Multiple Regression Model using Car Prices
Kagiso Mthimkulu
2026-02-28
1. Introduction
This report explores the variables influencing the price of cars using data from Kaggle. I conducted some exploratory data analysis in ordert to create a multiple linear regression model, and to evaluate its perfromance.
2. Loading the data
# file path
library(readr)
library(readxl)
CarPrice_Assignment_csv = read_csv("~/Desktop/CarPrice_Assignment.csv.xls")
# Display data structure and preview
glimpse(CarPrice_Assignment_csv)## Rows: 205
## Columns: 26
## $ car_ID <dbl> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16…
## $ symboling <dbl> 3, 3, 1, 2, 2, 2, 1, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0,…
## $ CarName <chr> "alfa-romero giulia", "alfa-romero stelvio", "alfa-ro…
## $ fueltype <chr> "gas", "gas", "gas", "gas", "gas", "gas", "gas", "gas…
## $ aspiration <chr> "std", "std", "std", "std", "std", "std", "std", "std…
## $ doornumber <chr> "two", "two", "two", "four", "four", "two", "four", "…
## $ carbody <chr> "convertible", "convertible", "hatchback", "sedan", "…
## $ drivewheel <chr> "rwd", "rwd", "rwd", "fwd", "4wd", "fwd", "fwd", "fwd…
## $ enginelocation <chr> "front", "front", "front", "front", "front", "front",…
## $ wheelbase <dbl> 88.6, 88.6, 94.5, 99.8, 99.4, 99.8, 105.8, 105.8, 105…
## $ carlength <dbl> 168.8, 168.8, 171.2, 176.6, 176.6, 177.3, 192.7, 192.…
## $ carwidth <dbl> 64.1, 64.1, 65.5, 66.2, 66.4, 66.3, 71.4, 71.4, 71.4,…
## $ carheight <dbl> 48.8, 48.8, 52.4, 54.3, 54.3, 53.1, 55.7, 55.7, 55.9,…
## $ curbweight <dbl> 2548, 2548, 2823, 2337, 2824, 2507, 2844, 2954, 3086,…
## $ enginetype <chr> "dohc", "dohc", "ohcv", "ohc", "ohc", "ohc", "ohc", "…
## $ cylindernumber <chr> "four", "four", "six", "four", "five", "five", "five"…
## $ enginesize <dbl> 130, 130, 152, 109, 136, 136, 136, 136, 131, 131, 108…
## $ fuelsystem <chr> "mpfi", "mpfi", "mpfi", "mpfi", "mpfi", "mpfi", "mpfi…
## $ boreratio <dbl> 3.47, 3.47, 2.68, 3.19, 3.19, 3.19, 3.19, 3.19, 3.13,…
## $ stroke <dbl> 2.68, 2.68, 3.47, 3.40, 3.40, 3.40, 3.40, 3.40, 3.40,…
## $ compressionratio <dbl> 9.00, 9.00, 9.00, 10.00, 8.00, 8.50, 8.50, 8.50, 8.30…
## $ horsepower <dbl> 111, 111, 154, 102, 115, 110, 110, 110, 140, 160, 101…
## $ peakrpm <dbl> 5000, 5000, 5000, 5500, 5500, 5500, 5500, 5500, 5500,…
## $ citympg <dbl> 21, 21, 19, 24, 18, 19, 19, 19, 17, 16, 23, 23, 21, 2…
## $ highwaympg <dbl> 27, 27, 26, 30, 22, 25, 25, 25, 20, 22, 29, 29, 28, 2…
## $ price <dbl> 13495.00, 16500.00, 16500.00, 13950.00, 17450.00, 152…cat("\nFirst few rows of the dataset:\n")##
## First few rows of the dataset:print(head(CarPrice_Assignment_csv))## # A tibble: 6 × 26
## car_ID symboling CarName fueltype aspiration doornumber carbody drivewheel
## <dbl> <dbl> <chr> <chr> <chr> <chr> <chr> <chr>
## 1 1 3 alfa-romer… gas std two conver… rwd
## 2 2 3 alfa-romer… gas std two conver… rwd
## 3 3 1 alfa-romer… gas std two hatchb… rwd
## 4 4 2 audi 100 ls gas std four sedan fwd
## 5 5 2 audi 100ls gas std four sedan 4wd
## 6 6 2 audi fox gas std two sedan fwd
## # ℹ 18 more variables: enginelocation <chr>, wheelbase <dbl>, carlength <dbl>,
## # carwidth <dbl>, carheight <dbl>, curbweight <dbl>, enginetype <chr>,
## # cylindernumber <chr>, enginesize <dbl>, fuelsystem <chr>, boreratio <dbl>,
## # stroke <dbl>, compressionratio <dbl>, horsepower <dbl>, peakrpm <dbl>,
## # citympg <dbl>, highwaympg <dbl>, price <dbl>Car_Price_Data1 = CarPrice_Assignment_csv
# clean columns
Car_Price_Data1 = Car_Price_Data1 %>%
