POLS6481-Lab 11 Interactions Among Variables in Linear Regression Models Solved

The primary objective is to explore interactions among variables in linear regression models. Interpretation of regression coefficients is complicated by the fact that each variable’s marginal effects will be conditional on the value of any variable it is interacted with, and by the fact that each coefficient can only be interpreted as the marginal effect of a variable when any variable it is interacted with equals zero.

 

II. Dataset:       alexseev.dta

 

III. Packages:   compactr, foreign, ggplot

 

IV. Preparation

1) Open RStudio by double-clicking the icon or selecting RStudio from the Windows Start menu.

2) Open the POLS6481-Spring2021-UH-lab Project and perform a Git pull. 

3) If not using Projects, Git, and here download files, place in working directory, and make changes as needed.

4) Open the R script by typing Ctrl+O or by clicking on File in the upper-left corner, using the dropdown menu, and navigating to the script in your working directory.

5) Open the R script by typing Ctrl+O or by clicking on File in the upper-left corner, using the dropdown menu, and navigating to the script in your working directory.

6) If you are missing any packages from the script, you should see a message similar to the one in the screenshot below (this screenshot is also available full size in the lab folder). Click install or install any needed packages manually:

 

7) Run line 1 in the script to allow easier loading of the data files or, alternately, manually edit line 5. 

8) Run lines 2-4 in the R script to load packages that you can use to make more attractive figures and the package that you will need to open the dataset. 

 

V. Instructions for Lab 11

 

You will use data on ethnocentric (anti-immigrant) voting during the 2003 Russian State Duma elections. This data was collected by Mikhail Alexseev and for his 2006 Political Behavior article, “Ballot-Box Vigilantism: Ethnic Population Shifts and Xenophobic Voting in Post-Soviet Russia.” William Berry, Matt Golder, and Daniel Milton re-analyzed the data on pages 665–668 of their 2012 Journal of Politics article, “Improving Tests of Theories Positing Interactions.” I posted both articles in Blackboard.

 

Load the dataset by running lines 5–6, changing the directory if needed. The dataset that can be downloaded from the internet has two additional lines of NA values, which create some trouble in trying to find means and other summary statistics. You will work with a data frame named alexseev.

 

The main dependent variable, e03ld, is the percent of votes cast for the extreme nationalist party, Zhirinovsky Bloc, in 2003. The main party in this bloc, LDPR, persistently ran on a campaign platform of “Russia for Ethnic Russians.”

 

Run lines 7–8 to examine the normality of the dependent variable (e03ld) visually. In line 7, you generate the normal quantile plot of e03ld against the normal distribution; any deviation from a straight line indicates a non-normal distribution. In line 8, you generate a histogram of e03ld; any deviation from a bell curve would indicate a non-normal distribution. 

            qqnorm(alexseev$e03ld)

            hist(alexseev$e03ld, freq = FALSE, breaks=19)

One reason for concern about the distribution of the dependent variable is that it is truncated below at 0. 

 

The two main independent variables are slav89, which is the percent of the population that belonged to the dominant ethnic group (Slavs) in 1989, and nonslav8, which the change in the percentage of the population accounted for by ethnic minorities between 1989 and 2002. 

 

Alexseev’s preferred theory, named the “Defended Nationhood” hypothesis, posits that a greater Slavic share of the population should be positively associated with the Zhirinovsky Bloc’s vote share and that this effect should be enhanced by a large influx of ethnic minorities, hence the inclusion of a multiplicative interaction term. 

 

Alexseev also argues that greater increases in the percentage of ethnic minorities should increase support for anti-immigrant parties regardless of Slavic population.[1]

 

Alexseev also includes six control variables: 

·        inc9903 is a measure of change in average personal income from 1999 to 2003; 

·        eduhi02 is the percent of the population with a college education in 2002; 

·        unemp02 is the percent of the working age population claiming unemployment benefits in 2002; 

·        apt9200 is the percent of apartments or houses converted to private ownership from the start of privatization in 1992 to 2000;

·        vsall03 is a region’s population size; and 

·        brdcont is a dummy variable identifying regions located along Russia’s borders over which it had a dispute with a neighboring state at any time between 1991 and 2003.

