Without examining interaction effects in your model, sometimes we are incorrect about the real relationship between variables.

This is particularly evident in political science when we consider, for example, the impact of regime type on the relationship between our dependent and independent variables. The nature of the government can really impact our analysis.

For example, I were to look at the relationship between anti-government protests and executive bribery.

I would expect to see that the higher the bribery score in a country’s government, the higher prevalence of people protesting against this corrupt authority. Basically, people are angry when their government is corrupt. And they make sure they make this very clear to them by protesting on the streets.

First, I will describe the variables I use and their data type.

With the dependent variable democracy_protest being an interval score, based upon the question: In this year, how frequent and large have events of mass mobilization for pro-democratic aims been?

The main independent variable is another interval score on executive_bribery scale and is based upon the question: How clean is the executive (the head of government, and cabinet ministers), and their agents from bribery (granting favors in exchange for bribes, kickbacks, or other material inducements?)

So, let’s run a quick regression to examine this relationship:

summary(protest_model <- lm(democracy_protest ~ executive_bribery, data = data_2010))

Examining the results of the regression model:

We see that there is indeed a negative relationship. The cleaner the government, the less likely people in the country will protest in the year under examination. This confirms our above mentioned hypothesis.

However, examining the R2, we see that less than 1% of the variance in protest prevalence is explained by executive bribery scores.

Not very promising.

Is there an interaction effect with regime type? We can look at a scatterplot and see if the different regime type categories cluster in distinct patterns.

The four regime type categories are

purple: liberal democracy (such as Sweden or Canada)

teal: electoral democracy (such as Turkey or Mongolia)

khaki green: electoral autocracy (such as Georgia or Ethiopia)

red: closed autocracy (such as Cuba or China)

The color clusters indicate regime type categories do cluster.

Liberal democracies (purple) cluster at the top left hand corner. Higher scores in clean executive index and lower prevalence in pro-democracy protesting.

Electoral autocracies (teal) cluster in the middle.

Electoral democracies (khaki green) cluster at the bottom of the graph.

The closed autocracy countries (red) seem to have a upward trend, opposite to the overall best fitted line.

So let’s examine the interaction effect between regime types and executive corruption with mass pro-democracy protests.

Plot the model and add the * interaction effect:

summary(protest_model_2 <-lm(democracy_protest ~ executive_bribery*regime_type, data = data_2010))

Adding the regime type variable, the R2 shoots up to 27%.

The interaction effect appears to only be significant between clean executive scores and liberal democracies. The cleaner the country’s executive, the prevalence of mass mobilization and protests decreases by -0.98 and this is a statistically significant relationship.

The initial relationship we saw in the first model, the simple relationship between clean executive scores and protests, has disappeared. There appears to be no relationship between bribery and protests in the semi-autocratic countries; (those countries that are not quite democratic but not quite fully despotic).

Let’s graph out these interactions.

In the plot_model() function, first type the name of the model we fitted above, protest_model.

Next, choose the type . For different type arguments, scroll to the bottom of this blog post. We use the type = "pred" argument, which plots the marginal effects.

Marginal effects tells us how a dependent variable changes when a specific independent variable changes, if other covariates are held constant. The two terms typed here are the two variables we added to the model with the * interaction term.

install.packages("sjPlot")
library(sjPlot)
plot_model(protest_model, type = "pred", terms = c("executive_bribery", "regime_type"), title = 'Predicted values of Mass Mobilization Index',
legend.title = "Regime type")

Looking at the graph, we can see that the relationship changes across regime type. For liberal democracies (purple), there is a negative relationship. Low scores on the clean executive index are related to high prevalence of protests. So, we could say that when people in democracies see corrupt actions, they are more likely to protest against them.

However with closed autocracies (red) there is the opposite trend. Very corrupt countries in closed autocracies appear to not have high levels of protests.

This would make sense from a theoretical perspective: even if you want to protest in a very corrupt country, the risk to your safety or livelihood is often too high and you don’t bother. Also the media is probably not free so you may not even be aware of the extent of government corruption.

It seems that when there are no democratic features available to the people (free media, freedom of assembly, active civil societies, or strong civil rights protections, freedom of expression et cetera) the barriers to protesting are too high. However, as the corruption index improves and executives are seen as “cleaner”, these democratic features may be more accessible to them.

If we only looked at the relationship between the two variables and ignore this important interaction effects, we would incorrectly say that as

Of course, panel data would be better to help separate any potential causation from the correlations we can see in the above graphs.

The blue line is almost vertical. This matches with the regression model which found the coefficient in electoral autocracy is 0.001. Virtually non-existent.

Different Plot Types

type = "std" – Plots standardized estimates.

type = "std2" – Plots standardized estimates, however, standardization follows Gelman’s (2008) suggestion, rescaling the estimates by dividing them by two standard deviations instead of just one. Resulting coefficients are then directly comparable for untransformed binary predictors.

type = "pred" – Plots estimated marginal means (or marginal effects). Simply wraps ggpredict.

type = "eff"– Plots estimated marginal means (or marginal effects). Simply wraps ggeffect.

type = "slope" and type = "resid" – Simple diagnostic-plots, where a linear model for each single predictor is plotted against the response variable, or the model’s residuals. Additionally, a loess-smoothed line is added to the plot. The main purpose of these plots is to check whether the relationship between outcome (or residuals) and a predictor is roughly linear or not. Since the plots are based on a simple linear regression with only one model predictor at the moment, the slopes (i.e. coefficients) may differ from the coefficients of the complete model.

type = "diag" – For Stan-models, plots the prior versus posterior samples. For linear (mixed) models, plots for multicollinearity-check (Variance Inflation Factors), QQ-plots, checks for normal distribution of residuals and homoscedasticity (constant variance of residuals) are shown. For generalized linear mixed models, returns the QQ-plot for random effects.

When one independent variable is highly correlated with another independent variable (or with a combination of independent variables), the marginal contribution of that independent variable is influenced by other predictor variables in the model.

And so, as a result:

Estimates for regression coefficients of the independent variables can be unreliable.

Tests of significance for regression coefficients can be misleading.

To check for multicollinearity problem in our model, we need the vif() function from the car package in R. VIF stands for variance inflation factor. It measures how much the variance of any one of the coefficients is inflated due to multicollinearity in the overall model.

As a rule of thumb, a vif score over 5 is a problem. A score over 10 should be remedied and you should consider dropping the problematic variable from the regression model or creating an index of all the closely related variables.

This blog post will look only at the VIF score. Click here to look at how to interpret various other multicollinearity tests in the mctest package in addition to the the VIF score.

Back to our model, I want to know whether countries with high levels of clientelism, high levels of vote buying and low democracy scores lead to executive embezzlement?

So I fit a simple linear regression model (and look at the output with the stargazer package)

summary(embezzlement_model_1 <- lm(executive_embezzlement ~ clientelism_index + vote_buying_score + democracy_score, data = data_2010))
stargazer(embezzlement_model_1, type = "text")

I suspect that clientelism and vote buying variables will be highly correlated. So let’s run a test of multicollinearity to see if there is any problems.

car::vif(embezzlement_model_1)

The VIF score for the three independent variables are :

Both clientelism index and vote buying variables are both very high and the best remedy is to remove one of them from the regression. Since vote buying is considered one aspect of clientelist regime so it is probably overlapping with some of the variance in the embezzlement score that the clientelism index is already explaining in the model

So re-run the regression without the vote buying variable.

summary(embezzlement_model_2 <- lm(v2exembez ~ v2xnp_client + v2x_api, data = vdem2010))
stargazer(embezzlement_model_2, embezzlement_model_2, type = "text")
car::vif(embezzlement_mode2)

Comparing the two regressions:

And running a VIF test on the second model without the vote buying variable:

car::vif(embezzlement_model_2)

These scores are far below 5 so there is no longer any big problem of multicollinearity in the second model.

Click here to quickly add VIF scores to our regression output table in R with jtools package.

Plus, looking at the adjusted R2, which compares two models, we see that the difference is very small, so we did not lose much predictive power in dropping a variable. Rather we have minimised the issue of highly correlated independent variables and thus an inability to tease out the real relationships with our dependent variable of interest.

tl;dr: As a rule of thumb, a vif score over 5 is a problem. A score over 10 should be remedied (and you should consider dropping the problematic variable from the regression model or creating an index of all the closely related variables).

If our OLS model demonstrates high level of heteroskedasticity (i.e. when the error term of our model is not randomly distributed across observations and there is a discernible pattern in the error variance), we run into problems.

Why? Because this means OLS will use sub-optimal estimators based on incorrect assumptions and the standard errors computed using these flawed least square estimators are more likely to be under-valued.

Since standard errors are necessary to compute our t – statistic and arrive at our p – value, these inaccurate standard errors are a problem.

Click here to check for heteroskedasticity in your model with the lmtest package.

To correct for heteroskedastcity in your model, you need the sandwich package and the lmtest package to employ the vcocHC argument.

First, let’s fit a simple OLS regression.

summary(free_express_model <- lm(freedom_expression ~ free_elections + deliberative_index, data = data_1990))

We can see that there is a small star beside the main dependent variable of interest! Success!

We have significance.

Thus, we could say that the more free and fair the elections a country has, this increases the mean freedom of expression index score for that country.

This ties in with a very minimalist understanding of democracy. If a country has elections and the populace can voice their choice of leadership, this will help set the scene for a more open society.

However, it is naive to look only at the p – value of any given coefficient in a regression output. If we run some diagnostic analyses and look at the relationship graphically, we may need to re-examine this seemingly significant relationship.

Can we trust the 0.087 standard error score that our OLS regression calculated? Is it based on sound assumptions?

First let’s look at the residuals. Can we assume that the variance of error is equal across all observations?

If we examine the residuals (the first graph), we see that there is actually a tapered fan-like pattern in the error variance. As we move across the x axis, the variance along the y axis gets continually smaller and smaller.

The error does not look random.

Let’s run a Breush-Pagan test (from the lmtest package) to check our suspicion of heteroskedasticity.

lmtest::bptest(free_exp_model)

We can reject the null hypothesis that the error variance is homoskedastic.

So the model does suffer from heteroskedasticty. We cannot trust those stars in the regression output!

In order to fix this and make our p-values more accuarate, we need to install the sandwich package to feed in the vcovHC adjustment into the coeftest() function.

vcovHC stands for variance covariance Heteroskedasticity Consistent.

With the stargazer package (which prints out all the models in one table), we can compare the free_exp_model alone with no adjustment, then four different variations of the vcovHC adjustment using different formulae (as indicated in the type argument below).

stargazer(free_exp_model,
coeftest(free_exp_model, vcovHC(free_exp_model, type = "HC0")),
coeftest(free_exp_model, vcovHC(free_exp_model, type = "HC1")),
coeftest(free_exp_model, vcovHC(free_exp_model, type = "HC2")),
coeftest(free_exp_model, vcovHC(free_exp_model, type = "HC3")),
type = "text")

Looking at the standard error in the (brackets) across the OLS and the coeftest models, we can see that the standard error are all almost double the standard error from the original OLS regression.

There is a tiny difference between the different types of Heteroskedastic Consistent (HC) types.

The significant p – value disappears from the free and fair election variable when we correct with the vcovHC correction.

The actual coefficient stays the same regardless of whether we use no correction or any one of the correction arguments.

Which HC estimator should I use in my vcovHC() function?

The default in the sandwich package is HC3.

STATA users will be familiar with HC1, as it is the default robust standard error correction when you add robust at the end of the regression command.

The difference between them is not very large.

The estimator HC0 was suggested in the econometrics literature by White in 1980 and is justified by asymptotic arguments.

For small sample sizes, the standard errors from HC0 are quite biased, usually downward, and this results in overly liberal inferences in regression models (Bera, Suprayitno & Premaratne, 2002). But with HC0, the bias shrinks as your sample size increases.

The estimator types HC1, HC2 and HC3 were put forward by MacKinnon and White (1985) to improve the performance in small samples.

Long and Ervin (2000) furthermore argue that HC3 provides the best performance in small samples as it gives less weight to influential observations in the model

In our freedom of expression regression, the HC3 estimate was the most conservative with the standard error calculations. however the difference between the approaches did not change the conclusion; ultimately the main independent variable of interest in this analysis – free and fair elections – can explain variance in the dependent variable – freedom of expression – does not find evidence in the model.

Click here to read an article by Hayes and Cai (2007) which discusses the matrix formulae and empirical differences between the different calculation approaches taken by the different types. Unfortunately it is all ancient Greek to me.

References

Bera, A. K., Suprayitno, T., & Premaratne, G. (2002). On some heteroskedasticity-robust estimators of variance–covariance matrix of the least-squares estimators. Journal of Statistical Planning and Inference, 108(1-2), 121-136.

Hayes, A. F., & Cai, L. (2007). Using heteroskedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation. Behavior research methods, 39(4), 709-722.

