1. Statistics 350 Lecture 21

2. Today Last Day: Tests and partial R 2 Today: Multicollinearity

3. Multicollinearity When explanatory variables are highly correlated, weird things can happen with the regression analysis Multicollinearity is said to exist among explanatory variables any time a regression of one of the explanatory variables against the rest yields a strong linear relationship, as measured by a high R 2 Can also attempt to visualize this relationship using a scatter-plot matrix

4. Multicollinearity Back to Example:

5. Multicollinearity Why might this matter? Consider the 3 variable linear regression model: Can view  1 in the model as the partial regression coefficient for X 1 What is its interpretation?

6. Multicollinearity If other variables tend to be correlated with X 1 this effect is difficult to isolate and estimate RESULT:

7. Multicollinearity Back to example:

8. Multicollinearity Back to example: Notice: If did a regression of X 1 on X 2 and X 3, the R 2 is 0.9986 Conclusion:

9. Multicollinearity Why exactly have we observed this phenomenon? Consider the 3 variable model in the body fat example:

10. Multicollinearity As a result:

11. Multicollinearity Detecting multicollinearity in practice: