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CANONICAL ANALYSIS
David J. Krus
Arizona State University

Canonical (from Gk.  bar, measuring rod, ruler) analysis belongs to the family of regression methods for data analysis. Regression analysis quantifies a relationship between a predictor variable and a criterion variable by the coefficient of correlation , coefficient of determination , and the standard regression coefficient . Multiple regression analysis expresses a relationship between a set of predictor variables and a single criterion variable by the multiple correlation , multiple coefficient of determination , and a set of standard partial regression weights  etc. Canonical variate analysis captures a relationship between a set of predictor variables and a set of criterion variables by the canonical correlations  etc. canonical coefficients of determination  etc., and by the sets of canonical weights C and D.

Canonical Analysis

Canonical analysis belongs to a group of methods which involve solving the characteristic equation for its latent roots and vectors. It describes formal structures in hyperspace, invariant with respect to the rotation of their coordinates. In this type of solution, rotation leaves many optimizing properties preserved, provided it takes place in certain ways and in a subspace of its corresponding hyperspace. This rotation from the maximum intervariate correlation structure into a different, simpler and more meaningful structure increases the interpretability of the canonical weights C and D. In this the canonical analysis differs from the Harold Hotelling’s (1936) canonical variate analysis (also called the canonical correlation analysis]) designed to obtain the maximum (canonical) correlations between the sets of predictor and criterion canonical variates. The difference between the canonical variate analysis and canonical analysis is analogous to the difference between the principal components analysis and factor analysis, each with its characteristic set of communalities, eigenvalues, and eigenvectors.

References

Cliff, N., & Krus, D. J. (1976) Interpretation of canonical variate analysis: rotated vs. unrotated solutions. Psychometrika, 41, 1, 35-42.

Hotelling, H. (1936) Relations between two sets of variates. Biometrika, 28, 321-377

Krus, D.J., et al. (1976) Rotation in canonical analysis. Educational and Psychological Measurement, 36, 725-730.

Liang, K. H., Krus, D. J., & Webb, J. M. (1995) K-fold cross-validation in canonical analysis. Multivariate Behavioral Research, 30, 539-545.

Note: The author would like to thank Michael Hardy for helpful editing of this encyclopedia entry.