Long, J. S., & Ervin, L. H. (2000). Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician, 54(3), 217-224.

MacKinnon, J. G., & White, H. (1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. Journal of econometrics, 29(3), 305-325.

One core assumption when calculating ordinary least squares regressions is that all the random variables in the model have equal variance around the best fitting line.

Essentially, when we run an OLS, we expect that the error terms have no fan pattern.

Example of homoskedasticiy

So let’s look at an example of this assumption being satisfied. I run a simple regression to see whether there is a relationship between and media censorship and civil society repression in 178 countries in 2010.

summary(repression_model <- lm(media_censorship ~ civil_society_repression, data = data_2010))
stargazer(repression_model, type = "text")

This is pretty common sense; a country that represses its citizens in one sphere is more likely to repress in other areas. In this case repressing the media correlates with repressing civil society.

We can plot the residuals of the above model with the autoplot() function from the ggplotigfy package.

library(ggplotify)
autoplot(repression_model)

Nothing unusual appears to jump out at us with regard to evidence for heteroskedasticity!

In the first Residuals vs Fitted plot, we can see that blue line does not drastically diverge from the dotted line (which indicates residual value = 0).

The third plot Scale-Location shows again that there is no drastic instances of heteroskedasticity. We want to see the blue line relatively horizontal. There is no clear pattern in the distribution of the residual points.

In the Residual vs. Leverage plot (plot number 4), the high leverage observation 19257 is North Korea! A usual suspect when we examine model outliers.

While it is helpful to visually see the residuals plotted out, a more objective test can help us find whether the model is indeed free from heteroskedasticity problems.

For this we need the Breusch-Pagan test for heteroskedasticity from the lmtest package.

The default in R is the studentized Breusch-Pagan. However if you add the studentize = FALSE argument, you have the non-studentized version

The null hypothesis of the Breusch-Pagan test is that the variance in the model is homoskedastic.

With our repression_model, we cannot reject the null, so we can say with confidence that there is no major heteroskedasticity issue in our model.

The non-studentized Breusch-Pagan test makes a very big assumption that the error term is from Gaussian distribution. Since this assumption is usually hard to verify, the default bptest() in R “studentizes” the test statistic and provide asymptotically correct significance levels for distributions for error.

Why do we care about heteroskedasticity?

If our model demonstrates high level of heteroskedasticity (i.e. the random variables have non-random variation across observations), we run into problems.

Why?

OLS uses sub-optimal estimators based on incorrect assumptions and

The standard errors computed using these flawed least square estimators are more likely to be under-valued. Since standard errors are necessary to compute our t – statistics and arrive at our p – value, these inaccurate standard errors are a problem.

Example of heteroskedasticity

Let’s look at an example of this homoskedasticity assumption NOT being satisfied.

I run a simple regression to see whether there is a relationship between democracy score and respect for individuals’ private property rights in 178 countries in 2010.

When you are examining democracy as the main dependent variable, heteroskedasticity is a common complaint. This is because all highly democratic countries are all usually quite similar. However, when we look at autocracies, they are all quite different and seem to march to the beat of their own despotic drum. We cannot assume that the random variance across different regime types is equally likely.

Next, let’s fit the model to examine the relationship.

summary(property_model <- lm(property_score ~ democracy_score, data = data_2010))
stargazer(property_model, type = "text")

To plot the residuals (and other diagnostic graphs) of the model, we can use the autoplot() functionto look at the relationship in the model graphically.

autoplot(property_model)

Graph number 1 plots the residuals against the fitted model and we can see that lower values on the x – axis (fitted values) correspond with greater spread on the y – axis. Lower democracy scores relate to greater error on property rights index scores. Plus the blue line does not lie horizontal and near the dotted line. It appears we have non-random error patterns.

Examining the Scale – Location graph (number 3), we can see that the graph is not horizontal.

Again, interpreting the graph can be an imprecise art. So a more objective approach may be to run the bptest().

bptest(property_model)

Since the p – value is far smaller than 0.05, we can reject the null of homoskedasticity.

Rather, we have evidence that our model suffers from heteroskedasticity. The standard errors in the regression appear smaller than they actually are in reality. This inflates our t – statistic and we cannot trust our p – value.

In the next blog post, we can look at ways to rectify this violation of homoskedasticity and to ensure that our regression output has more accurate standard errors and therefore more accurate p – values.

Click here to use the sandwich package to fix heteroskedasticity in the OLS regression.