PS 405 Week 7 Section: Interactions D.J. Flynn February 25, 2014 - - PowerPoint PPT Presentation

PS 405 – Week 7 Section: Interactions

D.J. Flynn February 25, 2014

Today’s plan

Review: the multiplicative interaction model Estimating/interpreting interaction models in R Plotting/visualizing interactive effects in R

Why interactions?

In political science, we ofen make conditional hypotheses (e.g., an increase in X leads to an increase in Y when condition Z is met, but not when condition Z is absent). For example, the abstract from Jerit and Barabas (2012) says: We know little about...how individual-level partisan motivation interacts with the information environment... Using survey data as well as an experiment with diverse subjects, we demonstrate...a selective pattern of learning in which partisans have higher levels of knowledge for facts that confirm their world view... This basic relationship is exaggerated on topics receiving high levels of media coverage.

Quick review of interactions

◮ We use interaction terms (“multiplicative interaction models”)

to test conditional hypotheses

◮ By using interactions, we’re testing moderation (NOT

mediation)

◮ Example from lecture: For public school students, higher

undergrad GPAs are associated with greater success in grad school – but not for private school students.

The basic interaction model Yi = β0 + β1Xi + β2Zi + β3XiZi + ǫi,

where Y is the DV, X and Z are regressors, XZ is the interaction, and ǫ is the residual.

What each term represents

Yi = β0 + β1Xi + β2Zi + β3XiZi + ǫi Notice:

◮ When Zi = 0, we have Yi = β0 + β1Xi + ǫ1. ◮ Thus, the marginal effect of X when Z = 0 is

δY δX = β1

◮ When Zi = 1, we have Yi = (β0 + β2) + (β1 + β3)Xi + ǫi. ◮ Thus, the marginal effect of X when Z = 1 is

δY δX = β1 + β3

Brambor, Clark, and Golder’s (2006) interaction rules1

• 1. Use interaction terms whenever the hypothesis is conditional

in nature.

• 2. Include all constitutive terms in the model.
• 3. Do NOT interpret the coefficients on constitutive terms as

unconditional marginal effects.

• 4. Do NOT forget to calculate substantively meaningful marginal

effects and standard errors. (coming...)

1Source: Brambor, Thomas, William Roberts Clark, and Matt Golder. 2006.

“Understanding Interaction Models: Improving Empirical Analyses.” Political Analysis 14: 63–82.

Constitutive terms

◮ Constitutive terms are those elements that “constitute” the

interaction term:

◮ if you include XZ, you should also include X and Z. ◮ if you include X2, you should also include X. ◮ If you include X3, you should also include X and X2, etc....

◮ Always include constitutive terms (even when the effect of X

when Z = 0 is not theoretically important)

If you omit a constitutive term, then there is potential for omitted variable bias: Suppose the true model is: Yi = β0 + β1Xi + β2Zi + β3XiZi + ǫi. But you estimate: Yi = β0 + β1Xi + β2XiZi + ǫi. If β2 = 0 and Z is correlated with an included regressor, then estimates of β0, β1, and β3 will be biased (and standard errors wrong).

Standard errors/hypothesis testing

◮ So far we’ve reviewed how to calculate marginal effects. But

what about the significance of these effects?

◮ Can’t just look at the significance of the interaction coefficient

– because effect of X on Y could be significant at some levels of Z, but not others.

◮ Easiest case: dummy variable interactions.

◮ Suppose we have the standard interaction model with a binary

Z: Yi = β0 + β1Xi + β2Zi + β3XiZi + ǫi

◮ The marginal effect of X when Z = 0 is β1, and its standard

error is just

• var( ˆ

β1).

◮ The marginal effect of X when Z = 1 is β1 + β3, and its standard

error is

• var( ˆ

β1) + Z2var( ˆ β3) + 2Zcov( ˆ β1 ˆ β3).

◮ However, if we’re interacting categorical/continuous terms,

then things get messy.

◮ Logic: Z could be at many different levels, so we need to know

significance of X at all these levels.

◮ To do this, we need marginal effect plots and confidence

intervals.

◮ Let’s estimate an interactive model and create a helpful plot.....

Interactions in R

There are two ways to run interactions in R. I think this way is best because it automatically includes constitutive terms for you: x<-rnorm(100,10,2) z<-rnorm(100,8,1) y<-x*z + rnorm(100,9,2) int.model<-lm(y∼x*z) summary(int.model) Example: library(car) prestige.model<-lm(prestige∼income*education + type, data=Prestige) summary(prestige.model)

Let’s interpret...

summary(prestige.model) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

• 1.780e+01

7.594e+00

• 2.344 0.021212 *

income 3.786e-03 9.445e-04 4.008 0.000124 *** education 5.104e+00 7.766e-01 6.572 2.93e-09 *** typeprof 5.479e+00 3.714e+00 1.475 0.143574 typewc

• 3.584e+00

2.428e+00

• 1.476 0.143303

income:education -2.102e-04 6.977e-05

• 3.012 0.003347 **
• Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’

Visualizing interactions with the effects package

install.packages("effects") library(effects) income.effect<-effect("income*education", prestige.model) plot(income.effect,as.table=T)

income*education effect plot

income prestige

30 40 50 60 70 80 90

education

5000 10000 15000 20000 25000

education

5000 10000 15000 20000 25000

education

30 40 50 60 70 80 90

education

We could look at the other side of the interaction: the effect of education on prestige at different levels of income: plot(income.effect, x.var="education", as.table=T)

income*education effect plot

education prestige

30 40 50 60 70 80 90

income

8 9 10 11 12 13 14

income income

8 9 10 11 12 13 14

income

30 40 50 60 70 80 90

income

Another example: using the same data, let’s look at the effect of income on prestige, conditional on the type of profession: prestige.model2 <- lm(prestige ∼ income*type + education, data=Prestige) income.effect <- effect("income*type", prestige.model2) plot(income.effect, layout=c(3,1))

income*type effect plot

income prestige

40 60 80 100 120 5000 10000 15000 20000 25000

: type bc

5000 10000 15000 20000 25000

: type prof

5000 10000 15000 20000 25000

: type wc

You might have noticed that none of these examples involved a continuous-by-continuous interaction. That’s because continuous by continuous interactions are very challenging to interpret. You have a few options in these situations2

◮ collapse one of the continuous variables into categories and

treat it as a factor

◮ complicated plots

2See Thomas Leeper’s page on this:

thomasleeper.com/Rcourse/Tutorials/olsinteractionplots2.html

Concluding notes on the effects package

◮ One of the more flexible packages you’ll encounter ◮ You can also use it to get marginal effect at different levels of X

and SEs (no plots): income.effect

◮ You can also make tons of alterations to the plots (e.g., change

from lines to confidence bars, re-label axes just like in standard R plots, etc.)

◮ The official effects manual is here . A more user-friendly intro

is here .

A final point: marginal effects vs. predictions

Everything we did today is about calculating marginal effects of X

• n Y (at diffferent levels of Z). Or, equivalently, marginal effects of Z
• n Y (at different levels of X). To get marginal effects, we don’t care

about other regressors in the model. If we were interested in coming up with predicted values for each

• bservation in the dataset (ˆ

Yi), we would care about the coefficients

• n the other regressors. This is easy. We just solve for the linear

prediction – just like in non-interactive models.