Quantitative Research Methods - Introduction to Applied Statistics

Eighth meeting

• Linear regression

• Todorov et. al. (2005) “Inferences of competence from faces predict election outcomes.” Science, vol. 308, no. 10 (June), pp. 1623–1626.

• How can one predict election results based on facial appearance?

## Read your data
## Download it from here: https://goo.gl/xYSjCr
face <- read.csv("face.csv")
## two-party vote share for Democrats and Republicans
face$d.share <- face$d.votes / (face$d.votes + face$r.votes)
face$r.share <- face$r.votes / (face$d.votes + face$r.votes)
face$diff.share <- face$d.share - face$r.share

• What is the correlation between perceived competence and vote share differential?

cor(face$d.comp, face$diff.share)

library(ggplot2)
ggplot(face, aes(x=d.comp, y=diff.share)) +
         geom_point(aes(color=w.party))+
        labs(title="Facial competence and vote share",
             x="Competence scores for Democrats",
             y="Competence scores for Republicans")+
      scale_color_manual(name="Winning party",
                           values=c("blue", "red"))

\( Y = \alpha + \beta X + \epsilon \), where \( \alpha \) is an intercept, \( X \) independent (explanatory) variable, \( \beta \) - regression coefficientS, \( \epsilon \) error term.

• All models are wrong, but some are useful;
• Our goal is to assess data-generating process
• Therefore, in all models, we present estimated coefficients

\( \hat{Y} = \hat{\alpha}+\hat{\beta} x \), where \( x \) is some value of \( X \);

\( \hat{\epsilon} = Y-\hat{Y} \), where \( x \) is some value of \( X \);

fit <- lm(diff.share ~ d.comp, data = face) # fit the model
fit

ggplot(face, aes(x=d.comp, y=diff.share)) +
    geom_point(aes(color=w.party))+
    geom_smooth(method='lm')+
    labs(title="Facial competence and vote share",
        x="Competence scores for Democrats",
        y="Competence scores for Republicans")+
    scale_color_manual(name="Winning party",
        values=c("blue", "red"))

• The best theoretical line
• We select those paramenters which produce the least errors
• Root-mean-squared error
  • \( RMSE = \sqrt{\frac{1}{n}SSR} \)

epsilon.hat <- resid(fit) # Residuals
sqrt(mean(epsilon.hat^2)) # RMSE

\( \hat{\alpha}=\bar{Y}-\hat{\beta}\bar{X} \)

\( \beta = corr(X, Y) * \frac{SD(Y)}{SD(X)} \)

outlierTest(fit)
qqPlot(fit, main="QQ Plot")
leveragePlots(fit)

# added variable plots 
av.Plots(fit)
# Cook's D plot
# identify D values > 4/(n-k-1) 
cutoff <- 4/((nrow(face)-length(fit$coefficients)-2)) 
plot(fit, which=4, cook.levels=cutoff)
# Influence Plot 
influencePlot(fit,  id.method="identify", main="Influence Plot", sub="Circle size is proportial to Cook's Distance" )

• MASS library

# Normality of Residuals
# qq plot for studentized resid
qqPlot(fit, main="QQ Plot")
# distribution of studentized residuals
library(MASS)
sresid <- studres(fit) 
hist(sresid, freq=FALSE, 
   main="Distribution of Studentized Residuals")
xfit<-seq(min(sresid),max(sresid),length=40) 
yfit<-dnorm(xfit) 
lines(xfit, yfit)

# Evaluate homoscedasticity
# non-constant error variance test
ncvTest(fit)
# plot studentized residuals vs. fitted values 
spreadLevelPlot(fit)

vif(fit) # variance inflation factors 
sqrt(vif(fit)) > 2 # problem?

# Evaluate Nonlinearity
# component + residual plot 
crPlots(fit)
# Ceres plots 
ceresPlots(fit)

durbinWatsonTest(fit)