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Multiple Linear Regression

This text explores multiple linear and polynomial regression, emphasizing that while the relationship between variables X and Y may be monotonic, it is not necessarily linear. By applying transformations to either or both variables, predictions can be refined. The analysis utilizes Copp's data on ladybugs, revealing that a polynomial regression (quadratic to cubic) offers a significantly better fit than a linear model. The results suggest that increased model complexity can be justified by superior explanatory power, while also addressing multicollinearity and the role of transformations in regression analysis.

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Multiple Linear Regression

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  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. R2 = .584

  4. R2 = .814

  5. Calories Log Model 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 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. R2 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 • 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.

  13. 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.

  14. 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.

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

  16. 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.

  17. For complete SPSS output, go here • Polynomial regression can also be used to conduct ANOVA.

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