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## PowerPoint Slideshow about ' Curvilinear Regression' - snowy

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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.

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.
- At each step, evaluate whether or not the last entered predictor should be retained.

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

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.

Phototaxis

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

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.

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.

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