Exploratory Factor Analysis (EFA)

Factor Analysis: Two Types
Exploratory Factor Analysis (EFA)
    • Explores new possible factorization
    • Theory discovery
    • Determine the number of factors
    • Determine factor correlation structure
    • Loadings may include any candidate variable

Confirmatory Factor Analysis (CFA)

    • Uses a priori factorization
    • Theory substantiating
    • A priori factors
    • Factor correlation structure established
    • Loadings variables established
    • Our interest is with EFA
Exploratory Factor Analysis (EFA or just FA)
Regression model linking observed variables to latent variables
Data collected on many correlated variables from one sample of subjects
Are there a small number of “”latent variables” (i.e. factors) that explain much of the correlations among a set of observed variables?
Purpose: attempt to study hypothetical, unobservable variables, i.e., Factors, latent variables, latent traits, constructs
E.g. intelligence, motivation, political \& social conservatism
Method: analyze the correlational structure of observed variables
FA: Example Correlation Matrix (Gorsuch, 1983)
\begin{align*}
&\hskip 10em \begin{matrix} 1 & \hskip 1em 2 & \hskip 2em 3 & \hskip 2em 4 & \hskip 2em 5 & \hskip 0.5em 6\\
\end{matrix}\\
&\begin{matrix} 1 & \text{Information}\\
2 & \text{Verbal Ability}\\
3 & \text{Verbal Analogies}\\
4 & \text{Ego Strength}\\
5 & \text{Guilt Proneness}\\
6 & \text{Tension}\\
\end{matrix}
\begin{bmatrix} 1 & & & & & & \\
0.67 & 1 & & & & \\
0.43 & 0.49 & 1 & & & \\
0.11 & 0.12 & 0.03 & 1 & & \\
-0.07 & -0.05 & -0.14 & -0.41 & 1 \\
-0.17 & -0.14 & -0.10 & -0.48 & 0.40 & 1 \\
\end{bmatrix}
\end{align*}
Notice the two sets of highly correlated variables
Conjecture:
    • The first set (var’s 1-3) can be explained by a single common factor that might be named “”verbal comprehension”
    • The second set (4-6) can be explained by a second common factor that could be named “”emotionality” or “anxiety”
FA: Number of Factors
Number of factors critical
k factors can be very different from k+1 factors
    • Too few factors \Rightarrow too high loadings
    • Too many factors \Rightarrow fragmented and uninterpretable factors
Skree plot
Sequential factor construction common
Chi-square statistic (ML estimation)
FA: Factor Rotation
No unique solution for the factor loading matrix
Introduce constraints on factor model
    • Require a diagonal matrix $\Rightarrow$ descending order of magnitude and factors in descending order of importance
    • Assures orthogonal factors
    • Solution unique but may be difficult to interpret

Factor rotation can simplify interpretation without altering the underlying mathematical properties

FA: Factor Rotation
Rotation improves by
    • Each variable is highly loaded on at most 1 factor
    • All factor loadings are either large and positive, or near zero with few intermediate values
    • Variables fall into mutually exclusive groups whose loadings are high on single factors, moderate to low on a few factors, near zero on remaining factors
Initial factoring generated
Rotation transformation is a rigid rotation of axes
    • Orthogonal rotation restricts the rotated factors to be uncorrelated (generalizable results)
    • Oblique rotation allows for correlated factors (best fit to data)
    • Varimax rotation have few large loadings and others near zero (orthogonal)
    • Promax rotation integer power on loadings (orthogonal)
    • Several other rotation types
FA Interpretation
Example: Do thermal inertia, location, crater diameter, dust coverage, number of crater layers, and type of crater layer suggest whether volatiles are entrained in the surface, mixed in the atmosphere, or both?
Two factors
    • Volatiles in surface and near-surface Mars
    • Volatiles in the atmosphere
Do the factors show volatiles location?