Regression Models Course Project
phobos
Sunday, November 23, 2014
The data
- Motor Trend Car Road Tests: 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).
head(mtcars,3)## 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 1A data frame with 32 observations on 11 variables.
| 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
- Graph data some variables according to the type of transmission. The automatic transmission in black and The manual transmission in blue.
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 ...summary(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. :8.000pairs(mtcars[,c(1,3:7)], main = "Motor Trend Car Road Tests", col = 4 + (mtcars$am == 0)*5, pch = 19,cex = 2)
Analysis
- We choose the best model according the “step” command that performs an automatic checks series and shows the best model.
model1<-step(lm(mpg~.,data=mtcars),trace = 0)
summary(model1)##
## Call:
## lm(formula = mpg ~ wt + qsec + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4811 -1.5555 -0.7257 1.4110 4.6610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.6178 6.9596 1.382 0.177915
## wt -3.9165 0.7112 -5.507 6.95e-06 ***
## qsec 1.2259 0.2887 4.247 0.000216 ***
## am 2.9358 1.4109 2.081 0.046716 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.459 on 28 degrees of freedom
## Multiple R-squared: 0.8497, Adjusted R-squared: 0.8336
## F-statistic: 52.75 on 3 and 28 DF, p-value: 1.21e-11We can see that a model has 3 variables; wt, qsec and am. This model captured 0.85 of total variance. - Optimize the model, placing as the variable to am.
modelF<- lm(mpg ~ factor(am):wt + factor(am):qsec, data = mtcars)
summary(modelF)##
## Call:
## lm(formula = mpg ~ factor(am):wt + factor(am):qsec, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.9361 -1.4017 -0.1551 1.2695 3.8862
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.9692 5.7756 2.419 0.02259 *
## factor(am)0:wt -3.1759 0.6362 -4.992 3.11e-05 ***
## factor(am)1:wt -6.0992 0.9685 -6.297 9.70e-07 ***
## factor(am)0:qsec 0.8338 0.2602 3.205 0.00346 **
## factor(am)1:qsec 1.4464 0.2692 5.373 1.12e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.097 on 27 degrees of freedom
## Multiple R-squared: 0.8946, Adjusted R-squared: 0.879
## F-statistic: 57.28 on 4 and 27 DF, p-value: 8.424e-13par(mfrow=c(1,4))
plot(modelF)
This final model captured 0.89 of total variance. Which predicts better mpg
- Graphed the variables most correlated wt (Weight (lb/1000)) and qsec (1/4 mile time)
par(mfrow=c(1,2))
plot(mtcars$mpg~mtcars$wt,col=4+(mtcars$am==0)*5,pch=19,xlab="Weight (lb/1000)",ylab="Miles/(US) gallon",frame=F)
abline(lm(mtcars$mpg[mtcars$am==1]~mtcars$wt[mtcars$am==1]),col=4,lwd = 2)
abline(lm(mtcars$mpg[mtcars$am==0]~mtcars$wt[mtcars$am==0]),col=1,lwd = 2)
legend("topright",pch=21,pt.bg=c(4,1),legend=c("Manual","Automatic"))
plot(mtcars$mpg~mtcars$qsec,col=4+(mtcars$am==0)*5,pch=19,xlab="1/4 mile time",ylab="Miles/(US) gallon",frame=F)
abline(lm(mtcars$mpg[mtcars$am==1]~mtcars$qsec[mtcars$am==1]),col=4,lwd = 2)
abline(lm(mtcars$mpg[mtcars$am==0]~mtcars$qsec[mtcars$am==0]),col=1,lwd = 2)
legend("bottomright",pch=21,pt.bg=c(4,1),legend=c("Manual","Automatic"))
-Relationship Miles/(US) gallon and transmission
mean(mtcars$mpg[mtcars$am==1])## [1] 24.39231mean(mtcars$mpg[mtcars$am==0])## [1] 17.14737mean(mtcars$mpg[mtcars$am==1]) - mean(mtcars$mpg[mtcars$am==0])## [1] 7.244939Conclusion
The manual transmission 24.39 better than the automatic transmission 17.15 for Miles/(US) gallon. Although manual cars have (on average) 7.24 better MPG (Miles/(US) gallon) than automatic cars.
This model captured 89.5% of total variance and adjusted variance is 0.879. When the weight increased 1000 lbs, the mpg decreased, -3.176 miles for automatic cars and -6.099 miles for manual cars and when the weight decreases 1000 lbs, the mpg increased, 3.176 miles for automatic cars and 6.099 miles for manual cars. And when the accelaration speed droped and 1/4 mile time increased 1 secs, the mpg increased 0.834 miles for automatic cars and 1.446 miles for manual cars.
In conclusion, the mpg is largely determined by the interplay between weight, accelaration and tramsmission.