1. Curvilinear Regression
Modeling Departures from the Straight Line
(Curves and Interactions)

2. Skill Set
How does polynomial regression model quadratic and cubic trends?
Describe the sequence of tests used to model curves in polynomial regression.
Why is collinearity likely to be a problem in using polynomial regression?

3. More skills
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 don’t 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 = X1+2*X2
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)
Y=X1+X2-.1*X1*X1. 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 = X1+X2-.2X1*X1-.2X2*X2
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?

14. Nonlinear Relations in Nonexperimental Research
In experimental research, you can fit curves with orthogonal polynomials, which have advantages. Not covered here.
Create power terms (IV taken to successive powers)
Test for increasing numbers of bends by adding terms
Quit when adding a term does not increase variance accounted for.

15. Polynomials to model bends in nonexperimental research
Rating
(DV)
Time
Time**2
Time**3 10 9 8 5 25
125 7
100
1000 6 15
225
3375

16. Correlations among terms
Excite
Time
Time**2
Time**3 1
-.72
-.62
.96
-.55
.91
.99
Note that terms with higher exponents are VERY highly correlated. There WILL be problems with collinearity.
Sequence of tests. Start with time, add time squared. If significant, add time cubed. Stop when adding a term doesn’t help. Each power adds a bend. Quadratic is one bend, cubic is two, and so forth.

17. Results of Polynomial Regression
Excite
Time
Time**2
Time**3 1
-.72
-.62
.96
-.55
.91
.99
Note that polynomial is a special case of hierarchical reg.
Model
Intercept b1 b2 b3 R2
R2 Ch
1 Time
8.90
-.18
.52
2 Time, Time2
9.25
-.39
.014
.58
.06
3 Time,
Time2,
Time3
9.20
-.23
-.02
.001
.59
.01

18. Polynomial Results (2)
Suppose it had happened that the term for time-squared had been significant. The regression equation is Y' = X X2. The results graphed:

19. Interpreting Weights in Polynomial Regression
All power terms for an IV work together to define the curve relating Y to X.
Do not interpret b weights for polynomials. They change if you subtract the mean from the raw data.
To estimate ‘importance’ look to the change in R-square for the block of variables that represent the IV.
Never use polynomials in a variable selection algorithm (e.g., stepwise regression).
Specialized literature on nonlinear terms in path analysis and SEM (hard to do).

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

21. Interactions
An interaction means that the ‘importance’ of one variable depends upon the value of another. Importance means ‘slope’ in regression with a continuous IV.
An interaction is also sometimes called a moderator, as in “Z moderates the relations between X and Y.”
In regression, we look to see if the slope relating the DV to the IV changes depending on the value of a second IV.

22. Example Interaction
For those with low cog ability, there is a small correlation between creativity and productivity.
As cognitive ability increases, the relations between creativity and productivity become stronger. The slope of productivity on creativity depends on cog ability.

23. Interaction Response Surface
The slope of X1 depends on the value of X2 and vice versa. Regression is looking to fit this response surface and no other when we do the customary analysis for interactions with continuous IVs. More restrictive than ANOVA.

24. Significance Tests for Interactions
Subtract means from each IV (optional).
Compute product of IVs.
Compute significance of change in R-square using interaction(s).
If R-square change is n.s., no interaction(s) present.
If R-square change is significant, find the significant interaction(s).
Graph the interaction(s)

25. ;
Data to test for interaction between cognitive ability and creativity on performance.

26. Correlation Matrix Prob > |r| under H0: Rho=0
Pearson Correlation Coefficients, N = 20
Prob > |r| under H0: Rho=0
person product create cog inter
person
product
< <.0001
create
<.0001
cog
< <.0001
inter
< < <.0001

27. Results for 2 IVs (Main Effects)

28. Result for interaction

29. Go to R code

30. Moderator and Mediator
Moderator Means Interaction. Slope of one depends on the value of the other. Use moderated regression (test for an interaction) to test.
Mediator means there is a causal chain of events. The mediating variable is the proximal cause of the DV. A more distal cause changes the mediator. Use path analysis* to test. In this graph, X2 is the mediator.
* Modern methods Sobel test; Preacher & Hayes; Bootstrap

31. Review
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?