1. Stat 512 – Lecture 18 Multiple Regression (Ch. 11)

2. Projects Guidelines handout (one per group)  Project Do’s and Don’ts  Presentations 5 minutes Will take volunteers for Tuesday Hit the highlights  Tons more detail for me in your paper  Sample hypothetical presentation Plan for technology in advance! Group evaluation form next week

3. Last Time – Inference for Regression Use residual plots to check technical conditions  Linearity: If residuals vs. EV/fitted values does not show pattern (e.g., curvature) will assume original relationship was linear  Independence: If have random sample or randomization, we will assume independence  Normality: If histogram of residuals is reasonably normal, will assume condition distributions at each x are all normal Will be a bit more forgiving on this condition with large n  Equal standard deviation: If residuals vs. EV/fitted values shows equal vertical spread across all the EV values, will assume conditions distributions at each x have same SD

4. Last Time – Inference for Regression Null hypothesis  H 0 : no association between RV and EV (identify)  H 0 : population slope  = 0  H o : no treatment effect from EV on RV (identify) Minitab/SAS output  Assumes two-sided alternative  t = observed slope – hypothesized slope SE(observed slope) d.f. = n-2  Equivalent to p-value reported by Minitab with correlation coefficient

5. PP – Money Making Movies box office = - 42.9 + 1.86 score

6. PP – Money Making Movies

7. Consequence: restrict population to movies earning less than $200 million

8. PP – Money Making Movies Is the relationship statistically significant? Highly significant (p-value <.001) But not all that useful (r 2 = 8.9%) Not a cause and effect relationship Not clear what population this represents

9. Can we improve on these models? Adding predictor variables to the model  Average response =  0 +  1 x 1 +   x 2 + …  Graphical displays  Interpreting coefficients  Interpreting R 2  Inference for model, coefficients  Checking technical conditions (the same!)

10. Three variables…

11. Summary Both mileage and number of stops appear useful in predicting cost, even after controlling for the other variable.  If number of stops held constant, each additional mile costs about 5 cents… The regression on mileage and number of stops allows us to explain 45.5% of the variability in airfares from LAX

12. Summary Can restrict population to “deal with” extreme outlier A statistically significant predictor individually, may not be significant when added to a model Overall F test just tells you that at least one of the slopes is non zero, use t tests to examine them individually

13. Summary If you want to remove variables from the model, do so one at a time as p-values will change each time Can use a 0-1 variable in the model. Interpret slope coefficient as average change in response between group 0 and group 1 (assuming the same relationship between response and explanatory)  Otherwise consider interactions….

14. Multicollinearity

15. For Tuesday Have a great Thanksgiving! Check Final Exam Schedule on Web  One Wed person back to Friday Submit PP (choice of procedure) HW 8 For Thursday  Submit last PP (review questions) Review sheet will be posted online  Presentations!