1. Multiple Linear Regression Polynomial Regression

2. Monotonic but Non-Linear The relationship between X and Y may be monotonic but not linear. The linear model can be tweaked to take this into account by applying a monotonic transformation to Y, X, or both X and Y. Predicting calories consumed from number of persons present at the meal.

3. R 2 =.584

4. R 2 =.814

5. Log Model Calories Persons

6. Polynomial Regression

7. A monotonic transformation will not help here. A polynomial regression will. Copp, N.H. Animal Behavior, 31, 424-430 Subjects = containers, each with 100 ladybugs Containers lighted on one side, dark on the other Y = number on the lighted side X = temperature

8. Polynomial Models Quadratic: Cubic: For each additional power of X added to the model, the regression line will have one more bend.

9. Using Copp’s Data Compute Temp 2, Temp 3 and Temp 4. Conduct a sequential multiple regression analysis, entering Temp first, then Temp 2, then Temp 3, and then Temp 4. At each step, evaluate whether or not the last entered predictor should be retained.

10. R 2 Linear =.137 Quadratic =.601

11. The Quadratic Model The quadratic model clearly fits the data better than does the linear model. Phototaxis is positive as temps rise to about 18 and negative thereafter.

12. A Cubic Model R 2 has increased significantly, from.601 to.753, p <.001 Does an increase of 15.2% of the variance justify making the model more complex? I think so.

14. Interpretation Ladybugs buried in leaf mold in Winter head up, towards light, as temperatures warm. With warming beyond 12, head for some shade – the aphids are in the shade under Karl’s tomato plant leaves. With warming beyond 32, this place is too hot, lets get out of here.

15. A Quartic Model  R 2 =.029, p =.030 Does this small increase in R 2 justify making the model more complex? Can you make sense of a third bend in the curve.

16. The quartic plot does not look much different than the cubic.

17. Multicollinearity May be a problem whenever you have products or powers of predictors in the model. Center the predictor variables, Or simply standardize all variables to mean 0, standard deviation 1.

18. For complete SPSS output, go herehere Polynomial regression can also be used to conduct ANOVA.