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

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

Polynomial Models Quadratic: 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. When deciding which model to adopt, consider whether making the model more complex is justified by the resulting increase in R2.

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.

Linear Model, R2 = .615

Quadratic Model, R2=.838

Cubic Model, R2= .861

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

SHIFT Shift to the OUTPUT PDF at this point, come back to the slideshow later.

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.

I am so Cute

SPSS See the document for an example of polynomial regression using SPSS.