clean_names()
head(Car_Price_Data1)## # A tibble: 6 × 26
## car_id symboling car_name fueltype aspiration doornumber carbody drivewheel
## <dbl> <dbl> <chr> <chr> <chr> <chr> <chr> <chr>
## 1 1 3 alfa-romer… gas std two conver… rwd
## 2 2 3 alfa-romer… gas std two conver… rwd
## 3 3 1 alfa-romer… gas std two hatchb… rwd
## 4 4 2 audi 100 ls gas std four sedan fwd
## 5 5 2 audi 100ls gas std four sedan 4wd
## 6 6 2 audi fox gas std two sedan fwd
## # ℹ 18 more variables: enginelocation <chr>, wheelbase <dbl>, carlength <dbl>,
## # carwidth <dbl>, carheight <dbl>, curbweight <dbl>, enginetype <chr>,
## # cylindernumber <chr>, enginesize <dbl>, fuelsystem <chr>, boreratio <dbl>,
## # stroke <dbl>, compressionratio <dbl>, horsepower <dbl>, peakrpm <dbl>,
## # citympg <dbl>, highwaympg <dbl>, price <dbl>## 3. Exploratory Data Analysis
### 3.1. Summary of statisticssummary(Car_Price_Data1)## car_id symboling car_name fueltype
## Min. : 1 Min. :-2.0000 Length:205 Length:205
## 1st Qu.: 52 1st Qu.: 0.0000 Class :character Class :character
## Median :103 Median : 1.0000 Mode :character Mode :character
## Mean :103 Mean : 0.8341
## 3rd Qu.:154 3rd Qu.: 2.0000
## Max. :205 Max. : 3.0000
## aspiration doornumber carbody drivewheel
## Length:205 Length:205 Length:205 Length:205
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
##
##
##
## enginelocation wheelbase carlength carwidth
## Length:205 Min. : 86.60 Min. :141.1 Min. :60.30
## Class :character 1st Qu.: 94.50 1st Qu.:166.3 1st Qu.:64.10
## Mode :character Median : 97.00 Median :173.2 Median :65.50
## Mean : 98.76 Mean :174.0 Mean :65.91
## 3rd Qu.:102.40 3rd Qu.:183.1 3rd Qu.:66.90
## Max. :120.90 Max. :208.1 Max. :72.30
## carheight curbweight enginetype cylindernumber
## Min. :47.80 Min. :1488 Length:205 Length:205
## 1st Qu.:52.00 1st Qu.:2145 Class :character Class :character
## Median :54.10 Median :2414 Mode :character Mode :character
## Mean :53.72 Mean :2556
## 3rd Qu.:55.50 3rd Qu.:2935
## Max. :59.80 Max. :4066
## enginesize fuelsystem boreratio stroke
## Min. : 61.0 Length:205 Min. :2.54 Min. :2.070
## 1st Qu.: 97.0 Class :character 1st Qu.:3.15 1st Qu.:3.110
## Median :120.0 Mode :character Median :3.31 Median :3.290
## Mean :126.9 Mean :3.33 Mean :3.255
## 3rd Qu.:141.0 3rd Qu.:3.58 3rd Qu.:3.410
## Max. :326.0 Max. :3.94 Max. :4.170
## compressionratio horsepower peakrpm citympg
## Min. : 7.00 Min. : 48.0 Min. :4150 Min. :13.00
## 1st Qu.: 8.60 1st Qu.: 70.0 1st Qu.:4800 1st Qu.:19.00
## Median : 9.00 Median : 95.0 Median :5200 Median :24.00
## Mean :10.14 Mean :104.1 Mean :5125 Mean :25.22
## 3rd Qu.: 9.40 3rd Qu.:116.0 3rd Qu.:5500 3rd Qu.:30.00
## Max. :23.00 Max. :288.0 Max. :6600 Max. :49.00
## highwaympg price
## Min. :16.00 Min. : 5118
## 1st Qu.:25.00 1st Qu.: 7788
## Median :30.00 Median :10295
## Mean :30.75 Mean :13277
## 3rd Qu.:34.00 3rd Qu.:16503
## Max. :54.00 Max. :45400# Create a heatmap
library(pheatmap)
numeric_cols = Car_Price_Data1[, sapply(Car_Price_Data1, is.numeric)]
scaled_data = scale(numeric_cols)
pheatmap(scaled_data,
main = "Car Features Heatmap",
show_rownames = FALSE,
clustering_method = "complete",
