Curvilinear regression
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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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Monotonic but non linear
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



Aggregation of ladybugs
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
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
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


Linear model r 2 615
Linear Model, R2 = .615



Cubic model r 2 861
Cubic Model, R2= .861


Which model to adopt
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
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

Quadratic = .601


The quadratic model
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
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
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
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
Multicollinearity cubic.

  • 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
I am so Cute cubic.


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