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A GENERALIZATION OF VECTOR CORRELATION AND ITS RELATION TO CANONICAL CORRELATION
Multivariate Behavioral Research
|January 26, 2016
Summary
This study links multivariate linear regression and canonical correlation. It shows vector correlation is the product of canonical correlations, offering a significance test for relationships between variable sets.
Area of Science:
- Multivariate statistics
- Linear regression analysis
- Correlation analysis
Background:
- Chow (1966) established that least-squares estimates in multivariate linear regression maximize squared vector correlation.
- Existing methods for assessing relationships between sets of variables have limitations.
Purpose of the Study:
- To demonstrate the close relationship between multivariate linear regression, vector correlation, and canonical correlation.
- To introduce a symmetric generalization of vector correlation applicable to matrices with varying numbers of variables and linear dependencies.
- To provide a significance test for vector correlation and related measures.
Main Methods:
- Analysis of the relationship between least-squares estimates in multivariate linear regression and vector correlation.
- Derivation of vector correlation as a product of canonical correlations.
- Development of a symmetric generalization of vector correlation.
Main Results:
- The vector correlation is shown to be the product of canonical correlations between independent and dependent variables.
- A symmetric generalization of vector correlation is presented, handling matrices with different variable counts and linear dependencies.
- This generalization is linked to canonical correlation and Rozeboom's (1965) measure of correlation between variable sets.
Conclusions:
- Vector correlation is fundamentally linked to canonical correlation.
- The proposed generalization provides a unified framework for understanding correlations between sets of variables.
- The findings offer a significance test for vector correlation, enhancing its utility in assessing linear relationships between variable sets.
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