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Curvilinear Regression

Curvilinear Regression. 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.

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Curvilinear Regression

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  1. Curvilinear 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. R2 = .584

  4. R2 = .814

  5. Calories Log Model Persons

  6. Polynomial Regression

  7. Aggregation of Ladybugs • 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 • Y = number of ladybugs free (not aggregated) • 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 Temp2, Temp3 and Temp4. • Conduct a sequential multiple regression analysis, entering Temp first, then Temp2, then Temp3, and then Temp4. • At each step, evaluate whether or not the last entered predictor should be retained.

  10. SAS • Curvi -- Polynomial Regression, Ladybugs. • Download and run the program. • Refer to it and the output as Professor Karl goes over the code and the output

  11. Linear Model, R2 = .615

  12. Quadratic Model, R2=.838

  13. Cubic Model, R2= .861

  14. Which Model to Adopt? • Adding Temp2 significantly increased R2, by .838-.615 = .223, keep Temp2. • Adding Temp3significantly increased R2, by .861-.838 = .023 – does this justify keeping Temp3? • Adding Temp4 did not significantly increase R2. • Somewhat reluctantly, I went cubic.

  15. Phototaxis • Subjects = containers, each with 100 ladybugs • Containers lighted on one side, dark on the other • Y = number on the lighted side • X = temperature

  16. R2 Linear = .137 Quadratic = .601

  17. 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.

  18. A Cubic Model • R2 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.

  19. 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.

  20. A Quartic Model • R2=.029, p = .030 • Does this small increase in R2 justify making the model more complex? • Can you make sense of a third bend in the curve.

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

  22. 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.

  23. I am so Cute

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