Motor Cars Analysis Using ‘mtcars’ Data Set
Dhawal Kapil
April 12, 2016
Overview
This report contains analysis done on a collection cars data. We are trying to explore here the relationship of miles per gallon(mpg) with other variables.
The data set we has used in this report is ‘mtcars’ from dataset package.
Summary
In this report we are doing analysis on mtcars dataset to explore the relation of mpg i.e. miles per gallon with other variables.
Basically we are trying to answer the following two questions
- “Is an automatic or manual transmission better for MPG”
- “Quantify the MPG difference between automatic and manual transmissions”
We will be doing throughout this report basic exploratory analysis, inferences, fitting and comparing several models to achieve to our final answer.
Data
Source : ?mtcars
The data was 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).
Structure of Data
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 ...The data set we have contains 32 observations and 11 regressors.
Variables Description
| mpg | Miles/(US) gallon |
| cyl | Number of cylinders |
| disp | Displacement (cu.in.) |
| hp | Gross horsepower |
| drat | Rear axle ratio |
| wt | Weight (lb/1000) |
| qsec | 1/4 mile time |
| vs | V/S |
| am | Transmission (0 = automatic, 1 = manual) |
| gear | Number of forward gears |
| carb | Number of carburetors |
Exploratory Analysis and Basic Inferences
Here first we will try to explore the variation in mpg over two typesam i.e. transmission using a simple plot.
library(ggplot2)## Warning: package 'ggplot2' was built under R version 3.2.3ggplot(mtcars,aes(x=factor(am),y=mpg))+geom_boxplot()+
xlab("Transmission Type")+ylab("Miles Per Gallon (mpg)")+
ggtitle("Box Plot Showing MPG vs AM")
Here from the box plot it gives an indication that for Automatic transmission results in less Mpg. But here we have considered only a single variable am and rest variables are ignored.
We will be doing further analysis by trying different models below.
Regression Models
Correlation Analysis
We will begin our analysis with finding the correlation among the variables in the mtcars dataset.
The correlation plot has been kept simple to find major (positive/negative) variables correlated with mpg first and then we will view the scatterplot of major variables correlated.
library(corrplot)
M<-cor(mtcars)
corrplot.mixed(M,lower="circle",upper="number")
Here we can see in the plot which variables are negatively or positively correlated with mpg.
We can also calculate them
#Setting Correlations of Variables with themselves as 0
diag(M)<-0
#Finding Variables that have corelation of more than 0.75 with mpg
which(abs(M[1,])>0.7)## cyl disp hp wt
## 2 3 4 6Linear Model Fits
1. The Generic Model
We will first try to fit our mpg variable with all the variables present in the dataset and do a coefficient analysis.
fit0<-lm(mpg ~ .,mtcars)
summary(lm(fit0))$coeff## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.30337416 18.71788443 0.6573058 0.51812440
## cyl -0.11144048 1.04502336 -0.1066392 0.91608738
## disp 0.01333524 0.01785750 0.7467585 0.46348865
## hp -0.02148212 0.02176858 -0.9868407 0.33495531
## drat 0.78711097 1.63537307 0.4813036 0.63527790
## wt -3.71530393 1.89441430 -1.9611887 0.06325215
## qsec 0.82104075 0.73084480 1.1234133 0.27394127
## vs 0.31776281 2.10450861 0.1509915 0.88142347
## am 2.52022689 2.05665055 1.2254035 0.23398971
## gear 0.65541302 1.49325996 0.4389142 0.66520643
## carb -0.19941925 0.82875250 -0.2406258 0.81217871coefam1<-as.numeric(fit0$coeff["am1"])Now here in the summary we can see the relative effect of all the variables on the outcome mpg given other variables are held constant.
Our interest variable here is am for which we can get the coefficient value as NA which is interpreted as
The relative effect of am=1 i.e. manual transmission on mpg is 2.52 greater than am=0 which is automatic transmission (given other variables are held constant)
Now this relative difference will hold true in default case but we will explore further models below to actually get an idea of effect of am on mpg in combination with other variables.
Below we will try to fit several models with mpg as outcome and highly correlated variables found above and on those models we will evaluate the effect of
3. Miles per gallon(mpg) versus number of Gross Horse Power (hp) related with Transmission (am)
hpandamwithout interaction
fit_hp<-lm(mpg ~ hp+factor(am),mtcars)
summary(lm(fit_hp))$coeff## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.5849137 1.425094292 18.654845 1.073954e-17
## hp -0.0588878 0.007856745 -7.495191 2.920375e-08
## factor(am)1 5.2770853 1.079540576 4.888270 3.460318e-05Here we can see that adding am is significant in this model.
hpandamwith interaction
fit_hp_intr<-lm(mpg ~ hp*factor(am),mtcars)
summary(lm(fit_hp_intr))$coeff## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.6248478696 2.18294320 12.19676624 1.014017e-12
## hp -0.0591369818 0.01294486 -4.56837583 9.018508e-05
## factor(am)1 5.2176533777 2.66509311 1.95777527 6.028998e-02
## hp:factor(am)1 0.0004028907 0.01646022 0.02447662 9.806460e-01Here we can see that interaction term is not significant which means the relationship of mpg vs hp is independent of am.
