Monotonic but NonLinear 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 ID: 254250
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Slide1
Curvilinear RegressionSlide2
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.Slide3
R
2
= .584Slide4
R
2
=
.814Slide5
Log Model
Calories
PersonsSlide6
Polynomial RegressionSlide7
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 = temperatureSlide8
Polynomial Models
Quadratic:
Cubic:
For each additional power of X added to the model, the regression line will have one more bend.Slide9
Using Copp’s Data
Compute Temp
2
, Temp
3
and Temp
4
.
Conduct a sequential multiple regression analysis, entering Temp first, then Temp
2
, then Temp
3
, and then Temp
4
.
When deciding which model to adopt, consider whether making the model more complex is justified by the resulting increase in
R
2
.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.Slide11
Linear Model, R
2
= .615Slide12
Quadratic Model, R
2
=.838Slide13
Cubic Model, R2
= .861Slide14
Which Model to Adopt?
Adding Temp
2
significantly increased
R
2
, by .838-.615 = .223, keep Temp
2
.
Adding
Temp
3
significantly increased
R
2
, by .
861-.838
= .023 – does this justify keeping
Temp3 ?Adding Temp
4 did not significantly increase R2.Somewhat reluctantly, I went cubic.Slide15
SHIFT
Shift
to the
OUTPUT PDF
at this point, come back to the slideshow
later.Slide16
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.Slide17
I am so CuteSlide18
SPSS
See
the document
for an example of polynomial regression using SPSS.Slide19
Curvilinear CattleSlide20
Just for Fun
https://xkcd.com/2048
/