Curvilinear Regression

1
Curvilinear Regression

• Modeling Departures from the Straight Line
• (Curves and Interactions)

2
Skill Set

• How does polynomial regression test for quadratic
  and cubic trends?
• What are orthogonal polynomials? When can they
  be used? Describe an advantage of using
  orthogonal polynomials over simple polynomial
  regression.

• Suppose we have one IV and we analyze this IV
  twice, once thru linear regression and once as a
  categorical variable. What does the test for the
  difference in R-square between the two tell us?
  What doesnt it tell us, that is, if the result
  is significant, what is left to do?

3
More skills

• Why is collinearity likely to be a problem in
  using polynomial regression?
• Describe the sequence of tests used to model
  curves in polynomial regression.

• How do you model interactions of continuous
  variables with regression?
• What is the difference between a moderator and a
  mediator? How do you test for the presence of
  each?

4
Linear vs. Nonlinear Models

• Typical linear Model
• Typical nonlinear models

We dont use models like this
Nonlinear means in the terms, not the
coefficients.
5
Curvilinear Regression
Uses Polynomial Regression to fit curves.
Polynomials are formed by taking IVs to
successive powers. Polynomial equation referred
to by its degree, determined by highest exponent.
Power terms introduce bends.
Linear
Quadratic
Cubic
6
Quadratic Function
Note the bend.
We can fit data with ceiling effects,
sensation/perception as a function of stimulus
intensity, performance as a function of practice,
etc.
7
Quadratic Function (2)
(original curve)
The graph shows the effect of changing the b
weight for the squared (quadratic) term.
8
Cubic Function
Note the two bends. We get a new bend for each
new power term. Sharpness of bend depends on
size of b weights.
9
Response surfaces (1)
With 1 IV, the relations between X and Y are
shown as a line. With linear regression and 2
IVs, the response surface relating Y to the X
variables will be a plane, e.g.
Y
Y X12X2
The response surface will be like a stiff sheet
of paper in a cardboard box.
X1
X2
10
Response Surface (2)
Y
X1
X2
The same surface from a different angle.
11
Response Surface (3)
YX1X2-.1X1X1. Relations between X2 and Y are
linear relations between X1 and Y are curved.
Response surface is like a section of a coffee
can.
12
Response Surface (4)
Y X1X2-.2X1X1-.2X2X2
In this graph, the relation between Y and both X
variables is curved. Response surface is like a
parachute.
13
Review
What is polynomial regression?
How does polynomial regression allow us to
include one or more bends into lines and response
surfaces?