September 8, 2009 Session 3 Slide 1

PSC 5940: Interactions as Multi-Level Models

Session 3

Fall, 2009

September 8, 2009 Session 3 Slide 2

Workshop: PRCs• Load EE data

• Run a simple model:

• Willingness to pay for an alternative energy tax

• Use randomly assigned “price” as IV

• Plot to relationship (use jitter)

• Now add: Income, Ideology

• Change in price variable? (Why?)

September 8, 2009 Session 3 Slide 3

Model Elaboration

• EE09 & NS09 Data: research thinking• Analysis of residuals

• Additions to the ERDF model:

• Belief in anthropogenic climate change

• Recodes?

• Understanding of GCC science

• Recode “What scientists’ believe…” variables

• Turn in 1 page summaries

September 8, 2009 Session 3 Slide 4

Dummy Intercept Variables• Dummy variables allow for tests of the differences

in overall value of the Y for different nominal groups in the data (akin to a difference of means)

• Coding: 0 and 1 values (e.g., men versus women)

Y

X1

X2,0

X2,1

September 8, 2009 Session 3 Slide 5

Modeling Belief in GCC as a function of knowledge and Republican Party Identification

Belief in GCC systematically lower for those who identify as Republicans

Call:lm(formula = gcc_bel ~ R_id + gcc_knowl)

Residuals: Min 1Q Median 3Q Max -16.158 -2.582 1.842 3.842 12.460

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.4467 0.5529 -2.617 0.00897 ** R_id -3.0135 0.3110 -9.688 < 2e-16 ***gcc_knowl 1.5210 0.1310 11.613 < 2e-16 ***---Residual standard error: 5.526 on 1507 degrees of freedom (188 observations deleted due to missingness)Multiple R-squared: 0.1468, Adjusted R-squared: 0.1457 F-statistic: 129.7 on 2 and 1507 DF, p-value: < 2.2e-16

September 8, 2009 Session 3 Slide 6

Dummy Variable Applications• Implies a comparison (the omitted group)

– Be clear about the “comparison category”• Multinomial Dummies

– When categories exceed 2• Importance of specifying the base category

• Examples of Category Variables– Experimental treatment groups– Race and ethnicity– Region of residence– Type of education– Religious affiliation– “Seasonality”

• Adds to modeling flexibility

September 8, 2009 Session 3 Slide 7

Interaction Effects• Interactions occur when the effect of one X is

dependent on the value of another• Modeling interactions:

– Use Dummy variables (requires categories)– Use multiplicative interaction effect

• Multiply an interval scale times a dummy (also known as a “slope dummy”)

• Example: the effect of GCC knowledge (gcc_knowl) on belief in climate change (gcc_bel) may be affected by whether the respondent identifies with the Republican Party (R_id)– Re-code the interaction; run it.

September 8, 2009 Session 3 Slide 8

Modeling belief in Climate Change with a Dummy Slope Variable: Knowledge*

Republican ID (=1) Call:lm(formula = gcc_bel ~ R_id + gcc_knowl + gcc_kn_R)

Residuals: Min 1Q Median 3Q Max -15.971 -2.610 1.691 4.029 13.826

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.6831 0.6696 -1.020 0.3079 R_id -5.1433 1.1007 -4.673 3.24e-06 ***gcc_knowl 1.3308 0.1613 8.252 3.37e-16 ***gcc_kn_R 0.5563 0.2758 2.017 0.0439 * ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.521 on 1506 degrees of freedom (188 observations deleted due to missingness)Multiple R-squared: 0.1491, Adjusted R-squared: 0.1474 F-statistic: 87.99 on 3 and 1506 DF, p-value: < 2.2e-16

September 8, 2009 Session 3 Slide 9

Illustration of Slope Interaction

Republicans

NotRepublicans

September 8, 2009 Session 3 Slide 10

BREAK

September 8, 2009 Session 3 Slide 11

Workshop