Curvilinear Regression - PowerPoint PPT Presentation

1 / 24

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Curvilinear Regression

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

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.

• Predicting calories consumed from number of persons present at the meal.

R2 = .584

R2 = .814

Calories

Log Model

Persons

Polynomial Regression

• 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

Polynomial Models

• Cubic:

• For each additional power of X added to the model, the regression line will have one more bend.

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.

SAS

• Curvi -- Polynomial Regression, Ladybugs.

• Refer to it and the output as Professor Karl goes over the code and the output

Cubic Model, R2= .861

• 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.

Phototaxis

• Subjects = containers, each with 100 ladybugs

• Containers lighted on one side, dark on the other

• Y = number on the lighted side

• X = temperature

R2

Linear = .137

• 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.

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.

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.

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.

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.