Regression Models - Course Project: Motor Trend from MPG Data Analysis
Htann
2017 8 19
Summary
This report analyzed the relationship . The report set out to determine which transmission type produces a higher MPG. The mtcars dataset was used for this analysis. A t-test between automatic and manual transmission vehicles shows that manual transmission vehicles have a 7.245 greater MPG than automatic transmission vehicles. According to this report, we explore the effects between transmission type (manual or automatic) and miles per gallon (MPG) from a set of 32 automobile models from 1973-74. The data was extracted from the 1974 Motor Trend US magazine, and is available from the R Package datasets. The analysis of this data showed: a) Exploratory Data Analysis, in which the data is loaded, preprocessed, and subject to an initial graphical examination b) Regression Analysis, in which a linear model is fit to the data. c) Appendix, wherein the plots that are used to support the discussion throughout this report are presented. After fitting multiple linear regressions, analysis showed that the manual transmission contributed less significantly to MPG, only an improvement of 1.81 MPG. Other variables, weight, horsepower, and number of cylinders contributed more significantly to the overall MPG of vehicles. The source code for this document can be retrieved from my GitHub account.
Load Data
library(ggplot2)
data(mtcars)
head(mtcars, n=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 1dim(mtcars)## [1] 32 11mtcars$cyl <- as.factor(mtcars$cyl)
mtcars$vs <- as.factor(mtcars$vs)
mtcars$am <- factor(mtcars$am)
mtcars$gear <- factor(mtcars$gear)
mtcars$carb <- factor(mtcars$carb)
attach(mtcars)## The following object is masked from package:ggplot2:
##
## mpgExploratory Analysis
The graph which was showed final result by Appendix, leads us to believe that there is a significant increase in MPG when for vehicles with a manual transmission vs automatic. T-Test transmission type and MPG
testResults <- t.test(mpg ~ am)
testResults$p.value## [1] 0.001373638testResults$estimate## mean in group 0 mean in group 1
## 17.14737 24.39231By resuly, we can find the difference estimate between the 2 transmissions is 7.24494 MPG in favor of manual.
Regression Analysis
To obtain a parsimonious model for this particular data, the following approach will be employed: The modified model has 4 variables (cylinders, horsepower, weight, transmission). a) The R-squared value of 0.8659 -> 87% of the variance in MPG b) The p-values: less than 0.05 c) The coefficients results; : Increasing the number of cylinders from 4 to 6 with decrease the MPG by 3.03 : Further increasing the cylinders to 8 with decrease the MPG by 2.16 : Increasing the horsepower is decreases MPG 3.21 for every 100 horsepower : Weight decreases the MPG by 2.5 for each 1000 lbs increase : A Manual transmission improves the MPG by 1.81
fullModelFit <- lm(mpg ~ ., data = mtcars)
summary(fullModelFit) ##
## Call:
## lm(formula = mpg ~ ., data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.5087 -1.3584 -0.0948 0.7745 4.6251
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 23.87913 20.06582 1.190 0.2525
