22. FTFC β€” Find Two-Factor Clusters (Sextad-Based)

Type: Latent cluster discovery
Output: Clusters of variables consistent with a latent factor having three observed indicators

FTFC extends FOFC by searching for clusters that satisfy sextad constraints instead of tetrads.
Where FOFC detects one-factor clusters using 2-by-2 covariance minors, FTFC targets three-indicator latent factors, using algebraic constraints that uniquely characterize a latent variable with exactly three children.

This makes FTFC useful when each latent factor is expected to have at least three indicators, or when tetrad structure alone is insufficient to separate overlapping factors.

22.1. Key Idea

A latent factor with three observed indicators satisfies a special set of algebraic equalities called sextads. Each sextad involves six variables arranged as three pairs. FTFC searches for such clusters by:

  1. Computing the correlation matrix of the observed variables.

  2. Testing whether all sextad constraints for a candidate 3-by-3 set of variables hold at significance level alpha.

  3. Applying a substitution test analogous to the one in FOFC:

    • Replacing any variable in the sextad with any outside variable should break the sextad equalities.

    • This ensures the cluster is pure (generated by one latent factor) rather than mixed.

  4. Growing the cluster by testing whether additional variables also satisfy the sextad constraints relative to the current cluster.

  5. Returning all valid sets as clusters with rank 2 (a latent with three indicators).

The test is based on rank constraints and vanishing minors rather than explicit SEM fitting.

22.2. Relation to FOFC and GFFC

  • FOFC finds one-factor clusters using tetrads (2-by-2 minors).

  • FTFC finds three-indicator clusters using sextads (3-by-3 minors).

  • GFFC (Generalized FOFC/FTFC) runs both tetrad- and sextad-based searches and integrates them.

FTFC is most effective when FOFC cannot separate latent factors because two-indicator structure is insufficient.

22.3. When to Use FTFC

  • When each latent factor is expected to have three or more indicators.

  • When tetrad-based tests alone do not recover pure clusters.

  • As part of a multi-stage pipeline for discovering measurement models.

  • When following the recommendations in Kummerfeld & Ramsey (2016) for latent factor recovery.

22.4. Strengths

  • Detects clusters defined by richer algebraic structure than FOFC.

  • More robust when tetrads alone are insufficient to distinguish latent factors.

  • Complements FOFC; often run together within GFFC.

22.5. Limitations

  • Requires larger sample size than tetrad-based clustering.

  • Pure sextad constraints are stricter; weak indicators may lead to missed clusters.

  • Only identifies single-factor groups; more complex factor structures require GFFC, BPC, or MimBuild methods.

22.6. Parameters in Tetrad

Parameter (camelCase)

Description

alpha

Significance level for sextad tests.

ess

Equivalent sample size used in rank/sextad calculations.

verbose

Whether to print detailed information during clustering.

22.7. Reference

FTFC is a generalization of FOFC in:

Kummerfeld, E., & Ramsey, J. (2016).
Causal clustering for 1-factor measurement models. Proceedings of KDD.

22.8. Summary

FTFC finds clusters corresponding to latent variables with at least three observed children by testing whether sextad constraints hold. It extends FOFC to richer measurement structures and forms the foundation of the GFFC latent clustering method.