Determining dimensionality

Factor Analysis in R

Jennifer Brussow

Psychometrician

How many dimensions does your data have?

dimensions

The bfi dataset

Big Five Inventory
2,800 subjects
25 questions
Data collected from the Synthetic Aperture Personality Assessment (SAPA)

bfi_structure

1 = Very Inaccurate ... 6 = Very Accurate

head(bfi)

      A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 E1 E2 E3 E4 E5 N1 N2 N3 N4 N5 O1  ...
61617  2  4  3  4  4  2  3  3  4  4  3  3  3  4  4  3  4  2  2  3  3  ...
61618  2  4  5  2  5  5  4  4  3  4  1  1  6  4  3  3  3  3  5  5  4  ...
61620  5  4  5  4  4  4  5  4  2  5  2  4  4  4  5  4  5  4  2  3  4  ...
61621  4  4  6  5  5  4  4  3  5  5  5  3  4  4  4  2  5  2  4  1  3  ...
61622  2  3  3  4  5  4  4  5  3  2  2  2  5  4  5  2  3  4  4  3  3  ...
61623  6  6  5  6  5  6  6  6  1  3  2  1  6  5  6  3  5  2  2  3  4  ...

names(bfi)

"A1" "A2" "A3" "A4" "A5" "C1" "C2" "C3" "C4" "C5" "E1" "E2" 
"E3" "E4" "E5" "N1" "N2" "N3" "N4" "N5" "O1" "O2" "O3" "O4" "O5"

Setup: split your dataset

# Establish two sets of indices to split the dataset
N <- nrow(bfi)
indices <- seq(1, N)
indices_EFA <- sample(indices, floor((.5*N)))
indices_CFA <- indices[!(indices %in% indices_EFA)]
# Use those indices to split the dataset into halves for your EFA and CFA
bfi_EFA <- bfi[indices_EFA, ]
bfi_CFA <- bfi[indices_CFA, ]

Setup: split your dataset

head(bfi_EFA, 2)

      A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 E1 E2 E3 E4 E5 N1 N2 N3 N4 N5 O1  ...
65237  3  4  4  4  4  4  4  5  2  3  3  4 NA  4  4  4  3  1  3  2  4  ...
61825  3  1  2  2  2  2  1  2  6  6  6  6  1  1  1  3  5  4  4  4  5  ...

head(bfi_CFA, 2)

      A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 E1 E2 E3 E4 E5 N1 N2 N3 N4 N5 O1  ...
61617  2  4  3  4  4  2  3  3  4  4  3  3  3  4  4  3  4  2  2  3  3  ...
61621  4  4  6  5  5  4  4  3  5  5  5  3  4  4  4  2  5  2  4  1  3  ...
...

An empirical approach to dimensionality

Imagine we have no theory...

no_theory

Without theory, use an empirical approach: Eigenvalues

Calculate the correlation matrix

# Calculate the correlation matrix first
bfi_EFA_cor <- cor(bfi_EFA, use = "pairwise.complete.obs")

           A1          A2          A3          A4          A5         C1 ...
A1  1.00000000 -0.31920397 -0.25651343 -0.12441523 -0.20083692  0.058252 
A2 -0.31920397  1.00000000  0.46698961  0.30599175  0.36599749  0.075002 
A3 -0.25651343  0.46698961  1.00000000  0.32762347  0.47616038  0.089720 
A4 -0.12441523  0.30599175  0.32762347  1.00000000  0.27182236  0.083987 
A5 -0.20083692  0.36599749  0.47616038  0.27182236  1.00000000  0.116890 
C1  0.05825219  0.07500228  0.08972097  0.08398741  0.11689059  1.000000 
C2  0.04236764  0.12843266  0.10471200  0.22697628  0.09639765  0.421518 
C3 -0.02289831  0.18618382  0.14009601  0.09975850  0.13797236  0.301556 
C4  0.09865372 -0.11178917 -0.11576273 -0.15035049 -0.10248897 -0.354081 
C5  0.04925038 -0.10820392 -0.15392300 -0.24998065 -0.15667123 -0.269701 
...

Eigenvalues

# Calculate the correlation matrix first
bfi_EFA_cor <- cor(bfi_EFA, use = "pairwise.complete.obs")

# Then use that correlation matrix to create the scree plot
scree(bfi_EFA_cor, factors = FALSE)

Scree plots

# Calculate the correlation matrix first
bfi_EFA_cor <- cor(bfi_EFA, use = "pairwise.complete.obs")

# Then use that correlation matrix to create the scree plot
scree(bfi_EFA_cor, factors = FALSE)

Scree_plot

Let's practice!

Factor Analysis in R