Motor Trend - Does automatic transmission improve miles per gallon performance?
Paolo Saracco
2024-05-10
Executive summary
We examine the effect of the transmission type (manual vs automatic) on the miles per gallon (mpg) in the mtcars dataset using linear regression. In particular, we focus on the following questions:
- Is an automatic or manual transmission better for mpg?
- Can we quantify the mpg difference between automatic and manual transmissions?
Regression analysis using transmission type, weight, and quarter mile time as explanatory variables leads to the conclusion that manual cars get on average 14 more mpg than automatic cars, holding the other effects constant.
Packages
We will use the dplyr package for working with data frames and the ggplot2
package for graphs.
library(dplyr)
library(ggplot2)Data
The data set comes from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973–74 models).
library(datasets)
data("mtcars")Exploratory data analysis
We are dealing with a
32 x 11
data set: we have 11 observations for
32 cars. Let us have a look at the data (we refer the reader
to the mtcars help page for the explanation of the variable names).
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 ...tail(mtcars)## mpg cyl disp hp drat wt qsec vs am gear carb
## Porsche 914-2 26.0 4 120.3 91 4.43 2.140 16.7 0 1 5 2
## Lotus Europa 30.4 4 95.1 113 3.77 1.513 16.9 1 1 5 2
## Ford Pantera L 15.8 8 351.0 264 4.22 3.170 14.5 0 1 5 4
## Ferrari Dino 19.7 6 145.0 175 3.62 2.770 15.5 0 1 5 6
## Maserati Bora 15.0 8 301.0 335 3.54 3.570 14.6 0 1 5 8
## Volvo 142E 21.4 4 121.0 109 4.11 2.780 18.6 1 1 4 2summary(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
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000To begin with, it is useful to look at a plot of how the different mpg measures (25th, 50th, and 75th percentile of mpg) vary across the two categories.
Figure 1: Box plot
It seems that manual transmission is better for mpg : cars with manual
transmission covers more miles per gallon. To quantify this effect we perform a
linear regression with mpg as outcome and the factor variable am as
predictor, without intercept (which is the same as a T-test)
naive_fit <- lm(mpg ~ as.factor(am)-1, data = mtcars)| mean | std error | 95% conf int | |
|---|---|---|---|
| automatic | 17.1474 | 1.1246 | [14.8506, 19.4441] |
| manual | 24.3923 | 1.3596 | [21.6157, 27.1689] |
However, we reasonably expect that other factors play a role. For example:
Figure 2: Pair plot
As we can see, mpg is also influenced by cyl, hp, wt, qsec,
and not all of them are uncorrelated.
Model selection
It may happen that the effect of transmission on miles per gallon is biased by the fact that we are not considering other factors. Therefore we start nesting models in order to see if adding regressors have a significant effect above and beyond the change in degrees of freedom.
data <- mtcars %>%
mutate(
cyl = as.factor(cyl),
vs = as.factor(vs),
am = as.factor(am),
gear = as.factor(gear),
carb = as.factor(carb)
)Since it is clear that the weight of the car influences its miles per gallon,
we start adding wt to our model and then we ask whether adding another
variable has a significant effect, by relying on the ANOVA test.
seek <- function(var, params = c()) {
formula <- reformulate(c(params, var), response = 'mpg')
fit_new <- lm(formula, data)
cat("p-value for adding", var, "is", anova(fit, fit_new)[2,6], "\n")
}
fit <- lm(mpg ~ am + wt, data)
invisible(sapply(names(select(data, !c(mpg,am,wt))), seek, params = c('am','wt')))## p-value for adding cyl is 0.003473216
## p-value for adding disp is 0.0678774
## p-value for adding hp is 0.0005464023
## p-value for adding drat is 0.2849371
## p-value for adding qsec is 0.0002161737
## p-value for adding vs is 0.008454158
## p-value for adding gear is 0.1200295
## p-value for adding carb is 0.2470563A posteriori, we will perform 20 tests. Therefore, to take into account the multiple hypothesis testing issue and to control the false positive
rate at level \(\alpha = 0.05\), we call significant the p-values below
\(0.05/20 = 0.0025\). It turns out that hp and qsec have a
significant effect. Since qsec is the one with weaker correlation with wt:
## qsec hp
## wt -0.17 0.66then we add qsec to our model and we ask ourselves the same question.
fit <- lm(mpg ~ am + wt + qsec, data)
invisible(sapply(names(select(data, !c(mpg,am,wt,qsec))), seek, params = c('am','wt','qsec') ))## p-value for adding cyl is 0.4582894
## p-value for adding disp is 0.4717085
## p-value for adding hp is 0.2230879
## p-value for adding drat is 0.6390028
## p-value for adding vs is 0.9931865
## p-value for adding gear is 0.9858944
## p-value for adding carb is 0.8960083We conclude that no other variable has a significant effect, so we search for potential cross-effects:
| ANOVA w.r.t. adding | am:wt |
am:qsec |
wt:qsec |
|---|---|---|---|
| p-value | 0.0018086 | 0.0384079 | 0.2823973 |
We can fit our linear model with mpg as output and am, wt, am:wt, qsec as regressors and view the inference results:
fit <- lm(mpg ~ am*wt + qsec, data)
summary(fit)##
## Call:
## lm(formula = mpg ~ am * wt + qsec, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.5076 -1.3801 -0.5588 1.0630 4.3684
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.723 5.899 1.648 0.110893
## am1 14.079 3.435 4.099 0.000341 ***
## wt -2.937 0.666 -4.409 0.000149 ***
## qsec 1.017 0.252 4.035 0.000403 ***
## am1:wt -4.141 1.197 -3.460 0.001809 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.084 on 27 degrees of freedom
## Multiple R-squared: 0.8959, Adjusted R-squared: 0.8804
## F-statistic: 58.06 on 4 and 27 DF, p-value: 7.168e-13Therefore, holding constant quarter mile time and weight, cars with manual transmissions get about 14 more miles per gallon:
| estimate | std error | 95% conf int | |
|---|---|---|---|
| manual | 14.0794 | 3.4353 | [7.0309, 21.128] |
Moreover, we are satisfied because our model significantly explains about 90% of the variance (R-squared is 0.8959).
We can check if regression assumptions are met, namely the normality assumption for the residuals, with a Shapiro-Wilk test and diagnostic plotting:
shapiro.test(fit$residuals)##
## Shapiro-Wilk normality test
##
## data: fit$residuals
## W = 0.94444, p-value = 0.1001
Figure 3: Diagnostic plotting
Normality assumptions don’t seem far off (approximately), and heteroskedasticity doesn’t seem to be an issue.
Conclusions
Regression analysis using transmission type, weight, and quarter mile time as explanatory variables leads to the conclusion that manual cars get on average 14.08 \(\pm\) 3.44 more mpg than automatic cars, holding the other effects constant.
However, the number of observations is relatively small, whence it is difficult to draw trustful conclusions. Figure 4 offers a glimpse of the problem. It is advisable to obtain a new and larger data set to challenge these results.
Appendices
Figure 4: 4-dim scatterplot