Regression Model : Motor Trend Analysis
Siti Salwani Yaacob
2/8/2020
Instruction
You work for Motor Trend, a magazine about the automobile industry. Looking at a data set of a collection of cars, they are interested in exploring the relationship between a set of variables and miles per gallon (MPG) (outcome). They are particularly interested in the following two questions:
“Is an automatic or manual transmission better for MPG”
“Quantify the MPG difference between automatic and manual transmissions”
Preprocessing
data(mtcars)
names(mtcars)## [1] "mpg" "cyl" "disp" "hp" "drat" "wt" "qsec" "vs" "am" "gear"
## [11] "carb"summary(cars)## speed dist
## Min. : 4.0 Min. : 2.00
## 1st Qu.:12.0 1st Qu.: 26.00
## Median :15.0 Median : 36.00
## Mean :15.4 Mean : 42.98
## 3rd Qu.:19.0 3rd Qu.: 56.00
## Max. :25.0 Max. :120.00Analysis
There are 11 variables in the dataset. We are interested in the relationship between mpg and other variables, so first we check the correlation between mpg and other variables by using the cor() function.
cor(mtcars$mpg, mtcars[, -1])## cyl disp hp drat wt qsec
## [1,] -0.852162 -0.8475514 -0.7761684 0.6811719 -0.8676594 0.418684
## vs am gear carb
## [1,] 0.6640389 0.5998324 0.4802848 -0.5509251From the correlation results, we can see that the cyl, hp, wt and carb are negatively correlated with mpg.
Is Automatic or Manual Transmission Better for MPG?
Checking at the help of dataset (?mtcars), we can identify the transmission type as :
0 = automatic
1 = manual
We need to make some data conversions.
mtcars$am <- as.factor(mtcars$am)
levels(mtcars $am) <- c("Automatic", "Manual")
boxplot(mtcars$mpg ~ mtcars$am, data = mtcars, outpch = 19, ylab="MPG (miles per gallon)", xlab = "Transmission Type", main = "MPG vs Transmission Type", col = "blue")
A boxplot can be seen with the relationship between MPG and Automatiic / Manual type. It seems like automatic cars has better MPG compare with manual cars. To support this hypothesis, we perform statistical analysis with a t-test.
t.test(mtcars$mpg ~ mtcars$am, conf.level = 0.95)##
## Welch Two Sample t-test
##
## data: mtcars$mpg by mtcars$am
## t = -3.7671, df = 18.332, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -11.280194 -3.209684
## sample estimates:
## mean in group Automatic mean in group Manual
## 17.14737 24.39231The p-value is 0.001374, we may reject the null hypothesis and conclude, that automatic transmission cars have lower MPG compared with manual transmission cars. But this assumption is based on all other characteristics of automatic transmission cars and manual transmission cars are same (both has same weight distribution). This needs to be further explored in multiple linear regression analysis.
Quantifying MPG Difference Between Automatic & Manual Transmissions
Here we try to quantify the MPG difference between transmission type, and find if there are other variables that account for the MPG differeneves. Here we can adopt a stepwise algorithm (step() function) to choose the best model.
step_model = step(lm(data = mtcars, mpg ~ .), trace = 0, steps = 10000)
summary(step_model)##
## 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 ***
## amManual 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-11At this point we have a model, which includes 3 variables; “wt” , “qsec”,“am”. This model has a 0.85 of total variance. To further optimize the model, we can examine mpg ~ wt + qsec correlation with am.
model <- lm(mpg ~ factor(am): wt + factor(am): qsec, data = mtcars)
summary(model)##
## 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)Automatic:wt -3.1759 0.6362 -4.992 3.11e-05 ***
## factor(am)Manual:wt -6.0992 0.9685 -6.297 9.70e-07 ***
## factor(am)Automatic:qsec 0.8338 0.2602 3.205 0.00346 **
## factor(am)Manual: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-13Summary
Interpreting the results, we can see that this model has a 89.5% total variance with an adjusted variance of 0.879. By adding the coefficients, we have the following conclusions;
When the weight increased by 1000 lbs, the MPG decreased by -3.176 for automatic transmission cars, and for manual transmission cars.
So, with increasing car weight, we should choose manual transmission cars.
When the acceleration speed dropped, and 1/4 mile time increased (by 1 sec), the MPG factor increased by 0.834 miles for automatic transmission cars, and 1.446 miles for manual transmission cars.
With lower acceleration speed, but same weight, manual transmission cars are better for MPG.
Conclusion
The MPG is largely determined by the interpay between weight, acceleration and transmission. Given the above analysis, the original question (Automatic Transmission vs Manual Transmission) is not really answered, and should considered in the context of weight and acceleration speed.
Appendices
1. Residual Check Diagnostic Plot
par(mfrow = c(2,2))
plot(model)
2. Further plot
plot(model)
