The goal of this study is to check whether automatic transmission has an impact on MPG in cars. We have a dataset of 32 cars with 11 measurement such as MPG, weight, displacement etc from each. We use regression analysis to check for the impact of transmission type on fuel consumption.
At the end we will see that, although it seems at the first look that automatic transmission decreases MPG but this is not an statistically meaningful conclusion.
The data was extracted from the 1974 Motor Trend US magazine, and includes fuel consumotion (mpg), Number of cylinders(cyl), Displacement (disp), Gross horsepower(hp), Rear axle ratio (drat), Weight (wt), 1/4 mile time (qsec), V/S (vs), Transmission (am, 0 = automatic, 1 = manual), Number of forward gears (gear), Number of carburetorsof (carb), for 32 automobiles (1973-74 models).
Comparing the MPG of automatic transmissited cars vs manaully transsmitted ones (17.1473684, 24.3923077) make us think that automatic transsmission has a negaive imoact on fuel ecomomy. But having a closer look at the data set (see Fig. 1), reveals that in particular in the case of weight(wt) and rear axle ratio (drat) variables, in one hand they both have impact on MPG and on the other hand, they can roughly separate automatic and manually transmitted cars. In other words more weight means less mpg also cars with automatic transmission are heavier in this data set. So we use multivariable regression analysis to face this lack of data in the dataset.
For start we fit a linear model for mpg vs all other variables and look at the coefficients of the model.
## (Intercept) cyl disp hp drat wt
## 12.30 -0.11 0.01 -0.02 0.79 -3.72
## qsec vs1 am1 gear carb
## 0.82 0.32 2.52 0.66 -0.20
It says that keeping other factors constant, the change from automatic transsmition to manuall increases mpg by 2.52 miles per gallon. But the confidence interval for this qoefiicent is (-1.76, 6.8) containg 0 which means that this improvement is not statisically significant. But there might be extra variables in the model which introduce extra variance. To seek for that we use the varible importance factor (VIF). These are the VIF for the model.
## cyl disp hp drat wt qsec vs am gear carb
## 15.37 21.62 9.83 3.37 15.16 7.53 4.97 4.65 5.36 7.91
cyl, disp have the most vif i.e. their impact on the model is respectively 15.37, 21.62 times more as if they were diagonal to other variables. Now we omit these variables and fit a new model.
## (Intercept) hp drat wt qsec vs1
## 13.81 -0.01 0.89 -2.61 0.64 0.09
## am1 gear carb
## 2.42 0.69 -0.61
The amt coefficient is a bit smaller but not that much changed. It is still insignificant with 95% confidence interval of (−1.53, 6.38). Let us take a look at the VIF's of the new model.
## hp drat wt qsec vs am gear carb
## 6.02 3.11 6.05 5.92 4.27 4.29 4.69 4.29
The VIF's of this last model, although not ideally small, are in a quite similar range. So there is no sense in ommiting any of them. Now for assurance we comare the models with anova function.
## Analysis of Variance Table
##
## Model 1: mpg ~ hp + drat + wt + qsec + vs + am + gear + carb
## Model 2: mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 23 151.47
## 2 21 147.49 2 3.9808 0.2834 0.7561
The P-value is quite high, so using these two varaibales was not actually necessary.
To get sure abput the validity of the last model I plot the residual plot (see Fig2).As you can see there is no special trend in the residual plot. So the model appears to be valid.
Although a manually transmitted car seems to show better fuel economy, this improvement does not appear to be significant. It should be noted that this analysis is done on an old and small dataset, so is not that realiable. For assurance more investigation needs to be done.
