Motor trend study (Manual vs Automatic cars)
by Renato Pedroso
Executive Summary
This study intends to explore more about manual and automatic transmission cars. At the end of this study we will be able to answer wich transmission, manual or automatic, is better for gas consumption as well as the quantified difference in miles per galoon between these two types of transmission.
To reach this goal we will use the mtcars dataset, extracted from 1974 Motor Trend US magazine, and perform linear regressions and technics to improve the model.
The summary of the results is:
- Manual transmission cars are more economic in gas consumption than automatic transmission cars
- Manual transmission cars have the mpg increased by 1.8 if compared to automatic transmission cars
- The medians of automatic and manual transmission cars, that could be observed on the boxplot, are considerable different
Exploratory Data Analises
To begin the study it is important to check the dataset and its variables:
The help page of the mtcars dataset has the format:
[, 1] mpg Miles/(US) gallon
[, 2] cyl Number of cylinders
[, 3] disp Displacement (cu.in.)
[, 4] hp Gross horsepower
[, 5] drat Rear axle ratio
[, 6] wt Weight (1000 lbs)
[, 7] qsec 1/4 mile time
[, 8] vs V/S
[, 9] am Transmission (0 = automatic, 1 = manual)
[,10] gear Number of forward gears
[,11] carb Number of carburetors
library(dplyr, quietly = TRUE, warn.conflicts = FALSE)
library(ggplot2, quietly = TRUE, warn.conflicts = FALSE)
data("mtcars")
head(mtcars, n = 5)## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2We can plot a preliminary graph comparing the mpg in an automatic transmission vs manual transmission in order to guide our study:
ggplot(mtcars, aes(x = factor(am, labels = c("Automatic","Manual")), y = mpg)) + geom_boxplot(aes(fill=factor(am, labels = c("Automatic","Manual")))) + ggtitle(label = "Gas Consumption on Automatic vs Manual Transmission") + xlab("Transmission Type") + ylab("Miles per Galon") + labs(fill = "Transmission")
This graph showed us that the gas consumption on automatic cars, on avarage, tend to be higher than the manual cars.
We can perform the t-test to reject, or don’t, the null hyphothesis:
t.test(as.numeric(mtcars$mpg) ~ as.factor(mtcars$am))$p.value## [1] 0.001373638The p-value < 0.05 let us to reject the null hypothesis that the mpg for automatic and manual cars are the same.
In order to discover wich variables have more impact on MPG we can study the Pearson correlation table:
cor(mtcars)[1,]## mpg cyl disp hp drat wt
## 1.0000000 -0.8521620 -0.8475514 -0.7761684 0.6811719 -0.8676594
## qsec vs am gear carb
## 0.4186840 0.6640389 0.5998324 0.4802848 -0.5509251The correlation table showed us that the variables cyl, disp, hp, drat, wt, vs and am are strong correlated with the outcome mpg
Data processing and transformation
As we could see in the preview of the dataset, we can transform some variables in factors:
mtcars <- mutate(mtcars, cyl = as.factor(cyl))
mtcars <- mutate(mtcars, vs = as.factor(vs))
mtcars <- mutate(mtcars, am = factor(am, labels = c("Automatic","Manual")))
mtcars <- mutate(mtcars, gear = as.factor(gear))
mtcars <- mutate(mtcars, carb = as.factor(carb))Model Construction
One approach to the linear model selection variables is the backward elimination, encapsulated in the step function in R. To use this technique we first fit a linear regression model with the outcome, in this case the mpg variable, and all other variables as predictors and, after that, we can use the step function as follows:
fit_initial <- lm(mpg ~ ., data = mtcars)
fit_step <- step(fit_initial, direction = "both", trace = FALSE)
summary(fit_step)##
## Call:
## lm(formula = mpg ~ cyl + hp + wt + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.9387 -1.2560 -0.4013 1.1253 5.0513
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.70832 2.60489 12.940 7.73e-13 ***
## cyl6 -3.03134 1.40728 -2.154 0.04068 *
## cyl8 -2.16368 2.28425 -0.947 0.35225
## hp -0.03211 0.01369 -2.345 0.02693 *
## wt -2.49683 0.88559 -2.819 0.00908 **
## amManual 1.80921 1.39630 1.296 0.20646
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.41 on 26 degrees of freedom
## Multiple R-squared: 0.8659, Adjusted R-squared: 0.8401
## F-statistic: 33.57 on 5 and 26 DF, p-value: 1.506e-10If we compare the Adjusted R^2 we can see a good improvement in it with less variables than the first model:
data.frame(initial_model = summary(fit_initial)$adj.r.squared, step_model = summary(fit_step)$adj.r.squared)## initial_model step_model
## 1 0.7790215 0.8400875We can also check for anova between the model only considering the transmission type as predictor and the model recomended from step function:
anova(lm(mpg ~ am, data = mtcars), fit_step)## Analysis of Variance Table
##
## Model 1: mpg ~ am
## Model 2: mpg ~ cyl + hp + wt + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 720.90
## 2 26 151.03 4 569.87 24.527 1.688e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This anova test, considering the p-value, let us to reject the null hypothesis that the variables included after the step function don’t help the model to be more precise.
Residuals
Let’s study the residuals, and others diagnostics, in this model:
par(mfrow = c(2,2), mar = c(4,4,2,2))
plot(fit_step)
- This residual plot show us an unbiased model (no pattern found)
- The Q-Q Plot show us some normality distribution on the residuals
- The scale-location plot is almost the same as the residual plot, no pattern could be observed so the model in unbiased
- The residual vs leverage plot show us some outliers that, if removed, will change the linear regression model
Inferences
There are differences between cars equiped with automatic transmission and manual transmission. As we could observe in the t-test and other tests performed in this study.
Conclusions
The summary of the linear model lead us to the following conclusions:
- If all the other variables, cyl, hp and wt, hold the same, manual cars will have an increase on mpg of 1.8
- If all the other variables, cyl, hp and am, hold the same, for each unit increased in wt the mpg will decrease by 2.5
- If all the other variables, cyl, am and wt, hold the same, for each unit increased in hp the mpg will decrease by 0.03
- If all the other variables, am, hp and wt, hold the same, for cyl change from 4 to 6 it will decrease the mpg in 3.03
- If all the other variables, am, hp and wt, hold the same, for cyl change from 4 to 8 it will decrease the mpg in 2.16