factormodeling

Factormodeling is a statistical approach for explaining the correlations among a set of observed variables with a smaller number of unobserved latent variables, or factors. It is used to uncover underlying constructs that give rise to observed patterns in data and to reduce dimensionality while preserving the structure of relationships.

The standard latent variable formulation represents an observed p-dimensional vector x as x = Lambda f + e, where Lambda is a p by k matrix of factor loadings, f is a k-dimensional vector of latent factors, and e is a vector of unique, observed-specific errors. In matrix terms, the model implies a covariance structure Sigma = Lambda Phi Lambda^T + Psi, where Phi is the covariance of the factors and Psi is a diagonal matrix of unique variances. Assumptions typically include linear relationships, normally distributed errors, and, depending on the method, specific forms for Phi and Psi.

Two main branches are exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). EFA seeks a factor structure without strong a priori constraints, often requiring rotation to achieve interpretability. CFA tests a hypothesized structure with prescribed zero loadings and, in many cases, specified factor correlations. Estimation methods include maximum likelihood, principal axis factoring, and semi-parametric or Bayesian approaches; model fit is assessed with statistics such as RMSEA, CFI, and SRMR.

Applications span psychology, education, finance, and marketing, where factor models help identify latent constructs (for example, intelligence, mood, or risk factors) and reduce data complexity. Typical challenges include model identifiability, sensitivity to specification, and sample size requirements.