Regression Models Course Project
Marcos Medeiros
1/16/2022
Executive Summary
This is a report as part of the grades required for approval in the Regression Models course. The instructions for this assignment are the following:
Is an automatic or manual transmission better for MPG?
Quantify the MPG difference between automatic and manual transmissions
I will use the mtcars built-in dataset to demonstrate:
Compared with automatic transmission, manual transmission will yield better MPG, achieving 24mpg, against 17 MPG for automatics.
There are at least two confounders (Simpson’s Paradox), between MPG and the following variables:
- wt (weight): The greater the weight of the car, the less MPG.
- cyl (number of engine cylinder).
Data Analysis
Loadind and exploring the data:
library(ggplot2)
data (mtcars)
dim(mtcars)## [1] 32 11head(mtcars)## 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 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1Setting factors
mtcars$vs <- factor(mtcars$vs)
mtcars$transm <- factor(mtcars$am, labels=c("Automatic","Manual"))
mtcars$gear <- factor(mtcars$gear)
mtcars$carb <- factor(mtcars$carb)
head(mtcars)## mpg cyl disp hp drat wt qsec vs am gear carb transm
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4 Manual
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4 Manual
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1 Manual
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1 Automatic
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2 Automatic
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1 AutomaticComparing MPG per Transmission Types in a boxplot
boxplot(mpg ~ transm, data = mtcars, col = (c("red","blue")), ylab = "MPG", xlab = "Transmission Type")
We can see the means of MPG by Type of transmission
aggregate(mtcars$mpg,by=list(mtcars$transm),FUN=mean)## Group.1 x
## 1 Automatic 17.14737
## 2 Manual 24.39231Regression Analysis
lmt <- lm(mpg ~ factor (am), data=mtcars)
summary(lmt)##
## Call:
## lm(formula = mpg ~ factor(am), data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.3923 -3.0923 -0.2974 3.2439 9.5077
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.147 1.125 15.247 1.13e-15 ***
## factor(am)1 7.245 1.764 4.106 0.000285 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.902 on 30 degrees of freedom
## Multiple R-squared: 0.3598, Adjusted R-squared: 0.3385
## F-statistic: 16.86 on 1 and 30 DF, p-value: 0.000285In the simple linear regression test, we can see that p-value is less than 0.001, so we will reject the null hypothesis. However, the R-squared test is 0.3598, suggesting that only 35% of MPG variance can be explained by transmission type alone.
Variance Analysis
vart <- aov(mpg ~ ., data= mtcars)
summary (vart)## Df Sum Sq Mean Sq F value Pr(>F)
## cyl 1 817.7 817.7 102.591 2.3e-08 ***
## disp 1 37.6 37.6 4.717 0.04525 *
## hp 1 9.4 9.4 1.176 0.29430
## drat 1 16.5 16.5 2.066 0.16988
## wt 1 77.5 77.5 9.720 0.00663 **
## qsec 1 3.9 3.9 0.495 0.49161
## vs 1 0.1 0.1 0.016 0.90006
## am 1 14.5 14.5 1.816 0.19657
## gear 2 2.3 1.2 0.145 0.86578
## carb 5 19.0 3.8 0.477 0.78789
## Residuals 16 127.5 8.0
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1In the Variance test we are looking for variables with p-value less than 0.05. We have cyl with strong p-value, disp and wt.
Multivariate Regression Analysis
multivart <- lm(mpg ~ cyl + disp + wt + am, data = mtcars)
summary(multivart)##
## Call:
## lm(formula = mpg ~ cyl + disp + wt + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.318 -1.362 -0.479 1.354 6.059
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 40.898313 3.601540 11.356 8.68e-12 ***
## cyl -1.784173 0.618192 -2.886 0.00758 **
## disp 0.007404 0.012081 0.613 0.54509
## wt -3.583425 1.186504 -3.020 0.00547 **
## am 0.129066 1.321512 0.098 0.92292
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.642 on 27 degrees of freedom
## Multiple R-squared: 0.8327, Adjusted R-squared: 0.8079
## F-statistic: 33.59 on 4 and 27 DF, p-value: 4.038e-10The multivariate regression test now shows R-squared value over 0.83, suggestion that 83% of variance can be explained by the multivariate model. P-values for cyl and weight are less than 0.05, suggesting that these are the counfounders in relation between MPG and Type of transmission.
So, we can fit the final model using just the relevant variables
multivart_final <- lm(mpg ~ cyl + wt + am, data = mtcars)
summary(multivart_final)##
## Call:
## lm(formula = mpg ~ cyl + wt + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.1735 -1.5340 -0.5386 1.5864 6.0812
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 39.4179 2.6415 14.923 7.42e-15 ***
## cyl -1.5102 0.4223 -3.576 0.00129 **
## wt -3.1251 0.9109 -3.431 0.00189 **
## am 0.1765 1.3045 0.135 0.89334
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.612 on 28 degrees of freedom
## Multiple R-squared: 0.8303, Adjusted R-squared: 0.8122
## F-statistic: 45.68 on 3 and 28 DF, p-value: 6.51e-11MPG = 39.4+0.17x1-1.51x2-3.12x3, where x1=type of transmission (0 for manual and 1 for automatic), x2=number of cylinders and x3=weight
The final proof will be performed with residual plot analysis.
Residual Plot Analysis
par (mfrow=c(2,2))
plot (multivart_final)
The variance of “Residuals vs Fitted” plot is homoscedastic. The Q-Q test shows normal theoretical quantiles.
Conclusion
The number of cylinders and weight are confounders in the relation between MPG and type of transmission and the best model that explains the relation is a multivariate model MPG = 39.4+0.17x1-1.51x2-3.12x3, where x1=type of transmission (0 for manual and 1 for automatic), x2=number of cylinders and x3=weight. On average, a manual car will achieve 24 mpg and an automatics 17 mpg.