More Topics in Regression

Correlations

1 .130** -.031 .108** .163** .023

. .000 .228 .000 .000 .334

1800 1771 1549 1800 1800 1797

.130** 1 -.168** .175** .034 -.070**

.000 . .000 .000 .154 .003

1771 1776 1525 1776 1776 1774

-.031 -.168** 1 -.135** -.003 .069**

.228 .000 . .000 .916 .007

1549 1525 1553 1553 1553 1550

.108** .175** -.135** 1 .078** .061**

.000 .000 .000 . .001 .010

1800 1776 1553 1807 1807 1804

.163** .034 -.003 .078** 1 .221**

.000 .154 .916 .001 . .000

1800 1776 1553 1807 1807 1804

.023 -.070** .069** .061** .221** 1

.334 .003 .007 .010 .000 .

1797 1774 1550 1804 1804 1804

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

EDU1

PID1

MEDTRU1

RAD1

PAPER1

NATNEW1

EDU1 PID1 MEDTRU1 RAD1 PAPER1 NATNEW1

Correlation is significant at the 0.01 level (2-tailed).**.

Simulation

• Imagine that you find yourself out of college and in a job. • Take a look at the sheet. See what your monthly income

is• Decide how to allocate money, and record values in

each category. • You must have shelter, food, transportation, and

clothing. You may have multiples of these things. • You cannot spend any more than you have• You have to allocate all of your money to something,

whether it is consumer goods or charity or savings or whatever.

Heteroskedasticity

• We will plot savings against income

• What I expect to see is more variance at high values

• If your income is low, all of your income goes to providing necessities

• If it is high, can go to fun or savings

Heteroskedasticity

An example of Heteroskedasticity

• Regression model assumes that errors are of equal average magnitude at all values of IV.

• Problem- sometimes in the real world, this does not happen

• Difference of variance at low and high values. That is, fit is tighter at one end than at the other

Linear Regression

5.00 10.00 15.00 20.00

var00001

0.00

10.00

20.00

30.00

het

ero

hetero = 0.21 + 0.91 * var00001R-Square = 0.52

Model Summaryb

.721a .520 .515 5.09748Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), VAR00001a.

Dependent Variable: HETEROb.

Coefficientsa

.210 1.059 .198 .844

.911 .088 .721 10.303 .000

(Constant)

VAR00001

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: HETEROa.

Scatterplot

Dependent Variable: HETERO

Regression Standardized Predicted Value

2.01.51.0.50.0-.5-1.0-1.5-2.0

Re

gre

ssio

n S

tan

da

rdiz

ed

Re

sid

ua

l

4

3

2

1

0

-1

-2

-3

Heteroskedasticity

• Causes

• There may be an underlying interactive relationship missing

• There may be different measurement error at different values

• Causes estimates of b’s to be less precise.

• Can lead to real results looking insignificant.

Modeling Interactive Relationships in Regression

• Income=b1(sex)+b2(education)+c• Income=b1(sex)+b2(education)+b3(sex x

education)+c• In both cases, b1 gives us the difference for

being a man or woman, b2 gives the impact of education. What does b3 tell us?

• b3 is the interaction, tells us if there is a differential impact of education among the genders

• For sex, let 0=m, 1=f, education in years, income in dollars

Education

income

Education

income

Men

Women

Education

income

Men

Women

Education

income

Women

Men

Using Interaction Terms

• Easiest when one is a dummy variable (0-1) other is either continuous or a dummy

• Multiply the two together• Include all lesser terms (that is, in a two way,

have x1,x2, and x1*x2.

• In a three way, include x1,x2,x3, x1*x2, x1,x3, x2,x3, and x1*x2*x3

• For interpretation, you can add the slope of the interaction term to the slope of the appropriate variable, use to create two sets of predicted values

An Example- Internet and Personality

• Research question- does internet have differential impact on political knowledge for people depending on their motivation to seek information.

• Look at Need for Cognition and Need to Evaluate

• Set up regression with political knowledge as DV. NE, NC, media use, and interactions as IVs

Looking at Results

• Model 2- No interactions, but includes important traits including media use.

• Model 3- includes interactions between NC and all media types. NC is significant and positive. Negative interaction with cable.

• Model 4- interactions with NE. NE is significant and positive, Negative interaction with cable and internet

NE

Knowledge

No internet

internet

Challenges to Interaction Terms

• Interpretation– Dummy*continuous is tricky– Continuous*continuous is trickier– Three ways are especially challenging

• Example- Miller and Krosnick 2000• Trust*knowledge*condition • Results- all three required for effect

Challenges to Interactions

• Multicolinearity

• Interaction terms are typically highly correlated with other terms

• Inflates errors of parameter estimates

• Makes potentially significant relationships appear non-significant