In this issue of the “Motor Trend”, we will explore the features that affect fuel consumption in miles per gallon (MPG). The main questions to be addressed are:
It seems that fuel consumption is larger for manual transmission vehicles compared to automatic.
The data to be used in this study are extracted 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).
The data frame consists of 32 observations on 11 variables.
data(mtcars)
head(mtcars)
## 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 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1
Initially we plot a matrix of scatter plots displaying how the variables affect each other. Then we calculate which of the variables are strongly correlated with each other.
The most important features seem to be disp - Displacement (cu.in.) and cyl - Number of cylinders.
The feature under discussion, am - Transmission (0 = automatic, 1 = manual) seems not between those strongly affecting others. We will combine it with the features with which am is least correlated:
require(graphics)
pairs(mtcars, main = "mtcars data")
M <- abs(cor(mtcars[, -1]))
diag(M) <- 0
which(M > 0.9, arr.ind = T)
## row col
## disp 2 1
## cyl 1 2
sort(M[8, -8])
## carb vs qsec hp cyl disp wt drat gear
## 0.05753 0.16835 0.22986 0.24320 0.52261 0.59123 0.69250 0.71271 0.79406
am is a binary variable. After several repetitions (starting with all candidate covariants) it seems that regression with covariates carb and vs produced the best results (regarding variance and p-values).
lm1 <- lm(mpg ~ am, mtcars)
lm2 <- lm(mpg ~ am + carb + vs, mtcars)
Residual standard error, p-values and standard and press residuals (for am and covariants) are displayed below:
summary(lm1)
##
## Call:
## lm(formula = mpg ~ am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.392 -3.092 -0.297 3.244 9.508
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.15 1.12 15.25 1.1e-15 ***
## am 7.24 1.76 4.11 0.00029 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.9 on 30 degrees of freedom
## Multiple R-squared: 0.36, Adjusted R-squared: 0.338
## F-statistic: 16.9 on 1 and 30 DF, p-value: 0.000285
rstandard(lm1)
## Mazda RX4 Mazda RX4 Wag Datsun 710
## -0.72028 -0.72028 -0.33809
## Hornet 4 Drive Hornet Sportabout Valiant
## 0.89130 0.32541 0.19966
## Duster 360 Merc 240D Merc 230
## -0.59677 1.52006 1.18472
## Merc 280 Merc 280C Merc 450SE
## 0.43021 0.13678 -0.15664
## Merc 450SL Merc 450SLC Cadillac Fleetwood
## 0.03199 -0.40814 -1.41416
## Lincoln Continental Chrysler Imperial Fiat 128
## -1.41416 -0.51294 1.70025
## Honda Civic Toyota Corolla Toyota Corona
## 1.27560 2.01874 0.91226
## Dodge Challenger AMC Javelin Camaro Z28
## -0.34527 -0.40814 -0.80636
## Pontiac Firebird Fiat X1-9 Porsche 914-2
## 0.43021 0.61738 0.34136
## Lotus Europa Ford Pantera L Ferrari Dino
## 1.27560 -1.82438 -0.99630
## Maserati Bora Volvo 142E
## -1.99424 -0.63535
resid(lm1)/(1 - hatvalues(lm1))