color = colorRampPalette(c("blue", "white", "red"))(50)
)
### 3.2. Distribution of the data
numeric_cols %>%
pivot_longer(cols = everything(), names_to = "variable", values_to = "value") %>%
ggplot(aes(x = value)) +
geom_histogram(bins = 10, fill = "steelblue", colour = "black") +
facet_wrap(~ variable, scales = "free") +
labs(title = "Histogram of all numeric values")
# Convert the appropriate columns to factors
Car_Price_Data1$fueltype = as.factor(Car_Price_Data1$fueltype)
Car_Price_Data1$aspiration = as.factor(Car_Price_Data1$aspiration)
Car_Price_Data1$doornumber = as.factor(Car_Price_Data1$doornumber)
Car_Price_Data1$enginelocation = as.factor(Car_Price_Data1$enginelocation)
Car_Price_Data1$drivewheel = as.factor(Car_Price_Data1$drivewheel)
Car_Price_Data1$carbody = as.factor(Car_Price_Data1$carbody)3.3. Creating a Correlation Matrix
numeric_vars = Car_Price_Data1 %>%
select(where(is.numeric), -price)
corr_matrix = cor(numeric_vars, use = "complete.obs")
print(round(corr_matrix, 2))## car_id symboling wheelbase carlength carwidth carheight
## car_id 1.00 -0.15 0.13 0.17 0.05 0.26
## symboling -0.15 1.00 -0.53 -0.36 -0.23 -0.54
## wheelbase 0.13 -0.53 1.00 0.87 0.80 0.59
## carlength 0.17 -0.36 0.87 1.00 0.84 0.49
## carwidth 0.05 -0.23 0.80 0.84 1.00 0.28
## carheight 0.26 -0.54 0.59 0.49 0.28 1.00
## curbweight 0.07 -0.23 0.78 0.88 0.87 0.30
## enginesize -0.03 -0.11 0.57 0.68 0.74 0.07
## boreratio 0.26 -0.13 0.49 0.61 0.56 0.17
## stroke -0.16 -0.01 0.16 0.13 0.18 -0.06
## compressionratio 0.15 -0.18 0.25 0.16 0.18 0.26
## horsepower -0.02 0.07 0.35 0.55 0.64 -0.11
## peakrpm -0.20 0.27 -0.36 -0.29 -0.22 -0.32
## citympg 0.02 -0.04 -0.47 -0.67 -0.64 -0.05
## highwaympg 0.01 0.03 -0.54 -0.70 -0.68 -0.11
## curbweight enginesize boreratio stroke compressionratio
## car_id 0.07 -0.03 0.26 -0.16 0.15
## symboling -0.23 -0.11 -0.13 -0.01 -0.18
## wheelbase 0.78 0.57 0.49 0.16 0.25
## carlength 0.88 0.68 0.61 0.13 0.16
## carwidth 0.87 0.74 0.56 0.18 0.18
## carheight 0.30 0.07 0.17 -0.06 0.26
## curbweight 1.00 0.85 0.65 0.17 0.15
## enginesize 0.85 1.00 0.58 0.20 0.03
## boreratio 0.65 0.58 1.00 -0.06 0.01
## stroke 0.17 0.20 -0.06 1.00 0.19
## compressionratio 0.15 0.03 0.01 0.19 1.00
## horsepower 0.75 0.81 0.57 0.08 -0.20
## peakrpm -0.27 -0.24 -0.25 -0.07 -0.44
## citympg -0.76 -0.65 -0.58 -0.04 0.32
## highwaympg -0.80 -0.68 -0.59 -0.04 0.27
## horsepower peakrpm citympg highwaympg
## car_id -0.02 -0.20 0.02 0.01
## symboling 0.07 0.27 -0.04 0.03
## wheelbase 0.35 -0.36 -0.47 -0.54
## carlength 0.55 -0.29 -0.67 -0.70
## carwidth 0.64 -0.22 -0.64 -0.68
## carheight -0.11 -0.32 -0.05 -0.11
## curbweight 0.75 -0.27 -0.76 -0.80
## enginesize 0.81 -0.24 -0.65 -0.68
## boreratio 0.57 -0.25 -0.58 -0.59
## stroke 0.08 -0.07 -0.04 -0.04
## compressionratio -0.20 -0.44 0.32 0.27
## horsepower 1.00 0.13 -0.80 -0.77
## peakrpm 0.13 1.00 -0.11 -0.05
## citympg -0.80 -0.11 1.00 0.97
## highwaympg -0.77 -0.05 0.97 1.00corrplot(corr_matrix, method = "color", type = "upper",
tl.cex = 0.7, tl.col = "black",
title = "Correlation Matrix of Numeric Predictions",
mar = c(0, 0, 2, 0)
)
3.4. Detecting Multicolinearity
high_cor = which(abs(corr_matrix) > 0.8 & upper.tri(corr_matrix), arr.ind = TRUE)
if(nrow(high_cor) > 0) {
for(i in 1:nrow(high_cor)) {
cat("High Correlation Between",
rownames(corr_matrix)[high_cor[i,1]], "and",
