12. Factor Analysis

Category: Latent Structure / Measurement Models
Type: Classical factor analysis (linear-Gaussian)

Factor Analysis is the standard statistical method for learning continuous measurement models. It assumes that each observed variable is a linear combination of one or more latent factors plus independent Gaussian noise. Tetrad provides a lightweight wrapper so that factor-analysis results can be integrated into causal discovery workflows.

12.1. Purpose

Use FactorAnalysis when you want a traditional measurement model for forming latent variables prior to structural modeling.

It estimates:

  • latent factors,

  • loadings of indicators on each factor,

  • unique variances,

  • and covariances among factors.

The resulting latent variables can then be handed off to FGES, PC, GFCI, or any other Tetrad algorithm.

12.2. When to Use

  • You want a classical factor model (linear, continuous, Gaussian).

  • You believe variables load cleanly onto latent factors.

  • You need latent scores for downstream causal modeling.

  • You prefer parametric estimation over trek-separation clustering.

12.3. How It Works (Conceptual)

  1. Compute the covariance matrix of the observed variables.

  2. Estimate loadings that best explain this covariance using linear factor decomposition.

  3. Determine factor scores (latent-variable estimates) via regression or ML.

  4. Return a measurement model where latent variables are linked to their indicators.

12.4. Strengths

  • Widely used and well-understood.

  • Produces interpretable loadings and factor scores.

  • Integrates naturally with SEM and MIM-building workflows.

12.5. Limitations

  • Assumes linearity and Gaussian noise.

  • Sensitive to model misspecification.

  • Does not identify clusters automatically—you must choose the number of factors.

12.5.1. Parameters

Parameter (camelCase)

Description

fa_threshold

Non-negative double. Threshold used when deciding which factor loadings are considered “large enough” to be substantive. Typical values are small (e.g., 0.1–0.4).

numFactors

Integer ≥ 1. The number of latent factors to extract. If set incorrectly, factors may be under- or over-extracted.

useVarimax

Boolean. If true, applies Varimax rotation to the loading matrix, producing a more interpretable factor structure. If false, uses the raw (unrotated) solution.

convergenceThreshold

Small positive double. Iterative fitting stops when the change in log-likelihood or parameter estimates falls below this threshold. Typical values: 1e-4 to 1e-8.

verbose

Boolean. If true, prints detailed progress messages during factor extraction and rotation.

12.6. Relation to Other Latent Tools

  • For pure-cluster discovery, use TSC, FOFC, FTFC, GFFC, or BPC.

  • For automated latent construction after clustering, see MimbuildBollen or MimbuildPca.

  • For non-Gaussian or nonlinear latent orientation, see GIN.

12.7. References

  • Classical SEM / factor-analysis literature (Harman 1976; Bollen 1989).

12.8. Summary

Factor Analysis extracts a classical linear-Gaussian measurement model in which observed variables load on a small number of latent factors. It provides interpretable loadings, latent scores, and unique variances, and integrates smoothly with downstream causal discovery (e.g., FGES, PC, GFCI, MIM-building). It is best suited to continuous data with approximately linear structure and is limited when indicators exhibit strong nonlinearity, non-Gaussianity, or complex latent clustering patterns.