Lesson 19 - Qualitative Predictors

In this lesson, we will see how work with qualitative predictors in which there are more than two possible values.

Load Packages and Data

require(ggplot2)

Our goal in this lesson will be to build a regression model for the purposes of predicting the sales price of an apartment. In addition to the response variable, price, our dataset contains two predictors: sqft and region. The predictor sqft is a quantitative variable that is equal to the square footage of the apartment. The predictor region is a qualitative variable that reports in which of three geographic regions the apartment is located. The regions are encoded A, B, and C. This is synthetic data.

df <- read.table("apt_prices.txt", sep="\t", header=TRUE)
summary(df)

     price             sqft        region 
 Min.   : 39067   Min.   : 548.4   A:154  
 1st Qu.:107526   1st Qu.: 762.5   B:117  
 Median :151706   Median : 870.6   C:156  
 Mean   :181081   Mean   : 903.7          
 3rd Qu.:221799   3rd Qu.:1021.1          
 Max.   :776368   Max.   :1735.8          

Exploratory Analysis

We will begin our analysis by plotting price against sqft.

ggplot(df, aes(x=sqft, y=price, col=region)) + geom_point()

We see that heteroskedasticity is likely a concern with this dataset. We will now plot ln(price) against ln(sqft).

ggplot(df, aes(x=log(sqft), y=log(price), col=region)) + geom_point()

The log-log transformation seems to have addressed the concerns regarding heteroskedasticity.

ggplot(df, aes(x=region, y=price, fill = region)) + geom_boxplot()

ggplot(df, aes(x=sqft, y=price, col=region)) + geom_point() + facet_wrap(~region)

Create Model

mod <- lm(log(price) ~ log(sqft) + region, df)
summary(mod)

Call:
lm(formula = log(price) ~ log(sqft) + region, data = df)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.83246 -0.17565 -0.02565  0.20028  0.78663 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.70981    0.44700  -1.588   0.1130    
log(sqft)    1.85212    0.06602  28.055   <2e-16 ***
regionB      0.46219    0.03580  12.909   <2e-16 ***
regionC     -0.07744    0.03311  -2.339   0.0198 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2911 on 423 degrees of freedom
Multiple R-squared:  0.7211,    Adjusted R-squared:  0.7191 
F-statistic: 364.6 on 3 and 423 DF,  p-value: < 2.2e-16

\[\large\ln(\hat {price}) = -0.70981 + 1.85212\cdot \ln(sqft) + 0.46219 \cdot regionB - 0.07744\cdot regionC\]

Residual Analysis

plot(mod$residuals ~ mod$fitted.values, pch=21, col="black", bg="salmon",
     xlab="Fitted Value", ylab="Residuals", main="Residual Plot")
abline(h=0, col="darkred", lwd=2)

res <- mod$residuals
par(mfrow=c(1,2))
hist(res, col='orchid', breaks=10)
qqnorm(res)

qqline(res)
par(mfrow=c(1,1))