Multiple Linear Regression example on Boston Dataset
Loading Required Libraries
Summary
The dataset for this project originates from the UCI Machine Learning Repository. The Boston housing data was collected in 1978 and each of the 506 entries represent aggregated data about 14 features for homes from various suburbs in Boston, Massachusetts. For the purposes of this project, the following preprocessing steps have been made to the dataset: 16 data points have an ‘MEDV’ value of 50.0. These data points likely contain missing or censored values and have been removed.
Data Load and Initial analysis
houseprice<-read.csv("housing.csv", header = TRUE)
str(houseprice)## 'data.frame': 489 obs. of 4 variables:
## $ RM : num 6.58 6.42 7.18 7 7.15 ...
## $ LSTAT : num 4.98 9.14 4.03 2.94 5.33 ...
## $ PTRATIO: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ MEDV : num 504000 453600 728700 701400 760200 ...is.na(houseprice)## RM LSTAT PTRATIO MEDV
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## [489,] FALSE FALSE FALSE FALSEsummary(houseprice)## RM LSTAT PTRATIO MEDV
## Min. :3.561 Min. : 1.98 Min. :12.60 Min. : 105000
## 1st Qu.:5.880 1st Qu.: 7.37 1st Qu.:17.40 1st Qu.: 350700
## Median :6.185 Median :11.69 Median :19.10 Median : 438900
## Mean :6.240 Mean :12.94 Mean :18.52 Mean : 454343
## 3rd Qu.:6.575 3rd Qu.:17.12 3rd Qu.:20.20 3rd Qu.: 518700
## Max. :8.398 Max. :37.97 Max. :22.00 Max. :1024800attach(houseprice)
plot1<-qplot(MEDV, RM)
plot2<-qplot(MEDV, LSTAT)
plot3<-qplot(MEDV, PTRATIO)
library(gridExtra)
grid.arrange(plot1,plot2,plot3, nrow = 1)
##Observation * All attributes are numeric in nature and none of the columns have NULL or NA values * The plot between Rooms and Mean value of house price show a positive coorelation, indicating an increase in price as the no of rooms increase The plot between ratio of home owners with low income and the house price is negatively correlated, implying that low income areas will not have expensive residency complexes The PTRATIO and MEDV have a negative correlation, indicating house prices will be higher in areas with a low student teacher ratio as students will get more attention
Run Regression Model
pricingmodel<-lm(MEDV~RM+LSTAT+PTRATIO, data = houseprice)
summary(pricingmodel)##
## Call:
## lm(formula = MEDV ~ RM + LSTAT + PTRATIO, data = houseprice)
##
## Residuals:
## Min 1Q Median 3Q Max
## -231330 -55228 -8137 41788 326444
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 415464.4 68845.7 6.035 3.17e-09 ***
## RM 86565.2 7888.9 10.973 < 2e-16 ***
## LSTAT -10849.3 732.1 -14.819 < 2e-16 ***
## PTRATIO -19492.1 2039.0 -9.559 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 88130 on 485 degrees of freedom
## Multiple R-squared: 0.7176, Adjusted R-squared: 0.7159
## F-statistic: 410.9 on 3 and 485 DF, p-value: < 2.2e-16houseAIC<-AIC(pricingmodel)
houseAIC## [1] 12529.78houseVIF<-vif(pricingmodel)
houseVIF## RM LSTAT PTRATIO
## 1.619920 1.689112 1.164402Observation
We find, that Multiple R-squared is 71.76% and value of Adjusted R-square is 71.59% This means about 72% of variation in target variable (MEDV) can be predicted by predictor variables, indicating a erasonably good fit. The Variable inflation factor values for all the predictor variables are under 2 , indicating very less multicolinearity. Hence this is a good model
Conclusion
egression equation is MEDV = 415464.4 + 86565.2 * RM - 10849.3 * LSTAT - 19492.1 * PTRATIO
The coefficient for the variable RMA says that for a fixed combination of LSTAT and PTRATIO, on average RM will be costing 86565.2 more than others.
The coefficient of about -10849 for LSTAT tells us that for a given RM and PTRATIO, the predicted MEDV decreases by about 10849 for every 1.0 unit increase in LSTAT.
The coefficient of about -19492 for PTRATIO tells us that for a given RM and LSTAT, the predicted MEDV decreases by about 19492 for every 1.0 unit increase in PTRATIO.