Regression models: Decision-making tools
Sebastian Barreto Rivera
September of 2020
1. Executive summary
4.1) The results indicate that statistically the automatic type transmission has a lower fuel consumption, compared to the manual type transmission.
4.2) The R-Squared value for the optimized model indicates that it is a good model, a reliable model
4.3) The type of transmission is not the only variable that determines fuel consumption. Fuel consumption is also determined by acceleration and the weight of the car
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2. Data pre-processing
Getting the data:
data(mtcars)See the variables that make up the data set:
names(mtcars)## [1] "mpg" "cyl" "disp" "hp" "drat" "wt" "qsec" "vs" "am" "gear"
## [11] "carb"See the main metrics of the variables:
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. :8.0002. Data analysys
Check correlation of the mpg variable and the other variables:
cor(mtcars$mpg, mtcars[,-1])## cyl disp hp drat wt qsec vs
## [1,] -0.852162 -0.8475514 -0.7761684 0.6811719 -0.8676594 0.418684 0.6640389
## am gear carb
## [1,] 0.5998324 0.4802848 -0.5509251Visually the correlation between variables can be seen like this:
pairs(mpg ~ ., data=mtcars)
The highest correlation is between 1) mpg and mt, 2) mpg and cyl, and 3) mpg and disp (negative correlations)
3. What type of transmission should I choose?
First it is necessary to change the type of variable for the case of ma, pass it to factor, in order to better interpret the information of this variable
mtcars$am <- as.factor(mtcars$am)
levels(mtcars$am) <- c("Automatic","Manual")We can use a diagram to visually get an idea of the relationship between the two types of transmission with respect to fuel consumption:
library(ggplot2)
ggplot(mtcars, aes(x=am, y=mpg, fill=am))+
geom_boxplot()+
geom_jitter(color="black", size=0.4, alpha=0.9) +
theme(legend.position="none",
plot.title = element_text(size=11)) +
ggtitle("Boxplot mgp vs transmission type") +
xlab("Transmission type")+
ylab("Gallons")
However, we need an analysis that statistically supports our decisions:
test.T <- t.test(mtcars$mpg ~ mtcars$am, conf.level=0.95)
test.T##
## 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 less than **0.05*, which allows us to state that the automatic type transmission has a lower fuel consumption, as opposed to the manual transmission.
4. Quantifyng mpg difference
We will build a stepwise model (using all the variables) through the following instructions (usin 10.000 steps:
stepwise.model <- step(lm(data=mtcars, mpg ~ .), trace=0, steps=10000)
summary(stepwise.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-11The above result allows us to identify 3 variables that contribute most to the model: wt, qsec and am. With this new information we can optimize our model:
optimized.model <- lm(mpg ~ factor(am): wt+ factor(am):qsec, data=mtcars)
summary(optimized.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-13Our optimized model presents a value for R-Squared of: 0.89, higher value than the initial model
par(mfrow=c(2,2))
plot(optimized.model)
4. Final choise
4.1) The results indicate that statistically the automatic type transmission has a lower fuel consumption, compared to the manual type transmission.
4.2) The R-Squared value for the optimized model indicates that it is a good model, a reliable model
4.3) The type of transmission is not the only variable that determines fuel consumption. Fuel consumption is also determined by acceleration and the weight of the car