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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.
slide5

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.
slide10
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
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
slide16

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
  • 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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