Motor Trend - Regression Models
Executive Summary
This assignmentx will analyse mtcars dataset and explore the relationship between a set of variables and miles per gallon (MPG) (outcome). The purpose of this assignment is to answer the following two questions:
Is an automatic or manual transmission better for MPG
Quantify the MPG difference between automatic and manual transmissions
Data Processing
# Load the data
data(mtcars)
# Check the structure of the data
str(mtcars)## 'data.frame': 32 obs. of 11 variables:
## $ mpg : num 21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
## $ cyl : num 6 6 4 6 8 6 8 4 4 6 ...
## $ disp: num 160 160 108 258 360 ...
## $ hp : num 110 110 93 110 175 105 245 62 95 123 ...
## $ drat: num 3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
## $ wt : num 2.62 2.88 2.32 3.21 3.44 ...
## $ qsec: num 16.5 17 18.6 19.4 17 ...
## $ vs : num 0 0 1 1 0 1 0 1 1 1 ...
## $ am : num 1 1 1 0 0 0 0 0 0 0 ...
## $ gear: num 4 4 4 3 3 3 3 4 4 4 ...
## $ carb: num 4 4 1 1 2 1 4 2 2 4 ...# Convert appropriate variables into factors
mtcars$am <- factor(mtcars$am, labels = c("Automatic", "Manual"))Data Exploring
head(mtcars)## mpg cyl disp hp drat wt qsec vs am gear
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 Manual 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 Manual 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 Manual 4
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 Automatic 3
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 Automatic 3
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 Automatic 3
## carb
## Mazda RX4 4
## Mazda RX4 Wag 4
## Datsun 710 1
## Hornet 4 Drive 1
## Hornet Sportabout 2
## Valiant 1# Plot variable "am" and "mpg" for visial representation
plot(mtcars$am,mtcars$mpg, ylab = "Miles per gallon (MPG)")
From the plot we can conclude that the Automatic and Manual transmission are very different regarding MPG. We can also find support in that conclusion by performing a t.test to test a null hypothesis for difference in the transmission mode in the am variable.
t.test(mpg ~ am, data = mtcars)##
## Welch Two Sample t-test
##
## data: mpg by am
## t = -3.7671, df = 18.332, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -11.280194 -3.209684
## sample estimates:
## mean in group Automatic mean in group Manual
## 17.14737 24.39231With p-value equal to 0.001374, we can reject the Null Hypothesis and firmly conclude that there is indeed difference b/n Automatic Transmission and Manual Transmission, regarding Miles Per Gallon.
Simple Linear Regression
# Perform Simple Linear Regression
fit <- lm(mpg ~ am, data = mtcars)
summary(fit)##
## Call:
## lm(formula = mpg ~ am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.3923 -3.0923 -0.2974 3.2439 9.5077
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.147 1.125 15.247 1.13e-15 ***
## amManual 7.245 1.764 4.106 0.000285 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.902 on 30 degrees of freedom
## Multiple R-squared: 0.3598, Adjusted R-squared: 0.3385
## F-statistic: 16.86 on 1 and 30 DF, p-value: 0.000285The Adjusted R-squared value of the linear model is 0.3385, which means that 33.8% of the response variable variation is explained by our model, and we should explore different models.
Multivariable Regression
In order to see which variables to include in our next model we can use the stepAIC function from MASS package to show us which are the optimal variables.
# step function to find variables that are optimal
library(MASS)
OptVariables <- stepAIC(lm(mpg ~., data = mtcars), direction = "both", trace = 0)
summary(OptVariables)##
## Call:
## lm(formula = mpg ~ wt + qsec + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4811 -1.5555 -0.7257 1.4110 4.6610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.6178 6.9596 1.382 0.177915
## wt -3.9165 0.7112 -5.507 6.95e-06 ***
## qsec 1.2259 0.2887 4.247 0.000216 ***
## amManual 2.9358 1.4109 2.081 0.046716 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.459 on 28 degrees of freedom
## Multiple R-squared: 0.8497, Adjusted R-squared: 0.8336
## F-statistic: 52.75 on 3 and 28 DF, p-value: 1.21e-11Adjusted R-squared = 0.8401, which means that 84% of the response variable variation is explained by our new model. That makes this model the better choice for our data.
# Perform an ANOVA test to compare the two models
anova(fit, OptVariables)## Analysis of Variance Table
##
## Model 1: mpg ~ am
## Model 2: mpg ~ wt + qsec + am
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 720.90
## 2 28 169.29 2 551.61 45.618 1.55e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Looking at the p-value = 1.688e-08 we can reject the null hypothesis and to conclude that the cyl, hp, wt and am variables, all contribute to the model, which is trying to predict miles per gallon variable.
Diagnostic Plots
par(mfrow = c(2, 2))
plot(OptVariables)
Residuals vs. Fitted plot is randomly scattered - verifies the independence condition
Normal Q-Q plot fall on the line - indicating normal distribution of the residuals
Scale-Location plot consists of scattered points - indicating constant variance
Cook’s distance plot consist of some points - outliers or leverage points - we may investigate further.
Chrysler Imperial, Fiat 128, and Toyota Corolla are influential points
Conclusion
Answers of the two questions:
1. Is an automatic or manual transmission better for MPG?
- Manual transmission is better for mpg
2. Quantify the MPG difference between automatic and manual transmissions
- Manual transmission cars delivers 2.94 more mpg than automatic transmission cars with an adjusted R-squared of 0.83 and a p-value below 0.05