## cyl6 -2.64870 3.04089 -0.871 0.3975
## cyl8 -0.33616 7.15954 -0.047 0.9632
## disp 0.03555 0.03190 1.114 0.2827
## hp -0.07051 0.03943 -1.788 0.0939 .
## drat 1.18283 2.48348 0.476 0.6407
## wt -4.52978 2.53875 -1.784 0.0946 .
## qsec 0.36784 0.93540 0.393 0.6997
## vs1 1.93085 2.87126 0.672 0.5115
## am1 1.21212 3.21355 0.377 0.7113
## gear4 1.11435 3.79952 0.293 0.7733
## gear5 2.52840 3.73636 0.677 0.5089
## carb2 -0.97935 2.31797 -0.423 0.6787
## carb3 2.99964 4.29355 0.699 0.4955
## carb4 1.09142 4.44962 0.245 0.8096
## carb6 4.47757 6.38406 0.701 0.4938
## carb8 7.25041 8.36057 0.867 0.3995
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.833 on 15 degrees of freedom
## Multiple R-squared: 0.8931, Adjusted R-squared: 0.779
## F-statistic: 7.83 on 16 and 15 DF, p-value: 0.000124summary(fullModelFit)$coeff ## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 23.87913244 20.06582026 1.19004018 0.25252548
## cyl6 -2.64869528 3.04089041 -0.87102622 0.39746642
## cyl8 -0.33616298 7.15953951 -0.04695316 0.96317000
## disp 0.03554632 0.03189920 1.11433290 0.28267339
## hp -0.07050683 0.03942556 -1.78835344 0.09393155
## drat 1.18283018 2.48348458 0.47627845 0.64073922
## wt -4.52977584 2.53874584 -1.78425732 0.09461859
## qsec 0.36784482 0.93539569 0.39325050 0.69966720
## vs1 1.93085054 2.87125777 0.67247551 0.51150791
## am1 1.21211570 3.21354514 0.37718957 0.71131573
## gear4 1.11435494 3.79951726 0.29328856 0.77332027
## gear5 2.52839599 3.73635801 0.67670068 0.50889747
## carb2 -0.97935432 2.31797446 -0.42250436 0.67865093
## carb3 2.99963875 4.29354611 0.69863900 0.49546781
## carb4 1.09142288 4.44961992 0.24528452 0.80956031
## carb6 4.47756921 6.38406242 0.70136677 0.49381268
## carb8 7.25041126 8.36056638 0.86721532 0.39948495stepFit <- step(fullModelFit)## Start: AIC=76.4
## mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb
##
## Df Sum of Sq RSS AIC
## - carb 5 13.5989 134.00 69.828
## - gear 2 3.9729 124.38 73.442
## - am 1 1.1420 121.55 74.705
## - qsec 1 1.2413 121.64 74.732
## - drat 1 1.8208 122.22 74.884
## - cyl 2 10.9314 131.33 75.184
## - vs 1 3.6299 124.03 75.354
## <none> 120.40 76.403
## - disp 1 9.9672 130.37 76.948
## - wt 1 25.5541 145.96 80.562
## - hp 1 25.6715 146.07 80.588
##
## Step: AIC=69.83
## mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear
##
## Df Sum of Sq RSS AIC
## - gear 2 5.0215 139.02 67.005
## - disp 1 0.9934 135.00 68.064
## - drat 1 1.1854 135.19 68.110
## - vs 1 3.6763 137.68 68.694
## - cyl 2 12.5642 146.57 68.696
## - qsec 1 5.2634 139.26 69.061
## <none> 134.00 69.828
## - am 1 11.9255 145.93 70.556
## - wt 1 19.7963 153.80 72.237
## - hp 1 22.7935 156.79 72.855
##
## Step: AIC=67
## mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am
##
## Df Sum of Sq RSS AIC
## - drat 1 0.9672 139.99 65.227
## - cyl 2 10.4247 149.45 65.319
## - disp 1 1.5483 140.57 65.359
## - vs 1 2.1829 141.21 65.503
## - qsec 1 3.6324 142.66 65.830
## <none> 139.02 67.005
## - am 1 16.5665 155.59 68.608
## - hp 1 18.1768 157.20 68.937
## - wt 1 31.1896 170.21 71.482
##
## Step: AIC=65.23
## mpg ~ cyl + disp + hp + wt + qsec + vs + am
##
## Df Sum of Sq RSS AIC
## - disp 1 1.2474 141.24 63.511
## - vs 1 2.3403 142.33 63.757
## - cyl 2 12.3267 152.32 63.927
## - qsec 1 3.1000 143.09 63.928
## <none> 139.99 65.227
## - hp 1 17.7382 157.73 67.044
## - am 1 19.4660 159.46 67.393
## - wt 1 30.7151 