colnames(corr_matrix)[high_cor[i,2]], ":",
corr_matrix[high_cor[i,1], high_cor[i,2]], "\n")
}
} else {
cat("No highly correlated pairs found.\n")
}## High Correlation Between wheelbase and carlength : 0.8745875
## High Correlation Between carlength and carwidth : 0.8411183
## High Correlation Between carlength and curbweight : 0.8777285
## High Correlation Between carwidth and curbweight : 0.8670325
## High Correlation Between curbweight and enginesize : 0.8505941
## High Correlation Between enginesize and horsepower : 0.8097687
## High Correlation Between horsepower and citympg : -0.8014562
## High Correlation Between citympg and highwaympg : 0.971337# Names of variables involved
vars_involved = unique(c(
rownames(corr_matrix)[high_cor[,1]],
colnames(corr_matrix)[high_cor[,2]]
))
cor_with_price = sapply(vars_involved, function(v) {
abs(cor(Car_Price_Data1[[v]], Car_Price_Data1$price, use = "complete.obs"))
})
sort(cor_with_price, decreasing = TRUE)## enginesize curbweight horsepower carwidth highwaympg citympg carlength
## 0.8741448 0.8353049 0.8081388 0.7593253 0.6975991 0.6857513 0.6829200
## wheelbase
## 0.5778156# Highlighting the variables that need to be dropped
vars_to_drop = c()
for (i in 1:nrow(high_cor)) {
v1 = rownames(corr_matrix)[high_cor[i,1]]
v2 = colnames(corr_matrix)[high_cor[i,2]]
if(cor_with_price[v1] < cor_with_price[v2]) {
vars_to_drop = c(vars_to_drop, v1)
cat("Drop", v1, "(cor with price =", round(cor_with_price[v1],3),
"(, keep", v2, "(cor =", round(cor_with_price[v2],3), ")\n")
} else{
vars_to_drop = c(vars_to_drop, v2)
cat("Drop", v2, "(cor with price =", round(cor_with_price[v2],3),
"(, keep", v1, "(cor =", round(cor_with_price[v1],3), ")\n")
}
}## Drop wheelbase (cor with price = 0.578 (, keep carlength (cor = 0.683 )
## Drop carlength (cor with price = 0.683 (, keep carwidth (cor = 0.759 )
## Drop carlength (cor with price = 0.683 (, keep curbweight (cor = 0.835 )
## Drop carwidth (cor with price = 0.759 (, keep curbweight (cor = 0.835 )
## Drop curbweight (cor with price = 0.835 (, keep enginesize (cor = 0.874 )
## Drop horsepower (cor with price = 0.808 (, keep enginesize (cor = 0.874 )
## Drop citympg (cor with price = 0.686 (, keep horsepower (cor = 0.808 )
## Drop citympg (cor with price = 0.686 (, keep highwaympg (cor = 0.698 )vars_to_drop = unique(vars_to_drop)
cat("\nVariables to drop:\n")##
## Variables to drop:print(vars_to_drop)## [1] "wheelbase" "carlength" "carwidth" "curbweight" "horsepower"
## [6] "citympg"Car_Price_Data1 = Car_Price_Data1 %>%
select(-car_id, -car_name, -citympg, -carlength, -wheelbase, -carwidth, -curbweight, -horsepower)
head(Car_Price_Data1)## # A tibble: 6 × 18
## symboling fueltype aspiration doornumber carbody drivewheel enginelocation
## <dbl> <fct> <fct> <fct> <fct> <fct> <fct>
## 1 3 gas std two convertible rwd front
## 2 3 gas std two convertible rwd front
## 3 1 gas std two hatchback rwd front
## 4 2 gas std four sedan fwd front
## 5 2 gas std four sedan 4wd front
## 6 2 gas std two sedan fwd front
## # ℹ 11 more variables: carheight <dbl>, enginetype <chr>, cylindernumber <chr>,
## # enginesize <dbl>, fuelsystem <chr>, boreratio <dbl>, stroke <dbl>,
## # compressionratio <dbl>, peakrpm <dbl>, highwaympg <dbl>, price <dbl>str(Car_Price_Data1)## tibble [205 × 18] (S3: tbl_df/tbl/data.frame)
