Is automatic or manual transmission better for MPG?

Regression Models Course Project

Summary

In this project, we look into “mtcars” data from the 1974 Motor Trend US magazine to investigate if an automatic or manual transmission is better for MPG. After fitting the model, we discovered that, with 95% confidence, automatic or manual transmission does not have significant difference for MPG.

Loading the data

library(datasets)
data(mtcars)

Looking at the data

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
cor(mtcars$mpg, mtcars)
##      mpg       cyl       disp         hp      drat         wt     qsec
## [1,]   1 -0.852162 -0.8475514 -0.7761684 0.6811719 -0.8676594 0.418684
##             vs        am      gear       carb
## [1,] 0.6640389 0.5998324 0.4802848 -0.5509251

Besides “am”, “mpg” has different degrees of correlation with all other variables. We select those with correlation coefficient greater than 0.8 to fit the model.

Fit the model

  1. Create the nested models
fit1 <- lm(mpg ~ factor(am), mtcars)
fit2 <- lm(mpg ~ factor(am)+I(wt-mean(wt)), mtcars)
fit3 <- lm(mpg ~ factor(am)+I(wt-mean(wt))+factor(cyl), mtcars)
fit4 <- lm(mpg ~ factor(am)+I(wt-mean(wt))+factor(cyl)+I(disp-mean(disp)), mtcars)
  1. Compare the models
anova(fit1, fit2, fit3, fit4)
## Analysis of Variance Table
## 
## Model 1: mpg ~ factor(am)
## Model 2: mpg ~ factor(am) + I(wt - mean(wt))
## Model 3: mpg ~ factor(am) + I(wt - mean(wt)) + factor(cyl)
## Model 4: mpg ~ factor(am) + I(wt - mean(wt)) + factor(cyl) + I(disp - 
##     mean(disp))
##   Res.Df    RSS Df Sum of Sq       F    Pr(>F)    
## 1     30 720.90                                   
## 2     29 278.32  1    442.58 62.9247 2.073e-08 ***
## 3     27 182.97  2     95.35  6.7784  0.004273 ** 
## 4     26 182.87  1      0.10  0.0141  0.906470    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Model 2 and 3 are significant. The summary of fit 3 (See in Appendix) show that the p-value of F-statistic is 2.73e-10, which means the model is significant. The adjusted R-squared is 0.8134, so the model explains 81.34% of the variance of the outcome(MPG).

Residual and Diagnostics

  1. Check the normality of the residuals(Residual plot please check in Appendix)
shapiro.test(fit3$residuals)$p
## [1] 0.2602345

The p-value is 0.26, greater than 0.05, we fail to reject the hypothesis. So the residual of fit3 is normal distribution.

  1. Check the hatvalues
summary(hatvalues(fit3))
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.07525 0.10942 0.15358 0.15625 0.20163 0.28563

The hatvalues are ranging from 0.075 to 0.286, with the mean value of 0.156. None of them has abnormal higher leverage value than the others.

Question 1: Is an automatic or manual transmission better for MPG?

Answer: No. According to the model, automatic or manual transmission does not have significant difference for MPG.

Question 2: Quantify the MPG difference between automatic and manual transmissions.

summary(fit3)$coef
##                    Estimate Std. Error    t value     Pr(>|t|)
## (Intercept)      23.6205485  1.1708629 20.1736253 8.181502e-18
## factor(am)1       0.1501031  1.3002231  0.1154441 9.089474e-01
## I(wt - mean(wt)) -3.1495978  0.9080495 -3.4685309 1.770987e-03
## factor(cyl)6     -4.2573185  1.4112394 -3.0167231 5.514697e-03
## factor(cyl)8     -6.0791189  1.6837131 -3.6105432 1.227964e-03
confint(fit3, 'factor(am)1')
##                 2.5 %   97.5 %
## factor(am)1 -2.517734 2.817941

Answer:

  • The average MPG of automatic car with 4 cylinders and average weight in the data set is 23.62.
  • The estimated difference MPG between automatic and manual car with 4 cylinders and average weight in the data set is 0.15, but the difference is not significant (p-value = 0.91).
  • The 95% confidence interval of average MPG difference between automatic and manual car is -2.52 to 2.82.

Appendix

Summary of fit3

summary(fit3)
## 
## Call:
## lm(formula = mpg ~ factor(am) + I(wt - mean(wt)) + factor(cyl), 
##     data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.4898 -1.3116 -0.5039  1.4162  5.7758 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       23.6205     1.1709  20.174  < 2e-16 ***
## factor(am)1        0.1501     1.3002   0.115  0.90895    
## I(wt - mean(wt))  -3.1496     0.9080  -3.469  0.00177 ** 
## factor(cyl)6      -4.2573     1.4112  -3.017  0.00551 ** 
## factor(cyl)8      -6.0791     1.6837  -3.611  0.00123 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.603 on 27 degrees of freedom
## Multiple R-squared:  0.8375, Adjusted R-squared:  0.8134 
## F-statistic: 34.79 on 4 and 27 DF,  p-value: 2.73e-10

Residual Plot

par(mfrow=c(2,2))
plot(fit3)

There is no clear pattern in the “Residual vs Fitted” plot, and most residuals are symmetrically distributed around the “0” line, which is good. The QQ plot looks OK too, the points form a roughly straight.