Introduction to Exploratory and Confirmatory Factor Analysis (Using R)
Princeton University
2024-04-22
What is factor analysis?
Let’s say we have 6 items in a scale:
Sleep disturbances (insomnia/hypersomnia)
Suicidal ideation
Lack of interest in normally engaging activities
Racing thoughts
Constant worrying
Nausea
Some of these could cross-load
FA considers this and items load on all factors
Why?
Allows you to summarize complex data with a smaller set of representative variables
- 6 variables to 2 variables
Why?
Can help identify/confirm underlying constructs
- Depression and anxiety
Partitioning variance
Variance common to other variables
- Communality \(h^2\): proportion of each variable’s/item’s variance that can be explained by the factors
- How much an item is related to other items in the analysis
- Communality \(h^2\): proportion of each variable’s/item’s variance that can be explained by the factors
Variance specific to that variable (unique variance)
Random measurement error
Common factor analysis
Common 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)
- Explaining the maximum amount of common variance in a correlation matrix
- Attempts to achieve parsimony (data reduction) by:
Common factor analysis
Common factor analysis
Common factor analysis
PCA
- Based on total variance!
- Goal: Find fewest components that accounts for the most varaiance among variables
PCA vs. FA
Run factor analysis if you assume or wish to test a theoretical model of latent factors causing observed variables
Run PCA If you want to simply reduce your correlated observed variables to a smaller set of important independent composite variables
Big 5
2800 participants
25 self-report items from big 5 inventory
- The personality items are split into 5 categories
Data visualization
Note
Always include correlation table in factor analysis!
Fitting factor model: # of factors
Scree plot
A plot of the Eigenvalues in order from largest to smallest
Look for the elbow (shared variability starting to level off)
- Above the elbow is how many components you want
Fitting factor model: # of factors
Parallel analysis
Run simulations pulling eigenvalues from randomly generated datasets (with same sample size and number of variables)
If eigenvalues > eigenvalues from random datasets more likely to represent meaningful patterns in the data
Method agreement procedure
Uses many methods to determine how many factor you should get
- This is the approach I would use
Extracting factor loadings
Runs another factor analysis to get the loading for each of the factors
- Principal axis factoring (PAF)
- Get initial estimates of communalities
- Squared multiple correlations (highest absolute correlation)
- Take correlation matrix and replace diagonal elements with communalities (reduced matrix)
- Principal axis factoring (PAF)
Path diagram
Rotation
Rotation
- Orthogonal
- Oblique
Rotation
- After rotation
Factor scores
Geller, J., Thye, M., & Mirman, D. (2019). Estimating effects of graded white matter damage and binary tract disconnection on post-stroke language impairment. NeuroImage, 189. https://doi.org/10.1016/j.neuroimage.2019.01.020
Plotting factor analysis
# correlated rotation
efa_obs <- psych::fa(data, nfactors = 5, rotate="oblimin", fm="pa") %>%
model_parameters()
efa_plot <- as.data.frame(efa_obs) %>%
pivot_longer(PA2:PA4) %>%
dplyr::select(-Complexity, -Uniqueness) %>% rename("Loadings" = value, "Personality" = name)
#For each test, plot the loading as length and fill color of a bar
# note that the length will be the absolute value of the loading but the
# fill color will be the signed value, more on this below
efa_fact_plot <- ggplot(efa_plot, aes(Variable, abs(Loadings), fill=Loadings)) +
facet_wrap(~ Personality, nrow=1) + #place the factors in separate facets
geom_bar(stat="identity") + #make the bars
coord_flip() + #flip the axes so the test names can be horizontal
#define the fill color gradient: blue=positive, red=negative
scale_fill_gradient2(name = "Loading",
high = "blue", mid = "white", low = "red",
midpoint=0, guide=F) +
ylab("Loading Strength") + #improve y-axis label
theme_bw(base_size=22)
Information to include in paper