## $ symboling : num [1:205] 3 3 1 2 2 2 1 1 1 0 ...
## $ fueltype : Factor w/ 2 levels "diesel","gas": 2 2 2 2 2 2 2 2 2 2 ...
## $ aspiration : Factor w/ 2 levels "std","turbo": 1 1 1 1 1 1 1 1 2 2 ...
## $ doornumber : Factor w/ 2 levels "four","two": 2 2 2 1 1 2 1 1 1 2 ...
## $ carbody : Factor w/ 5 levels "convertible",..: 1 1 3 4 4 4 4 5 4 3 ...
## $ drivewheel : Factor w/ 3 levels "4wd","fwd","rwd": 3 3 3 2 1 2 2 2 2 1 ...
## $ enginelocation : Factor w/ 2 levels "front","rear": 1 1 1 1 1 1 1 1 1 1 ...
## $ carheight : num [1:205] 48.8 48.8 52.4 54.3 54.3 53.1 55.7 55.7 55.9 52 ...
## $ enginetype : chr [1:205] "dohc" "dohc" "ohcv" "ohc" ...
## $ cylindernumber : chr [1:205] "four" "four" "six" "four" ...
## $ enginesize : num [1:205] 130 130 152 109 136 136 136 136 131 131 ...
## $ fuelsystem : chr [1:205] "mpfi" "mpfi" "mpfi" "mpfi" ...
## $ boreratio : num [1:205] 3.47 3.47 2.68 3.19 3.19 3.19 3.19 3.19 3.13 3.13 ...
## $ stroke : num [1:205] 2.68 2.68 3.47 3.4 3.4 3.4 3.4 3.4 3.4 3.4 ...
## $ compressionratio: num [1:205] 9 9 9 10 8 8.5 8.5 8.5 8.3 7 ...
## $ peakrpm : num [1:205] 5000 5000 5000 5500 5500 5500 5500 5500 5500 5500 ...
## $ highwaympg : num [1:205] 27 27 26 30 22 25 25 25 20 22 ...
## $ price : num [1:205] 13495 16500 16500 13950 17450 ...glimpse(Car_Price_Data1)## Rows: 205
## Columns: 18
## $ symboling <dbl> 3, 3, 1, 2, 2, 2, 1, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0,…
## $ fueltype <fct> gas, gas, gas, gas, gas, gas, gas, gas, gas, gas, gas…
## $ aspiration <fct> std, std, std, std, std, std, std, std, turbo, turbo,…
## $ doornumber <fct> two, two, two, four, four, two, four, four, four, two…
## $ carbody <fct> convertible, convertible, hatchback, sedan, sedan, se…
## $ drivewheel <fct> rwd, rwd, rwd, fwd, 4wd, fwd, fwd, fwd, fwd, 4wd, rwd…
## $ enginelocation <fct> front, front, front, front, front, front, front, fron…
## $ carheight <dbl> 48.8, 48.8, 52.4, 54.3, 54.3, 53.1, 55.7, 55.7, 55.9,…
## $ enginetype <chr> "dohc", "dohc", "ohcv", "ohc", "ohc", "ohc", "ohc", "…
## $ cylindernumber <chr> "four", "four", "six", "four", "five", "five", "five"…
## $ enginesize <dbl> 130, 130, 152, 109, 136, 136, 136, 136, 131, 131, 108…
## $ fuelsystem <chr> "mpfi", "mpfi", "mpfi", "mpfi", "mpfi", "mpfi", "mpfi…
## $ boreratio <dbl> 3.47, 3.47, 2.68, 3.19, 3.19, 3.19, 3.19, 3.19, 3.13,…
## $ stroke <dbl> 2.68, 2.68, 3.47, 3.40, 3.40, 3.40, 3.40, 3.40, 3.40,…
## $ compressionratio <dbl> 9.00, 9.00, 9.00, 10.00, 8.00, 8.50, 8.50, 8.50, 8.30…
## $ peakrpm <dbl> 5000, 5000, 5000, 5500, 5500, 5500, 5500, 5500, 5500,…
## $ highwaympg <dbl> 27, 27, 26, 30, 22, 25, 25, 25, 20, 22, 29, 29, 28, 2…
## $ price <dbl> 13495.00, 16500.00, 16500.00, 13950.00, 17450.00, 152…3.5. Sorting out rare categorial values
Some of the categorial variables have levels with very few observations, which can create unstable estimates. The rare levels will be combined (appearing fewer than five times) into a “other” category using ‘fct_lump_min()’
library(forcats)
# Get all factor columns
factor_cols = names(Car_Price_Data1)[sapply(Car_Price_Data1, is.factor)]
# Prepare data (combine rare levels)
library(dplyr)
library(forcats)
Car_Price_Data1 <- Car_Price_Data1 %>%
mutate(cylindernumber = fct_lump_min(cylindernumber, min = 5, other_level = "rare"))
# Combine rare levels (min frequency = 5)
Car_Price_Data1 = Car_Price_Data1 %>%
mutate(across(all_of(factor_cols),
~ fct_lump_min(.x, min = 5, other_level = "rare")))
# Quick summary of changes
cat("Factor columns processed:", length(factor_cols), "\n")## Factor columns processed: 6cat("Example - cylinder levels:\n")## Example - cylinder levels:print(table(Car_Price_Data1$cylindernumber))##
## eight five four six rare
## 5 11 159 24 64. Building the model
4.1. Split the data
Now the data will be split into a training set and a testing set.
set.seed(123)
n = nrow(Car_Price_Data1)
train_idx = sample(1:n, 0.7 * n)
train = Car_Price_Data1[train_idx, ]
test = Car_Price_Data1[-train_idx, ]4.2. Initial Linear Model
initial_model = lm(price ~ ., data = train)
