Chapter 3 Canonical Correlation Analysis (CCA)

3.1 What is CCA?

Seeks the weighted linear composit for each variate (sets of D.V. or I.V.) to maximize the overlap in their distributions.
Labeling of DV and IV is arbitrary. The procedure looks for relationships and not causation.
Goal is to maximize the correlation (not the variance extracted as in most other techniques).
Lacks specificity in interpreting results, that may limit its usefulness in many situations.

CCA helps us answer the questions:

What is the best way to understand how the variables in two sets are related? , mathematically speaking:
- what linear combinations of the set \(X\) variables (\(u\)) and the set Y variables (\(t\)) will maximize their correlation?

It represents the overlapping variance between two variates which are linear composites of each set of variables.

Multiple continuous variables for DVs and IVs or categorical with dummy coding.
Assumes linear relationship between any two variables and between variates.
Multivariate normality is necessary to perform CCA.
Multicollinearity in either variate confounds interpretation of canonical results.

Determine the magnitude of the relationships that may existe between two sets of variables.
Derive a variate(s) for each set of criterion and predictor variables such that the variate(s) of each set is maximally correlated.
Explain the nature of whatever relationships exist between the sets of criterion and predictor variables.
Seek the max correlation of shared variance between the two sides of the equation.

Canonical correlation: Correlation between two sets; the largest possible correlation that can be found between linear combinations.
Canonical variate: The linear combinations created from the IV set and DV set.
Canonical weights: weights used to create the liniear combinations; interpreted like regression coefficients.
Canonical loadings: correlations between each variable and its variate; interpreted like loadings in PCA.
Canonical cross-loadings: Correlation of each observed independent or dependent variable with opposite canonical variate.

Canonical weights
- larger wight contributes more to the function.
- negative weight indicates an inverse relationship with other variables.
- always look out for multicollinearity, it can skew the whole analysis.
Canonical Loadings.
- A direct assessment of each variable´s contribution to its respective canonical variate.
- Larger loadings are interpreted as more important to deriving the canonical variate.
- Correlation between the original variable and its canonoical variate.
Canonical Cross-Loadings
- Measure of correlation of each original D.V. with the independent canonical variate.
- Direct assessment of the relationship between each D.V. and the independent variate.
- Provides a more pure measure of the dependent and independent variable relationship.
- Preferred approach to interpretation.

Small samples sizes may have an adverse effect.
Suggested minimun sample size = 10 * # of values.
Selection of variables to be included:
Select them with domain knowledge or theoretical basis.
Inclusion of irrelevant or deletion of relevant variables may adversely affect the entire canonical solution.
All I.V.s must be interrelated and all D.V.s must be interrelated.
Composition of D.V. and I.V. variates is critical to producing practical results.

\(R_c\) (canonical R) reflects only the variance shared by the linear composites, not the variances extracted from the variables.
Canonical weights are subject to a great deal of instability, particularly when there is multicollinearity.
Interpretation difficult because rotation is not possible.
Precise statistics have not been developed to interpret canonical analysis.

This chapter is under construction.