Chapter 10 Canonical Correlation Analysis

Introduction Canonical correlation analysis focuses on the correlation between a linear combination of the variables in one set and a linear combination of variables in another set Variables Data Linear combination

where Without loss of generality, we can always assume that

Problem (1) becomes By the Lagrange’s method, we need to maximize Differentiating G with respect to and we have

we have

From the second equation should be an eigenvalue of (3) and be the related eigenvector. should be an eigenvalue of (3) and be the related eigenvector. Similarly, we have

Let be non-zero eigenvalues of (3) or (4), be the associated eigenvectors. We can prove that where is the positive square root of. The first pair of canonical variables, or first canonical variate pair, is the pair of linear combinations having unit variances, which maximize the correlation

We chooseas the projection directions. Denote 1. The first pair of canonical variables 2. The second pair of canonical variables

As part of a larger study of the effect of organizational structure on “job satisfaction”. Dunham investigated the extent to which measures of job satisfaction are related to job characteristics. Using a survey instrument. Dunham obtained measurement of 5 job characteristics and 7 job satisfaction variables for 784 executives from the corporate branch of a large retail merchandising corporation. Are measures of job satisfaction associated with job characteristics? Example 10.5: Job satisfaction

Job characteristics :feedback task significance task variety task identity autonomy Job satisfaction :supervisor satisfaction career-future satisfaction financial satisfaction workload satisfaction company identification kind-of-work satisfaction general satisfaction Observations: 784

Example 10.5: Job satisfaction feedback task significance task variety task identity autonomy supervisor satisfaction career-future satisfaction financial satisfaction workload satisfaction company identification kind-of-work satisfaction general satisfaction Correlation matrix

CANONICAL VARIATE COEFFICIENTS AND CANONICAL CORRELATIONS

Example 10.5: Job satisfaction Sample correlations between original variables and canonical variables

Example 10.5: Job satisfaction SAS output

Interpretation According to the coefficients, is primarily a feedback and autonomy variable, while represents supervisor, career-future, and kind-of-work satisfaction, along with company identification. might be interpreted as a job characteristic index might be interpreted as a job satisfaction- company identification index 1.The first pair of canonical variables

might be interpreted as a job robust index might be interpreted as a job workload satisfactory 2.The second pair of canonical variables