summary(initial_model)##
## Call:
## lm(formula = price ~ ., data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5285.5 -989.3 -52.8 714.6 9643.6
##
## Coefficients: (2 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.796e+03 1.716e+04 0.396 0.692864
## symboling -3.338e+02 2.701e+02 -1.236 0.219041
## fueltypegas -5.958e+03 7.902e+03 -0.754 0.452469
## aspirationturbo 3.442e+03 7.845e+02 4.387 2.65e-05 ***
## doornumbertwo -2.298e+02 6.685e+02 -0.344 0.731743
## carbodyhardtop -3.626e+03 1.779e+03 -2.038 0.043982 *
## carbodyhatchback -3.550e+03 1.489e+03 -2.384 0.018862 *
## carbodysedan -3.280e+03 1.606e+03 -2.042 0.043560 *
## carbodywagon -3.698e+03 1.777e+03 -2.081 0.039765 *
## drivewheelfwd 2.285e+02 1.229e+03 0.186 0.852881
## drivewheelrwd 1.077e+03 1.644e+03 0.655 0.513966
## enginelocationrare 4.877e+03 2.691e+03 1.813 0.072627 .
## carheight 2.240e+02 1.554e+02 1.442 0.152186
## enginetypedohcv -3.950e+03 4.284e+03 -0.922 0.358493
## enginetypel 1.469e+03 1.671e+03 0.879 0.381153
## enginetypeohc 2.602e+03 1.164e+03 2.236 0.027383 *
## enginetypeohcf 8.460e+02 2.023e+03 0.418 0.676535
## enginetypeohcv -6.704e+03 1.756e+03 -3.817 0.000224 ***
## enginetyperotor 8.026e+03 7.661e+03 1.048 0.297113
## cylindernumberfive -9.722e+03 3.643e+03 -2.669 0.008764 **
## cylindernumberfour -1.342e+04 4.077e+03 -3.291 0.001342 **
## cylindernumbersix -7.575e+03 2.663e+03 -2.845 0.005299 **
## cylindernumberrare -1.142e+04 4.071e+03 -2.804 0.005965 **
## enginesize 1.538e+02 2.337e+01 6.583 1.63e-09 ***
## fuelsystem2bbl 7.510e+02 1.013e+03 0.741 0.460095
## fuelsystem4bbl NA NA NA NA
## fuelsystemidi NA NA NA NA
## fuelsystemmfi -1.653e+03 2.810e+03 -0.588 0.557529
## fuelsystemmpfi 7.925e+02 1.081e+03 0.733 0.464877
## fuelsystemspdi -2.976e+02 1.818e+03 -0.164 0.870303
## boreratio -1.274e+03 2.163e+03 -0.589 0.556975
## stroke -3.950e+03 1.099e+03 -3.594 0.000490 ***
## compressionratio -2.527e+02 5.753e+02 -0.439 0.661267
## peakrpm 2.552e+00 6.718e-01 3.798 0.000239 ***
## highwaympg -2.244e+01 7.560e+01 -0.297 0.767153
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2246 on 110 degrees of freedom
## Multiple R-squared: 0.9406, Adjusted R-squared: 0.9233
## F-statistic: 54.43 on 32 and 110 DF, p-value: < 2.2e-16# checking for aliased coefficients
alias_info = alias(initial_model)
print(alias_info$Complete)## (Intercept) symboling fueltypegas aspirationturbo doornumbertwo
## fuelsystem4bbl 0 0 0 0 0
## fuelsystemidi 1 0 -1 0 0
## carbodyhardtop carbodyhatchback carbodysedan carbodywagon
## fuelsystem4bbl 0 0 0 0
## fuelsystemidi 0 0 0 0
## drivewheelfwd drivewheelrwd enginelocationrare carheight
## fuelsystem4bbl 0 0 0 0
## fuelsystemidi 0 0 0 0
## enginetypedohcv enginetypel enginetypeohc enginetypeohcf
## fuelsystem4bbl 0 0 0 0
## fuelsystemidi 0 0 0 0
## enginetypeohcv enginetyperotor cylindernumberfive
## fuelsystem4bbl 0 1 0
## fuelsystemidi 0 0 0
## cylindernumberfour cylindernumbersix cylindernumberrare
## fuelsystem4bbl 0 0 0
## fuelsystemidi 0 0 0
## enginesize fuelsystem2bbl fuelsystemmfi fuelsystemmpfi
## fuelsystem4bbl 0 0 0 0
## fuelsystemidi 0 0 0 0
## fuelsystemspdi boreratio stroke compressionratio peakrpm
## fuelsystem4bbl 0 0 0 0 0
## fuelsystemidi 0 0 0 0 0
## highwaympg
## fuelsystem4bbl 0
## fuelsystemidi 04.3. Stepwise selection
model_step = step(lm(price ~ ., data = train), direction = "both", trace = 0)
summary(model_step)##
## Call:
## lm(formula = price ~ symboling + aspiration + carbody + enginelocation +
## carheight + enginetype + cylindernumber + enginesize + stroke +
## compressionratio + peakrpm, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5376.3 -1074.5 -100.4 787.5 9597.2
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.587e+03 9.151e+03 -0.283 0.777871
