)) %% loadings(fa)’
- Bartlett’s method
- ML
v1 <- c(1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,5,6)
v2 <- c(1,2,1,1,1,1,2,1,2,1,3,4,3,3,3,4,6,5)
v3 <- c(3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,5,4,6)
v4 <- c(3,3,4,3,3,1,1,2,1,1,1,1,2,1,1,5,6,4)
v5 <- c(1,1,1,1,1,3,3,3,3,3,1,1,1,1,1,6,4,5)
v6 <- c(1,1,1,2,1,3,3,3,4,3,1,1,1,2,1,6,5,4)
m1 <- cbind(v1,v2,v3,v4,v5,v6)
fa <- factanal(m1, factors=3,scores="regression")
fa$scores # the correct solution
fac <- m1 %*% loadings(fa) # the answer on your question
> round(cor(fac),2)
Factor1 Factor2 Factor3
Factor1 1.00 0.79 0.81
Factor2 0.79 1.00 0.82
Factor3 0.81 0.82 1.00
> round(cor(fac2),2)
Factor1 Factor2 Factor3
Factor1 1 0 0
Factor2 0 1 0
Factor3 0 0 1
What happens with missing data
Next week: We will convert this model to CFA. Use a real data set so it has a purpose.
Refs
- http://www.psy.ed.ac.uk/people/iand/Bartholomew%20%282009%29%20Br%20J%20Math%20Stat%20Psychol%20factor%20scores%20Thomson%20Spearman%20Bartlett.pdf