Multiple Linear Regression
Polynomial 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
• 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
• Containers lighted on one side, dark on
the other
• Y = number on the lighted side
• X = temperature
Polynomial Models
• Quadratic:
• Cubic:
Yˆ  a  b1X  b2 X 2
Yˆ  a  b1X  b2 X 2  b3 X 3
• 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.
R2
Linear = .137
Quadratic = .601
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.
Lighted  33.7  3.37  Temp  .091 Temp
2
A Cubic Model
Lighted  27.6  6.64  Temp  .369  Temp 2  .006 * Temp3
• 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
Lighted  29.0  4.95  Temp  .072  Temp 2
 .010 * Temp  .00024  Temp
3
4
• 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.
The quartic plot does not look
much different than the cubic.
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.
• For complete SPSS output, go here
• Polynomial regression can also be used to
conduct ANOVA.