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Canonical Correlation

Canonical correlation is a statistical technique that is used to example the degree of the relationship between two canonical (Latent) variables. While in regression analysis, we can find the relationship between one dependent variable and many independent variables, canonical correlation, examples the relationship between many dependent variables and many independent variables. In canonical correlation, we make one variate from the many independent variables and one variate from the many dependent variables. Then, we compare those variates to find the degree of relationship between all variables. Wilks's lambda is used to test the significance of canonical correlation. Like simple correlation, canonical correlation coefficient square gives the percentages of variance that can be explained in the dependent variable by using the independent variable.

Key concepts and terms:

Canonical variable or variates: A canonical variable is the linear combination of a set of original variables in which correlation has been controlled. Canonical variate is the composite score in which common variance is removed. There are two variates for every canonical correlation.
 
Canonical correlation: Canonical correlation simply measures the degree of relationship between two variates(also called characteristic roots). In factor analysis, there may be more than two characteristic roots. One of these is the first characteristic root and it explains most of the relationship. Canonical correlation coefficients can be interpreted as a simple correlation coefficient. R-square in canonical correlation is the percentage of variance explained by the variate on a particular dimension. Eigenvalues also shows the variance explained by the variate. First, Eigenvalues shows the maximum variance explained by the variate, and second, shows the second variance explained by the variate. 
 
Pooled R-square: This is the sum of the square of all canonical correlation. Pooled r-square is used to know how one set of variables can predict the other set of variables.
 
Canonical Weight: Canonical weight is the linear equation coefficient which shows the importance of a single variable loading in a variate. There will be one canonical weight variable for each variable.
 
Significance test: Similar to a MANOVA test, Wilks’s Lambda is used in canonical correlation to test for statistical significance. If the probability value is less than .05, than it signifies a significant relationship between variates, otherwise the relationship will not be statistically related. The degree of freedom in canonical correlation is equal to P*Q where P equals the variable in the first variate and Q equals the variable in the second variate. Wilks’s lambda test is used for significance of the first canonical correlation, not for the second and other correlation. Bartlett's V test is used for the second and other canonical correlation. The likelihood ratio test is used for the overall significance of the two variables.

Assumptions:

Data: Interval data is preferred for canonical correlation, and there should be no missing values in the data.

Linearity: Linear relationship is assumed between the dependent and independent variates.

Multicollinearity: There should be no perfect multicollinearity between variables. Low multicollinearity may be considered, but perfect multicollinearity may cause problems with the results.

Homoscedasticity and correlation: Homogeneity and correlation assumptions are assumed in canonical correlation.

No Outlier: Canonical Correlation is sensitive to outliers.  Outliers are scores that fall outside of three (3) standard deviations. 

Sample Size: Stevens (1986) recommends that at least 20 cases per variable should be in a sample for the first canonical correlation. If we want to interpret two canonical correlations, then 40 to 60 cases per variable should be in a sample size.   

 
Canonical Correlation Resources
Alpert, M. I., & Peterson, R. A. (1972). On the interpretation of canonical analysis. Journal of Marketing Research, 9(2), 187-192.
Barcikowski, R. S., & Stevens, J. P. (1975). A Monte Carlo study of the stability of canonical correlations, canonical weights and canonical variate-variable correlations. Multivariate Behavioral Research, 10(3), 353-364.
Cliff, N., & Krus, D. J. (1976). Interpretation of canonical analysis: Rotated vs. unrotated solutions. Psychometrika, 41(1), 35-42.
Dunlap, W. P., Brody, C. J., & Greer, T. (2000). Canonical correlation and chi-square: Relationships and interpretation. The Journal of General Psychology, 127(4), 341-353.
Dunteman, G. H. (1989). Principal components analysis. Newbury Park, CA: Sage Publications.
Gifi, A. (1990). Non-linear multivariate analysis. New York: John Wiley & Sons.
Green, P. E., Halbert, M. H., & Robinson, P. J. (1966). Canonical analysis: An exposition and illustrative application. Journal of Marketing Research, 3, 32-39.
Levine, M. S. (1977). Canonical analysis and factor comparison. Newbury Park, CA: Sage Publications.
Lutz, J. G., & Eckert, T. L. (1994). The relationship between canonical correlation analysis and multivariate multiple regression. Educational and Psychological Measurement, 54(3), 666-675.
Stevens, J. (1986). Applied multivariate statistics for the social sciences. Hillsdale, NJ: Lawrence Erlbaum Associates.
Takane, Y., & Hwang, H. (2002). Generalized constrained canonical correlation analysis. Multivariate Behavioral Research, 37(2), 163-195.
Taskinen, S., Croux, C., Kankainen, A., Ollila, E., & Oja, H. (2006). Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices. Journal of Multivariate Analysis, 97(2), 359-384.
Thompson, B. (1984). Canonical correlation analysis: Uses and interpretation. Thousand Oaks, CA: Sage Publications.

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