170.71 69.574
##
## Step: AIC=63.51
## mpg ~ cyl + hp + wt + qsec + vs + am
##
## Df Sum of Sq RSS AIC
## - qsec 1 2.442 143.68 62.059
## - vs 1 2.744 143.98 62.126
## - cyl 2 18.580 159.82 63.466
## <none> 141.24 63.511
## - hp 1 18.184 159.42 65.386
## - am 1 18.885 160.12 65.527
## - wt 1 39.645 180.88 69.428
##
## Step: AIC=62.06
## mpg ~ cyl + hp + wt + vs + am
##
## Df Sum of Sq RSS AIC
## - vs 1 7.346 151.03 61.655
## <none> 143.68 62.059
## - cyl 2 25.284 168.96 63.246
## - am 1 16.443 160.12 63.527
## - hp 1 36.344 180.02 67.275
## - wt 1 41.088 184.77 68.108
##
## Step: AIC=61.65
## mpg ~ cyl + hp + wt + am
##
## Df Sum of Sq RSS AIC
## <none> 151.03 61.655
## - am 1 9.752 160.78 61.657
## - cyl 2 29.265 180.29 63.323
## - hp 1 31.943 182.97 65.794
## - wt 1 46.173 197.20 68.191summary(stepFit)##
## Call:
## lm(formula = mpg ~ cyl + hp + wt + am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.9387 -1.2560 -0.4013 1.1253 5.0513
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.70832 2.60489 12.940 7.73e-13 ***
## cyl6 -3.03134 1.40728 -2.154 0.04068 *
## cyl8 -2.16368 2.28425 -0.947 0.35225
## hp -0.03211 0.01369 -2.345 0.02693 *
## wt -2.49683 0.88559 -2.819 0.00908 **
## am1 1.80921 1.39630 1.296 0.20646
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.41 on 26 degrees of freedom
## Multiple R-squared: 0.8659, Adjusted R-squared: 0.8401
## F-statistic: 33.57 on 5 and 26 DF, p-value: 1.506e-10summary(stepFit)$coeff ## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.70832390 2.60488618 12.940421 7.733392e-13
## cyl6 -3.03134449 1.40728351 -2.154040 4.068272e-02
## cyl8 -2.16367532 2.28425172 -0.947214 3.522509e-01
## hp -0.03210943 0.01369257 -2.345025 2.693461e-02
## wt -2.49682942 0.88558779 -2.819404 9.081408e-03
## am1 1.80921138 1.39630450 1.295714 2.064597e-01Residuals & Diagnostics
To verify whether the assumption of Gaussian residuals is violated, a graphical examination of the residual plots is often suggested. These plots are shown in the Appendix, figure 2. The plots of figure 2 conclude: a.The randomness of the Residuals vs. Fitted plot supports the assumption of independence b.The points of the Normal Q-Q plot following closely to the line conclude that the distribution of residuals is normal c.The Scale-Location plot random distribution confirms the constant variance assumption d.Since all points are within the 0.05 lines, the Residuals vs. Leverage concludes that there are no outliers
sum((abs(dfbetas(stepFit)))>1)## [1] 0Conclusion
There is a difference in MPG based on transmission type. A manual transmission will have a slight MPG boost. However, it seems that weight, horsepower, & number of cylinders are more statistically significant when determining MPG.
Appendix Figures
Figure 1: Boxplot of MPG vs. Transmission
boxplot(mpg ~ am,
xlab="Transmission Type (0 = Automatic, 1 = Manual)",
ylab="MPG",
main="MPG by Transmission Type")
Figure 2: Pair Graph of Motor Trend Car Road Tests
pairs(mtcars, panel=panel.smooth, main="Pair Graph of Motor Trend Car Road Tests")
Figure 3: Scatter Plot of MPG vs. Weight by Transmission
ggplot(mtcars, aes(x=wt, y=mpg, group=am, color=am, height=3, width=3)) + geom_point() +
scale_colour_discrete(labels=c("Automatic", "Manual")) +
xlab("weight") + ggtitle("Scatter Plot of MPG vs. Weight by Transmission")
Figure 4: Residual Plots
par(mfrow = c(2, 2))
plot(stepFit)