Factor Analysis: Discovering Latent Structure

Factor analysis is a statistical technique for identifying latent (unobserved) constructs that explain the correlations among a set of observed variables (items or measures). The core idea: if many observed variables correlate highly with each other, they may all be manifestations of a smaller number of underlying factors. In psychology, 50 personality items might cluster into 5 underlying factors (the Big Five: Openness, Conscientiousness, Extraversion, Agreeableness, Neuroticism). Exploratory Factor Analysis (EFA) is data-driven — it discovers the factor structure empirically from the data without specifying it in advance. Key concepts: Factor loadings are the correlations between each observed variable and each factor (standardized regression coefficients of the factor on the variable). Loadings range from −1 to +1; loadings above |0.40| are typically considered meaningful. Communality (h²) for each item: the proportion of that item's variance explained by all extracted factors combined. A communality of 0.65 means 65% of the item's variance is shared with the factor solution. Unique variance (1 − h²) is specific to that item plus measurement error. Number of factors to retain: eigenvalues (>1 rule — retain factors with eigenvalue > 1), scree plot (retain factors before the 'elbow'), parallel analysis (gold standard — compare observed eigenvalues to eigenvalues from random data). Rotation: Varimax (orthogonal — factors uncorrelated, simple structure) or Oblimin/Promax (oblique — factors allowed to correlate, more realistic for psychological constructs).