Four Covariance Structure Models for Canonical Correlation Analysis: A COSAN Modeling Approach

Area of Science:

• Statistics
• Psychometrics
• Multivariate Analysis

Background:

• Canonical Correlation Analysis (CCA) and Covariance Structure Analysis (CSA) have a known mathematical link.
• The Multiple Indicators Multiple Causes (MIMIC) approach was used to connect CCA and CSA but presents technical and practical challenges.

Purpose of the Study:

• To propose a comprehensive COmprehensive Structure ANalysis (COSAN) modeling approach to address the limitations of the MIMIC approach in connecting CCA and CSA.
• To define four COSAN-CCA models based on data (unstandardized/standardized variables) and parameter estimation (weights/loadings).

Main Methods:

• Developed four COSAN-CCA models, estimating unique parameters (weights or loadings) and common parameters (canonical correlations).
• Compared standard error estimates from MIMIC and COSAN-CCA models using two numeric examples.
• Validated COSAN-CCA standard error estimates through simulation studies and asymptotic theory.

Main Results:

• The four COSAN-CCA models provide correct point and standard error estimates for commonly used CCA parameters.
• COSAN modeling approach yields more reliable standard error estimates compared to the MIMIC approach, as confirmed by numerical examples and simulations.
• The proposed COSAN-CCA models are mathematically sound and practically applicable.

Conclusions:

• The COSAN modeling approach offers a robust and accurate alternative to the MIMIC approach for integrating CCA and CSA.
• The developed COSAN-CCA models enhance the estimation of CCA parameters, providing validated standard errors.
• Future research can explore software implementation and further extensions of the COSAN-CCA framework.