Multi-set factor analysis by means of Parafac2

Area of Science:

• Multivariate statistics
• Psychometrics
• Data analysis

Background:

• Analyzing multi-set data with K samples and J variables presents challenges in modeling covariance structures.
• Existing methods may not adequately capture commonalities and unique variances across different datasets.

Purpose of the Study:

• To introduce a novel factor model for multi-set covariance matrices (Σk).
• To model the common part using the Parafac2 (Parallel Factor Analysis 2) model and unique variances (Uk) as diagonal matrices.
• To ensure proper estimation and interpretation of factor analysis components.

Main Methods:

• The proposed model incorporates Parafac2 for common factor modeling and minimum rank factor analysis for estimating unique variances (Uk) for each sample k.
• Factors can be specified as orthogonal or oblique.
• A new algorithm is developed for estimating the Parafac2 component.

Main Results:

• The model successfully estimates unique variances, a common factor correlation matrix, and communalities, guaranteeing proper solutions.
• A percentage of explained common variance can be computed for each sample.
• The Parafac2 component demonstrates rotational uniqueness under specific conditions.

Conclusions:

• The developed factor model is easy to estimate and interpret for multi-set data.
• The approach provides a robust framework for analyzing covariance structures across multiple samples.
• The method ensures statistically sound and interpretable factor analysis results.