## Mazda RX4 Mazda RX4 Wag Datsun 710
## -3.6750 -3.6750 -1.7250
## Hornet 4 Drive Hornet Sportabout Valiant
## 4.4889 1.6389 1.0056
## Duster 360 Merc 240D Merc 230
## -3.0056 7.6556 5.9667
## Merc 280 Merc 280C Merc 450SE
## 2.1667 0.6889 -0.7889
## Merc 450SL Merc 450SLC Cadillac Fleetwood
## 0.1611 -2.0556 -7.1222
## Lincoln Continental Chrysler Imperial Fiat 128
## -7.1222 -2.5833 8.6750
## Honda Civic Toyota Corolla Toyota Corona
## 6.5083 10.3000 4.5944
## Dodge Challenger AMC Javelin Camaro Z28
## -1.7389 -2.0556 -4.0611
## Pontiac Firebird Fiat X1-9 Porsche 914-2
## 2.1667 3.1500 1.7417
## Lotus Europa Ford Pantera L Ferrari Dino
## 6.5083 -9.3083 -5.0833
## Maserati Bora Volvo 142E
## -10.1750 -3.2417
summary(lm2)
##
## Call:
## lm(formula = mpg ~ am + carb + vs, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.280 -1.231 0.408 2.052 4.820
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.517 1.609 12.13 1.2e-12 ***
## am 6.798 1.101 6.17 1.2e-06 ***
## carb -1.431 0.408 -3.51 0.0016 **
## vs 4.196 1.325 3.17 0.0037 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.96 on 28 degrees of freedom
## Multiple R-squared: 0.782, Adjusted R-squared: 0.758
## F-statistic: 33.4 on 3 and 28 DF, p-value: 2.14e-09
rstandard(lm2)
## Mazda RX4 Mazda RX4 Wag Datsun 710
## 0.146331 0.146331 -2.267159
## Hornet 4 Drive Hornet Sportabout Valiant
## -0.315784 0.731453 -1.496817
## Duster 360 Merc 240D Merc 230
## 0.177655 1.269128 0.696874
## Merc 280 Merc 280C Merc 450SE
## 0.462716 -0.072659 0.412237
## Merc 450SL Merc 450SLC Cadillac Fleetwood
## 0.728007 -0.008789 -1.192344
## Lincoln Continental Chrysler Imperial Fiat 128
## -1.192344 0.318168 1.198392
## Honda Civic Toyota Corolla Toyota Corona
## 0.981444 1.739884 -0.279995
## Dodge Challenger AMC Javelin Camaro Z28
## -0.413585 -0.520932 -0.173627
## Pontiac Firebird Fiat X1-9 Porsche 914-2
## 0.910365 -0.642682 0.953467
## Lotus Europa Ford Pantera L Ferrari Dino
## 0.981444 -1.719689 0.740790
## Maserati Bora Volvo 142E
## 0.058333 -2.230000
resid(lm2)/(1 - hatvalues(lm2))
## Mazda RX4 Mazda RX4 Wag Datsun 710
## 0.46069 0.46069 -7.18033
## Hornet 4 Drive Hornet Sportabout Valiant
## -0.99152 2.29624 -4.69979
## Duster 360 Merc 240D Merc 230
## 0.54751 3.98232 2.18668
## Merc 280 Merc 280C Merc 450SE
## 1.55241 -0.24377 1.26893
## Merc 450SL Merc 450SLC Cadillac Fleetwood
## 2.24091 -0.02705 -3.67467
## Lincoln Continental Chrysler Imperial Fiat 128
## -3.67467 0.98056 3.79543
## Honda Civic Toyota Corolla Toyota Corona
## 3.07245 5.51039 -0.87914
## Dodge Challenger AMC Javelin Camaro Z28
## -1.29836 -1.63536 -0.53510
## Pontiac Firebird Fiat X1-9 Porsche 914-2
## 2.85790 -2.03544 3.13241
## Lotus Europa Ford Pantera L Ferrari Dino
## 3.07245 -5.41408 2.44473
## Maserati Bora Volvo 142E
## 0.22804 -6.98110
The following figure displays how fuel consumption varies with transmission. Fitted lines on am (red) and am+carb+vs (blue) are added.
plot(mtcars$mpg ~ mtcars$am, col = mtcars$hp, pch = 19, main = "Fuel Consumption (gmt) over Transmission (am)",
xlab = "Transmission (0 = automatic, 1 = manual)", ylab = "Miles/(US) gallon")
abline(lm1, lwd = 3, col = "red")
abline(lm2, lwd = 3, col = "blue")
## Warning: only using the first two of 4 regression coefficients
