1. Agenda of Week VII Review of Week VI Multiple regression Canonical correlation

2. Review of Week VI Inference on MV Mean Vector Two populations Inference on MV Mean Vector Multi-populations: MANOVA

3. Multiple Regression Basic assumptions Linearity X is an n by K matrix with rank K Regression Homoscedasticity

4. Multiple Regression Hypotheses By ANOVA When the null hypothesis is rejected, For all i.

5. Multiple Regression OLS model OLS solution

6. Multiple Regression ANOVA table

7. Multiple Regression t-statistics (d.f. = n-k) Where S kk is the k th diagonal element of (XX) -1 Standard error of the regression

8. Multiple Regression Standard error of coefficient

9. Canonical Correlation Correlation structure between a group of independent variables and a group of dependent variables A kind of multiple regression with more than two dependent variables Example: Physical size group vs. Exercise group How much explained the variation of a group of variables by other group of variables?

10. Canonical Correlation Basic model Group of independent variables Group of dependent variables Covariance matrixes

11. Canonical Correlation Basic model LC of two groups of variables Correlation between V and W

12. Canonical Correlation Correlation testing Hypotheses Correlation between V and W

13. Canonical Correlation Objective: To find α and β maximizing ρ or r.

14. Canonical Correlation 1 st canonical variable k th canonical variable

15. Canonical Correlation 1 st canonical correlation coefficient k th canonical coefficient coefficient

16. Canonical Correlation

17. Canonical loading matrix Canonical cross loading matrix