Three-mode factor analysis by means of Candecomp/Parafac

Area of Science:

• Multivariate statistics
• Psychometrics
• Data analysis

Background:

• Three-mode covariance matrices capture complex relationships across observations, variables, and conditions.
• Existing models may lack interpretability or proper estimation guarantees for unique variances.

Purpose of the Study:

• To develop a novel, interpretable model for three-mode covariance matrices.
• To ensure proper estimation of unique variances and factor covariance.
• To provide a method for calculating explained common variance.

Main Methods:

• Modeling a three-mode covariance matrix as a sum of a Candecomp/Parafac component and unique variances.
• Estimating unique variances using Minimum Rank Factor Analysis.
• Allowing for oblique or orthogonal factors.

Main Results:

• The proposed model is easy to estimate and interpret.
• Guaranteed proper estimation of unique variances, factor covariance matrix, and communalities.
• Obtained a percentage of explained common variance for each variable-condition combination.
• Demonstrated rotational uniqueness under mild conditions.

Conclusions:

• The new approach offers a robust and interpretable method for analyzing three-mode covariance data.
• The model's properties facilitate deeper understanding of complex data structures in various fields.
• Applicable to existing datasets and validated through simulation studies.