 

Run lines 10–11 to estimate Alexseev’s main model, which he labels as Test 1 in his Table 2 (2006: 225), and to inspect the results. (You may spend some time inspecting the constant and the control variables’ effects, but our focus in the remainder of lab will be on the slav89 and nonslav8 variables, and their interaction.) The remaining columns in Table 2 focus on specific non-slavic ethnic groups.

 

If you have run the model correctly, then you should see the following regarding his main hypothesis:  

·        slav89 has a positive regression coefficient, which is consistent with Alexseev’s theory, but it is statistically insignificant; 

·        nonslav8 has a negative regression coefficient, which is inconsistent with Alexseev’s theory, and it is statistically significant;

·        the interaction of slav89 and nonslav8 has a positive regression coefficient, which is consistent with Alexseev’s theory, but it is statistically insignificant. 

If you have kept up with the readings, then by now you should be able to answer the following question: Why are these three bulleted findings virtually meaningless – especially the first two?

 

Recall that when you are examining regression coefficients for variables that are in a multiplicative interaction, the coefficient only tells you the impact when the other variable in the interaction equals zero. 

That is, when including an interaction of x and z, the marginal effect of x on y equals βx + βxz×z; this implies that the marginal effect equals  only when z = 0. So, it is more useful to examine the marginal effect of each independent variable at their typical values instead. Moreover, multiplicative interactions have a tendency to inflate standard errors, so statements about statistical significance are dubious.

 

Run lines 13–15 to examine the distribution of the interaction term, sl89nsl8, and its two constituent variables, slav89 and nonslav8; save the mean values of the constituent terms, named s.bar and ns.bar: 

            s.bar = summary(alexseev$slav89)[4]

            ns.bar = summary(alexseev$nonslav8)[4]

Run line 16 to calculate the interaction of the mean values (which is not equal to the mean of the interaction term). Next, run line 18 to find the marginal effect of slav89 when nonslav8 is set at its mean value:

            model$coef[2] + model$coef[4]*ns.bar

Then, run line 19 to find the marginal effect of nonslav8 when slav89 is set at its mean value:

            model$coef[3] + model$coef[4]*s.bar 

Later in this lab, we will come back to whether these marginal effects are statistically significant.

 

To get more insight into the marginal effects, let us explore how the predicted values of vote for the Zhirinovsky Bloc changes when the values of the slav89 and nonslav8 variables change from their 25th percentile to their median, and from their median to the 75th percentile.

 

Begin by running line 21 to define the mean value of all the control variables, and then run line 22 to create a baseline value of e03ld by adding the intercept to the products found by multiplying each coefficient times its respective control variable’s mean value. If you were to inspect the summary of e03ld, then you might discover that the average is about 12.5 percent; our baseline value is a bit low because it implicitly holds the slav89 and nonslav8 variables, as well as the interaction term, sl89nsl8, equal to zero. You can run line 23 to check an equality that we found back in week 2: the fitted value of the dependent variable holding all independent variables at their averages equals the average value of the dependent variable.

 

When calculating a bunch of predicted values in order to inspect marginal effects, it is often extremely useful to create a new dataset of inputs. The approach we are about to take is not the most elegant, but it is practical. We are going to have three variables (slav89, nonslav8, and their interaction) set at three different values (first quartile, median, and third quartile), and then calculate predicted values by multiplying each model coefficient times a particular value of the corresponding variable, and adding the baseline value.

 

If you run lines 25–28, you will create the new data frame. The slav89 variable is set at first quartile, median, and third quartile – in that order – three times. The nonslav8 variable is set at its first quartile three times, at its median three times, and at its third quartile three times. The interaction is created from the product of slav89 and nonslav8. (Therefore, the fifth observation out of nine has all three variables at their median value.)

 

Run line 29 to create a column of predicted values in the data frame, which is named new.data. If you simply type new.data in the Console window, then RStudio will display the data frame for you to view. The first three columns show you the values that were set for the explanatory variables, and the fourth column shows you the predicted value of the dependent variable.

 

If you focus on the fourth, fifth and sixth predicted values, this shows you how the predicted value of e03ld changes in response to increases in slav89, holding nonslav8 at its median. You should observe that increases in slav89 correspond to increasing support for the Zhirinovsky Bloc, which is consistent with Alexseev’s theory.

 

If you focus on the second, fifth and eighth predicted values, this shows you how the predicted value of e03ld changes in response to increases in nonslav8, holding slav89 at its median. You should observe that increases in nonslav8 actually correspond to slight decreases in support for the Zhirinovsky Bloc, which is inconsistent with Alexseev’s theory. 

 

Demonstrating that predicted support for the Zhirinovsky Bloc increased with increases in slav89 or decreased with increases in nonslav8 does not prove that either variable has a statistically significant effect on the value of the dependent variable. We will postpone a discussion of statistical significance until after we perform two more tasks and then use R to calculate standard errors. Also, we have not directly addressed the main question of whether increases in nonslav8 tend to heighten the effect of changes in slav89.  We can view this indirectly by seeing the following: 

·        when the change in percent minority population is low (nonslav8 = 0.1568), yhat increases from 12.96 to 13.45 as slav89 increases from 82.14 to 97.16, a difference of _____

·        when the change in percent minority population is high (nonslav8 = 2.2550), yhat increases from 12.59 to 13.34 as slav89 increases from 82.14 to 97.16, a difference of _____

 

Three major tasks remain: to put R’s power to work by graphing predicted values, to put the power of R to work by graphing marginal effects, and to perform tests of statistical significance on marginal effects. 

 

You are about to create a new variable with many values of slav89, and then use the basic equation for predicted values, i.e.,  , to inspect how the value of e03ld is expected to vary as slav89 increases, at the same three values of nonslav8 identified above.

 

Run the code in line 31 to create a variable with 101 values, evenly spaced between the minimum and the maximum values of slav89. Next, run lines 32–34 to create predicted vote shares for the Zhirinovsky Bloc for all values in the domain of slav89, using the baseline value we created earlier (line 22), and setting nonslav8 equal to its first quartile (line 32), its median (line 33), and its third quartile (line 34). I have included the code from line 33 for your inspection:

yhat.q2 <- baseline + model$coef[2]*index + model$coef[3]*sumns[3] + model$coef[4]*index*sumns[3]

Thus, you are adding the (a) baseline value, plus (b) the product of the coefficient for slav89 and a value of slav89, plus (c) the product of the coefficient for nonslav8 and the median value of nonslav8, plus (d) the product of the interaction term and the median value for nonslav8 and a value of slav89.

 

Run the code in lines 35-36 to plot the three lines generated when nonslav8’s value is set at its first quartile (shown in red), at its median (shown in black), and at its third quartile (shown in blue). You should have expected to see all three of these lines being upward sloping. 

 

If the posited interaction effect for nonslav8 is correct, then the steepest line should be the blue line (more rapid increase in minority population) and the shallowest line should be the red line (slower increase in minority population). Are these expectations met? 

 

If the posited its main effect for nonslav8 is correct, the blue line’s value should diverge above the black line as slav89’s value increases, and the red line’s value should diverge below the black line as slav89’s value increases. Are these expectations met?

 

The next step is to investigate the interaction effect at all values in the domain of nonslav8. To foreshadow, you are about to create a new variable with many values of nonslav8, and then use the basic equation for marginal effects for x, i.e.,  , to inspect how the value of the partial derivative with respect to slav89 (¶ e03ld /¶ slav89) changes as the value of nonslav8 increases.

 

Run the code in line 38 to create a variable with 101 values, evenly spaced between the minimum and the maximum values of nonslav8. Next, run line 39 to create marginal effects for slav89 at all values in the domain of nonslav8. I have included the code from line 39 for your inspection:

dydsl89 <- model$coef[2]+model$coef[4]*smooth

 

You can run lines 40–45 or type the following code to plot marginal effects over the domain of nonslav8:

plot(smooth, dydsl89, cex=.25, xlim=c(min(smooth),max(smooth)))

 

Take the time to examine closely the equations for creating this plot. Students who have learned calculus will have an easier time understanding that the marginal effect of x is the first derivative with respect to x of the function y = f(x,z) = b0 + b1x + b2z + b3xz; using the rules for derivatives yields ¶y/¶x = b1 + b3z.

So, you must add the coefficient for slav89 and the coefficient for the interaction times a given value of nonslav8.

 

Nine lines of code (47–51 plus 54–57) provide one way to plot marginal effects and confidence intervals around those marginal effects. Recall that in a linear model, the marginal effect equals the regression coefficient, and there is a single confidence interval around that effect. For a model with an interaction, however, the marginal effect will depend on the value of z (as you saw above), but the confidence interval also depends on the value of z. You will start by saving the variance-covariance matrix of the estimators:

vce=vcov(model)

 

Next you will save three entries: the variance of the main effect coefficient for slav89, the variance of the interaction effect coefficient, and the covariance of the aforementioned main effect coefficient and the interaction effect coefficient:

varsl89<-vce[2,2]

varsl89nsl8<-vce[4,4]

cova<-vce[2,4]

 

Next you will combine these three terms using the equation for the standard error for a marginal effect, i.e.,  , at all 1000 values of nonslav8 that you generated earlier:

            sedydsl89<-sqrt(varsl89 + (smooth^2)*varsl89nsl8 + 2*smooth*cova)

If you wish to see how the standard error varies with the value of nonslav8, then you can run line 52.

 

To generate the confidence intervals around the marginal effect, you will need to multiply a t critical value times the standard error. Many of the scholarship about interaction terms (Kam and Franzese; Matt Golder and his co-authors) advise using a 90% confidence interval. Line 54 provides code for finding the t critical value associated with 5% in the upper tail of the t distribution, for degrees of freedom equal to n–k–1 from the regression model run earlier.

            t.star = qt(.95,model$df.residual)

 

Run lines 55–56 to create the upper and lower bounds for the confidence interval for the slope; you are adding and subtracting 1.96 times the standard error (computed in line 69) to/from the slope.

            upperci <- dydsl89 + t.star*sedydsl89 

            lowerci <- dydsl89 - t.star*sedydsl89

 

Finally, run line 57 to plot the marginal effect and its confidence interval, with a horizontal line for zero. The curve shown by running line 57 is not the predicted value of e03ld given the value of slav89; rather the curve shows the slope of the function relating share of the population belonging to the ethnic majority to support for the Zhirinovsky Bloc. 

 

When the marginal effect is above zero, it indicates that in this range, increases in slav89 have a positive effect on support for the Zhirinovsky Bloc. When the marginal effect plot slopes upward, it indicates that the impact of ethnic majority population on support for the Zhirnovsky Bloc increases as the change in the percent of ethnic minorities increases.

 

You might be wondering whether someone has automated this process. The answer is yes; the interplot package is one way of doing this. Run line 59 to install the package and run line 60 to activate it.

Run line 61 of to re-estimate the model with an interaction rather than the interaction variable; take a moment to inspect the difference between model (which you ran in line 10) and model2.

 

You can run lines 62–69 to show a basic plot (most of the code has to do with labels), or alternatively, you can run lines 71–78 to include a histogram of values of nonslav8 (the rest of the code is identical). Note that the default is to use a 95% confidence level; this can be changed easily by typing ci = 0.90, inside the first set of parentheses. 

 

Pretty cool, right? 

 

To clear the Environment, type rm(list=ls()) or click on the broom icon.

To clear the Console window, type Ctrl-l


 
[1] Alexseev’s theory is based on studies of anti-migrant vigilantism in American cities and anti-minority hate crimes in New York’s 51 boroughs, so this is an interesting application of American scholarship to comparative politics.