Glossary
Factor Analysis
What is Factor Analysis?
Factor analysis models relationships among observed variables using a smaller set of unobserved factors. In a common-factor model, variation shared across measures is separated from variation specific to each measure. Researchers use it to investigate the structure of questionnaires and other measurements.
How does factor analysis work?
Suppose several questionnaire items tend to receive similar responses. Factor analysis asks whether a smaller number of dimensions can account for those relationships. The researcher chooses the measures, estimates a model, decides how many factors to retain and interprets the resulting pattern.
- Factor loadings describe relationships between measures and factors.
- Communality concerns how much of a measure's variance the common factors explain;
- uniqueness is the part specific to that measure.
The labels given to factors require interpretation, not just a calculation.
Exploratory vs. confirmatory factor analysis
- Exploratory factor analysis investigates possible structure.
- Confirmatory factor analysis evaluates a structure specified in advance.
Rotation can make exploratory results easier to interpret: orthogonal solutions constrain factors to be uncorrelated, while oblique solutions allow correlations.
Factor analysis vs. PCA
Principal component analysis summarizes total variance using components. Common-factor analysis models shared variation. The methods are related, but PCA is not an interchangeable factor-extraction method.
Methods, factor retention and assumptions
Principal axis factoring and maximum likelihood are two estimation methods. A scree plot displays ordered eigenvalues to help assess dimensionality; parallel analysis compares observed eigenvalues with those from random data. Neither a cutoff nor a plot replaces examining whether a solution makes sense.
The usual common-factor model expresses observed measures as linear combinations of factors plus residual variation. Conventional maximum-likelihood estimation assumes multivariate normality; assumptions vary with the estimator and data type. Michael Hallquist’s factor-model notes explain these methods and distinctions.
What are its limitations?
Results depend on variable selection, factor retention, estimation and rotation. Sample adequacy depends on features such as communalities and how well the factors are measured. A rule such as five observations per variable cannot guarantee a sound analysis.
Why it matters
Factor analysis helps test whether a set of questions measures the dimensions it is intended to measure. That matters in personality research, scale development and other work with complex constructs. An interpretable pattern is useful evidence about measurement; it is not, by itself, proof of a causal mechanism or a fully validated scale.