1. Lecture for Multiple Regression HSPM J716

2. Data

3. Fertilizer-Rain chart • The two X variables graphed.

4. Yield-Fertilizer • Projection of 3-D graph, seen looking across the Fertilizer axis.

5. Yield-Rain • Projection looking across Rain axis

6. Simple regression line • Yield vs. Fertilizer • Like fitting a plane that is horizontal (no slope) in the Rain direction. No slope in the rain direction means that Rain is assumed to have no effect.

7. Multiple regression • Like fitting a grid on the Yield-fertilizer graph • The Rain lines all have to have the same slope. • The Rain lines have to be equidistant. • Linear assumption is why. • Minimize the sum of squares of distances from each point to the regression line that corresponds to that point’s rain amount.

8. Simple regression prediction

9. Multiple regression prediction

10. Collinearity • Two of your X variables are correlated with each other • = One of your X variables can be well predicted from another X variable • Multicollinearity – one of your X variables is well predictable from a linear combination of other X variables

11. Stupid examples • Collinearity – Two of your X variables measure the same thing, like height in inches and height in feet. • Multicollinearity – The X variables are scores made on questions on a test. Another X variable is the total score on the test.

12. Collinearity • Why you need multiple regression • But too much collinearity makes separation of causes impossible

13. Collinearity example

15. All slope in fertilizer direction

16. All slope in rain direction

17. F-test