## symboling -4.219e+02 2.048e+02 -2.060 0.041600 *
## aspirationturbo 3.652e+03 5.890e+02 6.200 8.26e-09 ***
## carbodyhardtop -3.818e+03 1.678e+03 -2.275 0.024679 *
## carbodyhatchback -3.931e+03 1.394e+03 -2.821 0.005610 **
## carbodysedan -3.525e+03 1.452e+03 -2.428 0.016684 *
## carbodywagon -3.969e+03 1.595e+03 -2.488 0.014225 *
## enginelocationrare 4.956e+03 2.520e+03 1.967 0.051548 .
## carheight 2.213e+02 1.231e+02 1.799 0.074610 .
## enginetypedohcv -5.510e+03 3.259e+03 -1.690 0.093561 .
## enginetypel 1.834e+03 1.402e+03 1.308 0.193208
## enginetypeohc 2.278e+03 1.084e+03 2.101 0.037692 *
## enginetypeohcf -4.586e+02 1.492e+03 -0.307 0.759129
## enginetypeohcv -7.218e+03 1.595e+03 -4.525 1.43e-05 ***
## enginetyperotor 9.122e+03 4.925e+03 1.852 0.066439 .
## cylindernumberfive -9.731e+03 2.757e+03 -3.530 0.000591 ***
## cylindernumberfour -1.404e+04 2.888e+03 -4.863 3.54e-06 ***
## cylindernumbersix -7.520e+03 2.064e+03 -3.643 0.000399 ***
## cylindernumberrare -1.312e+04 3.027e+03 -4.335 3.05e-05 ***
## enginesize 1.563e+02 1.235e+01 12.657 < 2e-16 ***
## stroke -4.663e+03 8.731e+02 -5.340 4.46e-07 ***
## compressionratio 1.285e+02 5.920e+01 2.171 0.031893 *
## peakrpm 2.463e+00 5.274e-01 4.670 7.91e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2179 on 120 degrees of freedom
## Multiple R-squared: 0.939, Adjusted R-squared: 0.9278
## F-statistic: 83.99 on 22 and 120 DF, p-value: < 2.2e-164.4. Multicollinearity check (VIF)
vif_values = vif(model_step)
print(vif_values)## GVIF Df GVIF^(1/(2*Df))
## symboling 1.846560 1 1.358882
## aspiration 1.460088 1 1.208341
## carbody 4.313462 4 1.200475
## enginelocation 2.638824 1 1.624446
## carheight 2.450903 1 1.565536
## enginetype 78.345865 6 1.438251
## cylindernumber 56.391677 4 1.655396
## enginesize 8.300746 1 2.881101
## stroke 2.132191 1 1.460202
## compressionratio 1.769727 1 1.330311
## peakrpm 2.047784 1 1.431008# Remove any factors with only one level
for(col in names(train)[sapply(train, is.factor)]) {
if(length(unique(train[[col]])) < 2) {
train = train %>% select(-all_of(col))
test = test %>% select(-all_of(col))
cat("Removed", col, "-only one level\n")
}
}
# Fit the model again
model_temp = lm(price ~ ., data = train)
# Stepwise again
model_step2 = step(lm(price ~ ., data = train), direction = "both", trace = 1)## Start: AIC=2235.48
## price ~ symboling + fueltype + aspiration + doornumber + carbody +
## drivewheel + enginelocation + carheight + enginetype + cylindernumber +
## enginesize + fuelsystem + boreratio + stroke + compressionratio +
## peakrpm + highwaympg
##
##
## Step: AIC=2235.48
## price ~ symboling + aspiration + doornumber + carbody + drivewheel +
## enginelocation + carheight + enginetype + cylindernumber +
## enginesize + fuelsystem + boreratio + stroke + compressionratio +
## peakrpm + highwaympg
##
## Df Sum of Sq RSS AIC
## - fuelsystem 5 9079873 563829230 2227.8
## - drivewheel 2 3969260 558718617 2232.5
## - highwaympg 1 444355 555193712 2233.6
## - doornumber 1 595681 555345038 2233.6
## - compressionratio 1 973497 555722854 2233.7
## - boreratio 1 1750415 556499772 2233.9
## - carbody 4 31476863 586226220 2235.4
## - symboling 1 7706126 562455483 2235.4
## <none> 554749357 2235.5
## - carheight 1 10484384 565233740 2236.2
## - enginelocation 1 16568705 571318062 2237.7
## - stroke 1 65124193 619873549 2249.3
## - peakrpm 1 72764509 627513865 2251.1
## - aspiration 1 97059903 651809260 2256.5
## - cylindernumber 4 132146339 686895696 2258.0
## - enginetype 5 176178631 730927987 2264.9
## - enginesize 1 218564558 773313915 2281.0
##
## Step: AIC=2227.8
## price ~ symboling + aspiration + doornumber + carbody + drivewheel +
## enginelocation + carheight + enginetype + cylindernumber +
## enginesize + boreratio + stroke + compressionratio + peakrpm +
## highwaympg
##
## Df Sum of Sq RSS AIC
## - drivewheel 2 4672711 568501941 2225.0
## - doornumber 1 568560 564397790 2225.9
## - highwaympg 1 941321 564770552 2226.0
## - boreratio 1 1305794 565135024 2226.1
## <none> 563829230 2227.8
## - symboling 1 9766685 573595915 2228.3
## - carbody 4 36476882 600306112 2228.8
## - compressionratio 1 13091616 576920846 2229.1
## - carheight 1 13881206 577710436 2229.3
## + fueltype 1 889197 562940034 2229.6
## - enginelocation 1 17612808 581442039 2230.2
## + fuelsystem 5 9079873 554749357 2235.5
## - peakrpm 1 74561578 638390808 2243.6
## - stroke 1 83688385 647517616 2245.6
## - aspiration 1 139124708 702953939 2257.3
## - enginetype 6 205813222 769642453 2260.3
## - cylindernumber 4 201230349 765059579 2263.4
## - enginesize 1 230303099 794132330 2274.8
##
## Step: AIC=2224.98
## price ~ symboling + aspiration + doornumber + carbody + enginelocation +
## carheight + enginetype + cylindernumber + enginesize + boreratio +
## stroke + compressionratio + peakrpm + highwaympg
##
## Df Sum of Sq RSS AIC
## - boreratio 1 4427 568506368 2223.0
## - doornumber 1 74576 568576517 2223.0
## - highwaympg 1 904484 569406425 2223.2
## <none> 568501941 2225.0
## - carheight 1 12020772 580522713 2226.0
## - carbody 4 39239801 607741743 2226.5
## - symboling 1 15421677 583923618 2226.8
## + fueltype 1 221301 568280640 2226.9
## - enginelocation 1 18053164 586555106 2227.4
## - compressionratio 1 19188779 587690721 2227.7
## + drivewheel 2 4672711 563829230 2227.8
## + fuelsystem 5 9783324 558718617 2232.5
## - peakrpm 1 88404094 656906035 2243.7
## - stroke 1 127240335 695742276 2251.9
## - aspiration 1 139344945 707846886 2254.3
## - enginetype 6 249447080 817949022 2265.0
## - cylindernumber 4 244573761 813075702 2268.2
## - enginesize 1 235652916 804154857 2272.6
##
## Step: AIC=2222.98
## price ~ symboling + aspiration + doornumber + carbody + enginelocation +
## carheight + enginetype + cylindernumber + enginesize + stroke +
## compressionratio + peakrpm + highwaympg
##
## Df Sum of Sq RSS AIC
## - doornumber 1 78958 568585326 2221.0
## - highwaympg 1 905001 569411369 2221.2
## <none> 568506368 2223.0
## - carheight 1 12028502 580534870 2224.0
## - carbody 4 39630610 608136978 2224.6
## + fueltype 1 225056 568281313 2224.9
## + boreratio 1 4427 568501941 2225.0
## - symboling 1 16769748 585276116 2225.1
## - enginelocation 1 18067464 586573832 2225.5
## - compressionratio 1 19568336 588074704 2225.8
## + drivewheel 2 3371344 565135024 2226.1
## + fuelsystem 5 9619130 558887238 2230.5
## - peakrpm 1 89775945 658282313 2241.9
## - stroke 1 128326762 696833131 2250.1
## - aspiration 1 140688921 709195289 2252.6
## - enginetype 6 249910048 818416416 2263.1
## - cylindernumber 4 297601167 866107535 2275.2
## - enginesize 1 443801378 1012307746 2303.5
##
## Step: AIC=2221
## price ~ symboling + aspiration + carbody + enginelocation + carheight +
## enginetype + cylindernumber + enginesize + stroke + compressionratio +
## peakrpm + highwaympg
##
## Df Sum of Sq RSS AIC
## - highwaympg 1 921311 569506636 2219.2
## <none> 568585326 2221.0
## - carheight 1 11994201 580579527 2222.0
## - carbody 4 40371654 608956980 2222.8
## + fueltype 1 198898 568386428 2222.9
## + doornumber 1 78958 568506368 2223.0
## + boreratio 1 8809 568576517 2223.0
## - enginelocation 1 18854567 587439893 2223.7
## - compressionratio 1 19489471 588074797 2223.8
## - symboling 1 19916965 588502291 2223.9
## + drivewheel 2 2933258 565652068 2224.3
## + fuelsystem 5 9589196 558996130 2228.6
## - peakrpm 1 90814686 659400012 2240.2
## - stroke 1 128255752 696841077 2248.1
## - aspiration 1 140634478 709219804 2250.6
## - enginetype 6 250169522 818754848 2261.1
## - cylindernumber 4 299125163 867710489 2273.4
## - enginesize 1 443893695 1012479020 2301.5
##
## Step: AIC=2219.23
## price ~ symboling + aspiration + carbody + enginelocation + carheight +
## enginetype + cylindernumber + enginesize + stroke + compressionratio +
## peakrpm
##
## Df Sum of Sq RSS AIC
## <none> 569506636 2219.2
## + highwaympg 1 921311 568585326 2221.0
## - carheight 1 15351271 584857908 2221.0
## + fueltype 1 366936 569139701 2221.1
## + doornumber 1 95267 569411369 2221.2
## + boreratio 1 1698 569504939 2221.2
## - carbody 4 41794069 611300706 2221.4
## - enginelocation 1 18353113 587859750 2221.8
## - symboling 1 20131841 589638477 2222.2
## + drivewheel 2 3118652 566387985 2222.4
## - compressionratio 1 22370435 591877071 2222.7
## + fuelsystem 5 10110312 559396324 2226.7
## - peakrpm 1 103523078 673029715 2241.1
## - stroke 1 135349977 704856614 2247.7
## - aspiration 1 182449358 751955994 2257.0
## - enginetype 6 256588487 826095123 2260.4
## - cylindernumber 4 304557161 874063797 2272.5
## - enginesize 1 760280536 1329787172 2338.5# Compare Original vs. Stepwise
cat("Original variable:", length(coef(lm(price ~ ., data = train))), "\n")## Original variable: 35cat("Stepwise variables:", length(coef(model_step2)), "\n")## Stepwise variables: 23cat("Variables dropped due to aliasing:",
length(coef(lm(price ~ ., data = train))) - length(coef(model_step2)), "\n")## Variables dropped due to aliasing: 124.5. Prediction
pred_step = predict(model_step2, newdata = test)
rmse_step = sqrt(mean((test$price - pred_step)^2))
r2_step = cor(test$price, pred_step)^2
cat("RMSE:", rmse_step, "\nR²:", r2_step, "\n")## RMSE: 2920.561
## R²: 0.8599762Residuals = test$price - pred_step
# Plotting the graph
plot(test$price, pred_step, xlab = "Actual Price", ylab = "Predicted Price")
abline(0, 1, col = "red")