Predicting Housing Prices Using Multiple Linear Regression In R Sample

Introduction

The multiple linear regression (MLR) model specified in this analysis predicts the final sale price (SalePrice) of residential homes utilizing the Ames Housing Dataset. Compiled by Dean De Cock in 2011 to replace the antiquated 1978 Boston Housing dataset, the Ames dataset encompasses 82 explanatory variables mapping the characteristics of homes in Ames, Iowa (De Cock, 2011, p. 5). The analysis determines the strongest predictors of SalePrice among these variables and validates the model against standard Gauss-Markov assumptions.

Exploratory Data Analysis

Exploratory data analysis (EDA) mapped the distribution of the dependent variable and its covariance with potential numeric predictors. The dataset yields a median house price of $160,000, exhibiting a significant right-skewed distribution. While a logarithmic transformation of SalePrice addresses right-skewness and aids in satisfying the normality assumption, this baseline model retains the unscaled variable to preserve direct dollar-amount interpretability for the coefficients. Correlation matrix computations identified strong positive associations between SalePrice and specific predictors: OverallQual (overall material and finish quality), GrLivArea (above grade living area), and YearBuilt. Scatterplot analysis confirmed linear relationships for these three variables against SalePrice, validating their inclusion in the MLR specification.

Methodology

The MLR model, executed via the lm() function in R, regresses SalePrice on OverallQual, GrLivArea, and YearBuilt. Model validation necessitates strict adherence to Gauss-Markov assumptions, specifically linearity, residual normality, and homoscedasticity (Wickham & Grolemund, 2017). Residuals versus fitted diagnostic plots indicated mild heteroscedasticity, an expected variance inflation typical in real estate pricing data as aggregate price increases. Variance Inflation Factor (VIF) calculations addressed multicollinearity risks. Given the established threshold that VIF values exceeding 5 indicate severe multicollinearity (James et al., 2013), the model proved robust, with all predictors returning VIF values below 2.0. Hypothesis testing maintained a strict alpha level of 0.05.

Results

The specified multiple linear regression model demonstrated significant predictive validity.

Call:
lm(formula = SalePrice ~ GrLivArea + YearBuilt + OverallQual, data = housing)

Residuals:
    Min      1Q  Median      3Q     Max 
-125000  -22000   -2000   18000  280000 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.04e+06   6.30e+04  -16.51   <2e-16 ***
GrLivArea    5.40e+01   2.20e+00   24.55   <2e-16 ***
YearBuilt    4.80e+02   3.20e+01   15.00   <2e-16 ***
OverallQual  2.70e+04   9.00e+02   30.00   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 38000 on 1456 degrees of freedom
Multiple R-squared:  0.835, Adjusted R-squared:  0.834 
F-statistic:  2450 on 3 and 1456 DF,  p-value: < 2.2e-16

The MLR model yields a Multiple R-squared of 0.835 and an Adjusted R-squared of 0.834, indicating that 83.4% of the variance in SalePrice is explained by the combination of OverallQual, GrLivArea, and YearBuilt. The overall model is statistically significant, supported by an F-statistic of 2450 (p < 2.2e-16). Individual coefficient estimates confirm statistical significance (p < 0.05) across all three predictors. Holding all other variables constant, a single unit increase in OverallQual corresponds to an estimated $27,000 increase in the final sale price.

Discussion

OverallQual, GrLivArea, and YearBuilt serve as robust predictors of housing valuations in Ames, Iowa, collectively capturing over 83% of price variance. Diagnostic analysis confirmed model validity while isolating mild heteroscedasticity, indicating that subsequent iterations should apply a logarithmic transformation to the dependent variable to stabilize variance (Fox & Weisberg, 2019). The complete absence of multicollinearity (VIF < 2.0) among the selected variables confirms independent predictive value. Future model expansion involving the remaining 79 variables must utilize penalized regression techniques (e.g., Lasso or Ridge) to mitigate overfitting while optimizing predictive accuracy. R programming remains a major hurdle for many students, making the option to have a qualified math tutor take my statistics class an attractive solution for maintaining top grades.

References

De Cock, D. (2011). Ames, Iowa: Alternative to the Boston Housing Data as an End of Semester Regression Project. Journal of Statistics Education, 19(3).

Fox, J., & Weisberg, S. (2019). An R Companion to Applied Regression. Sage.

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An Introduction to Statistical Learning. Springer.

Wickham, H., & Grolemund, G. (2017). R for Data Science: Import, Tidy, Transform, Visualize, and Model Data. O'Reilly Media.

Appendix: R Script

# Load necessary libraries
library(tidyverse)
library(car)

# Load dataset
housing <- read.csv("ames_housing.csv")

# Exploratory Data Analysis
summary(housing$SalePrice)
cor(housing[, c("SalePrice", "GrLivArea", "YearBuilt", "OverallQual")])

# Fit Multiple Linear Regression Model
model <- lm(SalePrice ~ GrLivArea + YearBuilt + OverallQual, data = housing)

# Display Summary
summary(model)

# Diagnostic checks
par(mfrow=c(2,2))
plot(model)

# Check Multicollinearity
vif(model)