Chapter 10

1. Chapter 10 Canonical Correlation Analysis

2. 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 Introduction Variables Data Linear combination

3. where Without loss of generality, we can always assume that

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

5. we have

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

7. 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

8. We choose as the projection directions. Denote The first pair of canonical variables The second pair of canonical variables

9. Example 10.5: Job satisfaction 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?

10. 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

11. Example 10.5: Job satisfaction Correlation matrix 1.00 . . . . . . . . . . . 0.49 1.00 . . . . . . . . . . 0.53 0.57 1.00 . . . . . . . . . 0.49 0.46 0.48 1.00 . . . . . . . . 0.51 0.53 0.57 0.57 1.00 . . . . . . . 0.33 0.30 0.31 0.24 0.38 1.00 . . . . . . 0.32 0.21 0.23 0.22 0.32 0.43 1.00 . . . . . 0.20 0.16 0.14 0.12 0.17 0.27 0.33 1.00 . . . . 0.19 0.08 0.07 0.19 0.23 0.24 0.26 0.25 1.00 . . . 0.30 0.27 0.24 0.21 0.32 0.34 0.54 0.46 0.28 1.00 . . 0.37 0.35 0.37 0.29 0.36 0.37 0.32 0.29 0.30 0.35 1.00 . 0.21 0.20 0.18 0.16 0.27 0.40 0.58 0.45 0.27 0.59 0.31 1.00 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

12. CANONICAL VARIATE COEFFICIENTS AND CANONICAL CORRELATIONS

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

14. Example 10.5: Job satisfaction SAS output

15. Interpretation • The first pair of canonical variables 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

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