Plotting the effect of mpg vs hp and am.
ggplot(mtcars,aes(x=hp,y=mpg,color=factor(am)))+geom_point()+
geom_abline(intercept = fit_hp$coeff[1],slope = fit_hp$coeff[2])+geom_abline(intercept = fit_hp$coeff[1],slope = fit_hp$coeff[2],color="salmon")+
geom_abline(intercept = fit_hp$coeff[1]+fit_hp$coeff[3],slope = fit_hp$coeff[2],color="cyan")
4. Miles per gallon(mpg) versus Weight (wt) related with Transmission (am)
wtandamwithout interaction
fit_wt<-lm(mpg ~ wt+factor(am),mtcars)
summary(lm(fit_wt))$coeff## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 37.32155131 3.0546385 12.21799285 5.843477e-13
## wt -5.35281145 0.7882438 -6.79080719 1.867415e-07
## factor(am)1 -0.02361522 1.5456453 -0.01527855 9.879146e-01Here we can see that adding am is not significant in this model.
wtandamwith interaction
fit_wt_intr<-lm(mpg ~ wt*factor(am),mtcars)
summary(lm(fit_wt_intr))$coeff## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 31.416055 3.0201093 10.402291 4.001043e-11
## wt -3.785908 0.7856478 -4.818836 4.551182e-05
## factor(am)1 14.878423 4.2640422 3.489277 1.621034e-03
## wt:factor(am)1 -5.298360 1.4446993 -3.667449 1.017148e-03Here we can see that interaction term is very much significant which means the relationship of mpg vs hp is dependent on am.
Plotting the effect of mpg vs hp and am.
ggplot(mtcars,aes(x=wt,y=mpg,color=factor(am)))+geom_point()+
geom_abline(intercept = coef(fit_wt_intr)[1], slope = coef(fit_wt_intr)[2],color="salmon")+
geom_abline(intercept = coef(fit_wt_intr)[1] + coef(fit_wt_intr)[3],
slope = coef(fit_wt_intr)[2] + coef(fit_wt_intr)[4], color="cyan")
5. Miles per gallon(mpg) versus Displacement (disp) related with Transmission (am)
dispandamwithout interaction
fit_disp<-lm(mpg ~ disp+factor(am),mtcars)
summary(lm(fit_disp))$coeff## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.84808111 1.834071377 15.183750 2.452658e-15
## disp -0.03685086 0.005781896 -6.373490 5.747528e-07
## factor(am)1 1.83345825 1.436099585 1.276693 2.118396e-01Here we can see that adding am is not significant in this model.
dispandamwith interaction
fit_disp_intr<-lm(mpg ~ disp*factor(am),mtcars)
summary(lm(fit_disp_intr))$coeff## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 25.15706407 1.925052971 13.068245 1.941478e-13
## disp -0.02758360 0.006218951 -4.435410 1.295371e-04
## factor(am)1 7.70907298 2.502677027 3.080331 4.600532e-03
## disp:factor(am)1 -0.03145482 0.011457373 -2.745378 1.043728e-02Here we can see that interaction term is very much significant which means the relationship of mpg vs disp is dependent on am.
Plotting the effect of mpg vs disp and am.
ggplot(mtcars,aes(x=disp,y=mpg,color=factor(am)))+geom_point()+
geom_abline(intercept = coef(fit_disp_intr)[1], slope = coef(fit_disp_intr)[2],color="salmon")+
geom_abline(intercept = coef(fit_disp_intr)[1] + coef(fit_disp_intr)[3], slope = coef(fit_disp_intr)[2] + coef(fit_disp_intr)[4], color="cyan")
Comparision Of Models
Variance explained by lower three models are very high which may be a sign of a good model.
The highest amongst these is explained by weight model but
summary(fit_wt_intr)$r.squared## [1] 0.8330375summary(fit_hp)$r.squared## [1] 0.7820346summary(fit_disp_intr)$r.squared## [1] 0.7898895Residual Plot
Plotting residual plot for model explained by ‘wt’ and ‘am’ in interaction
par(mfrow=c(2,2))
plot(fit_wt_intr)
Results Summary
“Is an automatic or manual transmission better for MPG”
When all the variables are held constant we can surely observe the difference between the ‘mpg’ in manual and automatic transmission and it was found out to be 2.52 more for manual.
But the basic model containing all the variables was not a good model to explain our system so we had to find out the best model and choose parameters that had high correlation with mpg.
At the end this question is not answerable because there are many other parameters that are present and may change the effect altogether.
“Quantify the MPG difference between automatic and manual transmissions”
For the best model that was found which was able to explain 83% of the variance it was find out that difference between manual and automatic transmission was 14.87 per unit change on mpg which was very less when other variables were adjusted.
But again this depends on the model that we choose and will vary as per the model.