Fuel Efficiency and Transmission
Aman Jindal
29 Aug 2020
1. Executive Summary:
This report analyzes the relationship between fuel efficiency and transmission type for cars using the mtcars dataset. The following questions are answered:
- Is an automatic or manual transmission better for MPG
- Quantify the MPG difference between automatic and manual transmissions
First, exploratory data analysis has been done to identify the relationship between mpg (fuel efficiency) and am (transmission) and confounding variables affecting the relationship. Second, several multivariable linear regression models are run to identify the model that best answers our questions. Third, residual diagnostics are run on the chosen model.
Our analysis has revealed that transmission type has no effect on MPG and thus there is no mpg difference between automatic and manual transmissions.
2. Exploratory Data Analysis (EDA):
i) Steps: First, we make the variables am and vs (engine type) as factor variables. Second, we summarize the dataset to view the summary statistics (Appendix A). Third, we plot mpg against am (Appendix B), and mpg against am controlling for cyl. Fourth, we plot mpg against other variables and color the points according to am (Appendix C)
ii) Inferences: At first glance, Manual transmission seems to have a better mpg. However, after examining the effect of other variables (cyl, disp, hp, wt, drat), the relationship seems to vanish.
data("mtcars"); setDT(mtcars)
mtcars[,`:=`(am = factor(am, labels=c('automatic','manual')),
vs = factor(vs, labels=c('vshaped','straight')))]
ggplot(data=mtcars) + geom_boxplot(mapping = aes(x=am, y=mpg)) + facet_grid(~cyl)
3. Model Fitting and Selection:
i) Steps: We start by fitting a linear model between mpg and am and then we add other variables that were diagnosed as confounders in EDA. We first add wt, then we add cyl (disp, hp and cyl are similar variables for cars. Hence, we have taken one of these). Then, we add drat and vs. We then perform analysis of variance (ANOVA) for our five models.
ii) Inferences: Based on the results of ANOVA, we choose Model 3 as it is statistically significant at 0.01% level and has low Residual sum of squares (RSS).
fit1 <- lm(mpg~am, data=mtcars)
fit2 <- lm(mpg~am+wt, data=mtcars)
fit3 <- lm(mpg~am+wt+cyl, data=mtcars)
fit4 <- lm(mpg~am+wt+cyl+drat, data=mtcars)
fit5 <- lm(mpg~am+wt+cyl+drat+vs, data=mtcars)
anova(fit1, fit2, fit3, fit4, fit5)[,1:6] Res.Df RSS Df Sum of Sq F Pr(>F)
1 30 720.90
2 29 278.32 1 442.58 60.6635 2.911e-08 ***
3 28 191.05 1 87.27 11.9624 0.001884 **
4 27 191.00 1 0.05 0.0070 0.933857
5 26 189.69 1 1.31 0.1796 0.675214
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 14. Analyzing the chosen Model:
i) am variable: The model reveals a p value for 0.89 for am variable. Also, the 95% confidence interval for am includes 0.
ii) Residuals diagnostics (Appendix D): The distribution of residuals is random about the fitted values. Residuals are i.i.d. and are normally distributed as evident from the Q-Q plot.
summary(fit3)$coefficients Estimate Std. Error t value Pr(>|t|)
(Intercept) 39.4179334 2.6414573 14.9227979 7.424998e-15
ammanual 0.1764932 1.3044515 0.1353007 8.933421e-01
wt -3.1251422 0.9108827 -3.4308942 1.885894e-03
cyl -1.5102457 0.4222792 -3.5764148 1.291605e-03confint.lm(fit3) # par(mfrow = c(2,2)); plot(fit3) 2.5 % 97.5 %
(Intercept) 34.007153 44.8287134
ammanual -2.495555 2.8485408
wt -4.991001 -1.2592836
cyl -2.375245 -0.64524595. Conclusion:
- There is not enough evidence at 5% significance level to conclude that manual transmission’s
mpgis different from that of automatic transmission. - Owing to the result above, we cannot conclude that at 5% significance level, the difference in
mpgfor two transmission types is different from 0.
Appendix A: Summary Statistics of Variables
summary(mtcars) mpg cyl disp hp
Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
Median :19.20 Median :6.000 Median :196.3 Median :123.0
Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
drat wt qsec vs am
Min. :2.760 Min. :1.513 Min. :14.50 vshaped :18 automatic:19
1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 straight:14 manual :13
Median :3.695 Median :3.325 Median :17.71
Mean :3.597 Mean :3.217 Mean :17.85
3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90
Max. :4.930 Max. :5.424 Max. :22.90
gear carb
Min. :3.000 Min. :1.000
1st Qu.:3.000 1st Qu.:2.000
Median :4.000 Median :2.000
Mean :3.688 Mean :2.812
3rd Qu.:4.000 3rd Qu.:4.000
Max. :5.000 Max. :8.000 Appendix B: mpg vs am boxplot
g <- ggplot(data=mtcars)
g + geom_boxplot(mapping = aes(x=am, y=mpg))
Appendix C: mpg vs other variables scatterplot (colour coded by am)
g1 <- g + geom_point(mapping = aes(x=wt, y=mpg, col=am)) + theme_gray(base_size = 16) + theme(legend.position = 'none')
g2 <- g + geom_point(mapping = aes(x=cyl, y=mpg, col=am)) + theme_gray(base_size = 16) + theme(legend.position = 'none')
g3 <- g + geom_point(mapping = aes(x=disp, y=mpg, col=am)) + theme_gray(base_size = 16) + theme(legend.position = 'none')
g4 <- g + geom_point(mapping = aes(x=hp, y=mpg, col=am)) + theme_gray(base_size = 16) + theme(legend.position = 'none')
g5 <- g + geom_point(mapping = aes(x=drat, y=mpg, col=am)) + theme_gray(base_size = 16) + theme(legend.position = 'bottom')
g6 <- g + geom_point(mapping = aes(x=vs, y=mpg, col=am)) + theme_gray(base_size = 16) + theme(legend.position = 'bottom')
grid.arrange(g1, g2, g3, g4, g5, g6, nrow=3, ncol=2)
Appendix D: Residual Diagnostics
par(mfrow = c(2,2)); plot(fit3)