Factor Analysis Calculator

What is Factor Analysis?

Factor Analysis is a multivariate statistical technique that examines correlation patterns among observed variables to identify a smaller number of latent variables—called factors—that explain the shared variance in the data. It is widely used in psychology, education, marketing, and the social sciences for survey development, construct validation, and data reduction.

The technique originated in 1904 when Charles Spearman proposed a general intelligence factor (g) to explain positive correlations among mental tests. In the 1930s, Louis L. Thurstone advanced multiple factor analysis by introducing simple structure and rotation methods, laying the foundation for modern factor analysis. This calculator performs Exploratory Factor Analysis (EFA), which discovers latent structure without prior hypotheses, as opposed to Confirmatory Factor Analysis (CFA), which tests a pre-specified factor model.

Suitability Tests: KMO and Bartlett's Test

Before extracting factors, verify that your data is suitable. The Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy ranges from 0 to 1, with values ≥ .60 considered acceptable and ≥ .80 meritorious. Bartlett's test of sphericity tests whether the correlation matrix differs significantly from an identity matrix using a chi-square test. A significant result (p < .05) confirms that meaningful correlations exist among variables.

Extraction: PCA vs PAF

Principal Component Analysis (PCA) extracts components that explain total variance and is best for data reduction. Principal Axis Factoring (PAF) extracts factors explaining only shared variance and is preferred when seeking underlying latent constructs. PCA places 1.0 on the diagonal of the correlation matrix; PAF uses communality estimates instead.

Determining the Number of Factors

The Kaiser criterion retains factors with eigenvalues > 1. The scree plot identifies the "elbow" where eigenvalues level off. Parallel analysis compares actual eigenvalues to those from random data and is considered the most accurate method. Using multiple criteria together is recommended.

Rotation: Varimax vs Promax

Varimax (orthogonal) assumes uncorrelated factors and produces a single loading matrix—simple to interpret. Promax (oblique) allows factors to correlate and produces both a pattern matrix (unique contributions) and a structure matrix (total correlations). Use Varimax when factors are theoretically independent; use Promax when factor correlations are expected (common in social science). A factor inter-correlation ≥ .32 suggests oblique rotation is appropriate.

Interpreting Factor Analysis Results

Factor loadings represent the correlation between each variable and a factor; values ≥ .40 are meaningful and ≥ .70 are strong. Cross-loadings occur when a variable loads ≥ .32 on multiple factors, complicating interpretation—consider removing such items. Communalities indicate the proportion of each variable's variance explained by the retained factors; values below .20 suggest the variable does not fit the factor structure.

Reporting in APA Format

Common Mistakes to Avoid

• Insufficient sample size: A minimum of 5–10 observations per variable (N/p ≥ 5) or at least 100 total cases is recommended. Small samples produce unstable loadings.
• Using EFA when CFA is appropriate: If you have a theoretically established factor structure to confirm, use CFA (structural equation modeling), not EFA.
• Over-extracting factors: Relying solely on the Kaiser criterion can over-extract factors when many variables are present. Supplement with scree plot and parallel analysis.
• Ignoring cross-loadings: Variables loading ≥ .32 on multiple factors compromise the clarity of the solution. Consider item removal or rewording.
• Skipping KMO before analysis: A KMO below .50 means the correlation patterns are inadequate for factor extraction. Always check KMO and Bartlett's test first.