Motor Trend Car Road Tests - Effects of Transmission Types on MPG
Synopsis
In this project, we analyze the mtcars dataset and explore the linear relationship between a set of variables and miles per gallon.
The main objectives of this research are as follows:
- Is an automatic or manual transmission better for MPG.
- Quantifying how different is the MPG between automatic and manual transmissions.
The results of this analysis are:
- Manual transmission is better than automatic transmission for MPG.
- Manual transmission is 2.94 mpg more fuel efficient than automatic transmission while holding other regressors constant.
Exploratory Data Analyses
t <- t.test(mpg ~ am, data = mtcars, paired = FALSE, var.equal = FALSE)
t##
## 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 0 mean in group 1
## 17.14737 24.39231The exploratory data analysis shows that:
- The t-test has a P-value of 1.374e-03 (< 5%) and 95% confidence interval of (-11.28 -3.21) for mean(automatic)-mean(manual), we reject the null hypothesis that there is no significant difference in MPG between the two transmission types.
- The boxplot in the appendix below shows that manual transmission has better MPG than automatic transmission.
- The Pairs plot in the appendix below shows that there are several variables that have high correlation with mpg.
Model Selection
Single variable linear regression model - Model #1:
model_1 = lm(mpg ~ am, data = mtcars)
summary(model_1)$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.147368 1.124603 15.247492 1.133983e-15
## am 7.244939 1.764422 4.106127 2.850207e-04summary(model_1)$adj.r.squared## [1] 0.3384589Multivariable linear regression model (Stepwise Regression) - Model #2:
fitAll <- lm(mpg ~ . , data = mtcars)
model_2 <- step(fitAll, direction = "both") # stepwise regressionsummary(model_2)$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.617781 6.9595930 1.381946 1.779152e-01
## wt -3.916504 0.7112016 -5.506882 6.952711e-06
## qsec 1.225886 0.2886696 4.246676 2.161737e-04
## am 2.935837 1.4109045 2.080819 4.671551e-02summary(model_2)$adj.r.squared## [1] 0.8335561Model #2 obtained from the above computations consists of the variables: “wt” and “qsec” as confounders and “am” as the independent variable.
Compare the Adjusted R-squared values of the two models:
## Adjusted R-squared
## model_1 0.3385
## model_2 0.8300Now, let’s look at the Analysis of Variance Table of the models:
varTbl <- anova(model_1, model_2)
varTbl## 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 ' ' 1Finally, the VIF (variance inflation factors) of model #2:
vif(model_2)## wt qsec am
## 2.482952 1.364339 2.541437Findings:
- Model #2 shows that the value of Adjusted R-squared is 0.8336; That is, 83.36% of the response variable variation is explained by this linear model, which is way better than that of model #1.
- Analysis of Variance Table shows that model #2 has P-value of 1.55e-09 (< 5%), which indicates that the confounders are significance.
- The VIF (variance inflation factors) shows that model #2 has pretty acceptable variance inflation values. Hence model #2 is chosen as the regression model of this analysis.
Residual Plot and Diagnostics
Refer to Residual Plots and Diagnostics sections in Appendix below for more details.
Findings:
- No systematic patterns or large outlying observations found in the Residual Plots.
- There are 2-3 observations that show low leverage but have fairly high influence.
Interpretation of Coefficients
For meaningful interpretation of coefficients, we applied centering on both “wt” and “qsec” variables:
fitCentered <- lm(mpg ~ I(wt - mean(wt)) + I(qsec - mean(qsec)) + factor(am),
data = mtcars)
summary(fitCentered)$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.897941 0.7193542 26.270704 2.855851e-21
## I(wt - mean(wt)) -3.916504 0.7112016 -5.506882 6.952711e-06
## I(qsec - mean(qsec)) 1.225886 0.2886696 4.246676 2.161737e-04
## factor(am)1 2.935837 1.4109045 2.080819 4.671551e-02Interpretation:
- 18.90 mpg with average weight, average quarter mile time, and automatic transmission (am = 0).
- 3.92 mpg less fuel efficiency with every kilo-pound of weight increase, while holding other regressors constant.
- 1.23 mpg more fuel efficiency with every second of quarter mile time increase, while holding other regressors constant.
- 2.94 mpg more fuel efficiency with manual transmission compared to automatic transmission, while holding other regressors constant.
Quantification of the Uncertainty
Below are confidence levels of intercept and predictors with 95% confidence:
confint(fitCentered)## 2.5 % 97.5 %
## (Intercept) 17.42441087 20.371471
## I(wt - mean(wt)) -5.37333423 -2.459673
## I(qsec - mean(qsec)) 0.63457320 1.817199
## factor(am)1 0.04573031 5.825944Conclusions
- The multivariable regression model was chosen as the regression model of this analysis due to the fact that more response variable variation was explained.
- Manual transmission is better than automatic transmission for MPG.
- Manual transmission is 2.94 mpg more fuel efficient than automatic transmission while holding other regressors constant.
Appendix
Exploratory Data Analysis
Boxplot
mtcars2 <- mtcars
mtcars2$txType <- factor(mtcars$am, labels = c("Automatic","Manual"))
ggplot(mtcars2, aes(x = txType, y = mpg, fill = txType)) +
geom_boxplot() +
labs(title = "Miles Per Gallon by Transmission Type",
x = "Transmission Type",
y = "Miles Per Gallon") +
scale_fill_discrete(name = "Transmission")
Pairs Plot
g = ggpairs(mtcars, lower = list(continuous = "smooth"))
g
Residual Plots
par(mfrow = c(2, 2))
plot(model_2)
Diagnostics
par(mfrow = c(1, 2))
# Leverage
plot(hatvalues(model_2), main="Leverage")
# Influence
#plot(rstandard(model_2))
plot(rstudent(model_2), main="Studentized Residuals")
par(mfrow = c(1, 2))
plot(dffits(model_2), main="Influence - dffits")
plot(cooks.distance(model_2), main="Influence - Cook's Distance")
par(mfrow = c(1, 2))
plot(dfbetas(model_2)[, 2], main="Influence - dfbetas of \"wt\"")
plot(dfbetas(model_2)[, 3], main="Influence - dfbetas of \"qsec\"")