Regression Model: Final Project
Hazim Hanif
12/19/2016
Regression Models: Final Project
Instructions
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?
Executive Summary: An analysis of the automobile industry
This study is focusing on relationship between the Miles Per Gallon (MPG) and Transmission type (automatic or manual) from mtcars dataset. This dataset is extracted from the 1974 Motor Trend US magazine. With the analysis of the both stated metrics, the performance of the cars are evaluated thus distinguishing the MPG between automatic and manual transmission cars.
Exploratory Data Analysis
- 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).
- The data consist of 32 oberservations with 11 variables
data("mtcars")
head(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 1summary(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. :8.000mtcars$cyl <- as.factor(mtcars$cyl)
mtcars$vs <- as.factor(mtcars$vs)
mtcars$gear <- factor(mtcars$gear)
mtcars$carb <- factor(mtcars$carb)
mtcars$am <- factor(mtcars$am, labels = c("Automatic","Manual"))- This observation is based on the figure plotted in Appendix 1 and Appendix 2.
- There is a higher throughput of MPG in manual transmission as compared to automatic transmission.
- Most of the readings for automatic transmission are around 15-20 MPGs.
In pair graph, the cyl, disp, hp & wt variables seems like having a correlation with MPGs.
- The mean MPG for automatic transmission is 17.1473684 Miles/gallon
- The mean MPG for manual transmission is 24.3923077 Miles/gallon
- The highest MPG for automatic transmission is 24.4 Miles/gallon
The highest MPG for manual transmission is 33.9 Miles/gallon
Statistical inference
Testing the null hypothesis that the MPG of automatic and manual transmisison came from the same population.
result <- t.test(mpg ~ am,data = mtcars)
result$p.value## [1] 0.001373638result$estimate## mean in group Automatic mean in group Manual
## 17.14737 24.39231From this, we get the value of 0.0013736 for p-value thus reject the null hypothesis. It is because the p-value shows that the MPGs of automatic and manual trasmission are indeeed came from different population.
Model Analysis
- Fit the full model by taking into account all the variables
model_full <- lm(mpg ~ .,data = mtcars)
summary(model_full)##
## 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
## amManual 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: 0.000124- Residual standard error: 2.65 on 21 degress of freedeom.
- Adjusted R-square values: 0.8066
- The model can explain about 80.6% of the variance of the MPG variable
- Fit the base model by take into account the transmission type variable
model_base <- lm(mpg ~ factor(am), data = mtcars)
summary(model_base)##
## 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)Manual 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: 0.000285- Residual standard error: 4.902 on 30 degress of freedeom.
- Adjusted R-square values: 0.3385
- The model can explain about 33.8% of the variance of the MPG variable
- Fit the extended model by take into account the variable that is likely to have a correlation with MPG(from the pair graphs). The variables are cyl, hp, wt and am.
model_extended <- lm(mpg ~ am + hp + wt + cyl, data = mtcars)
summary(model_extended)##
## Call:
## lm(formula = mpg ~ am + hp + wt + cyl, 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 ***
## amManual 1.80921 1.39630 1.296 0.20646
## hp -0.03211 0.01369 -2.345 0.02693 *
## wt -2.49683 0.88559 -2.819 0.00908 **
## cyl6 -3.03134 1.40728 -2.154 0.04068 *
## cyl8 -2.16368 2.28425 -0.947 0.35225
## ---
## 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-10- Residual standard error: 2.509 on 27 degress of freedeom.
- Adjusted R-square values: 0.8267
- The model can explain about 82.7% of the variance of the MPG variable
With all 3 different regression models fitted, we choose the third model which is the “extended model” based on its highest adjusted r-square values.
Now is comparing between all the three models.
anova(model_full,model_base,model_extended)## Analysis of Variance Table
##
## Model 1: mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb
## Model 2: mpg ~ factor(am)
## Model 3: mpg ~ am + hp + wt + cyl
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 15 120.40
## 2 30 720.90 -15 -600.49 4.9874 0.001759 **
## 3 26 151.03 4 569.87 17.7489 1.476e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(model_extended)$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.70832390 2.60488618 12.940421 7.733392e-13
## amManual 1.80921138 1.39630450 1.295714 2.064597e-01
## hp -0.03210943 0.01369257 -2.345025 2.693461e-02
## wt -2.49682942 0.88558779 -2.819404 9.081408e-03
## cyl6 -3.03134449 1.40728351 -2.154040 4.068272e-02
## cyl8 -2.16367532 2.28425172 -0.947214 3.522509e-01P-value in the Model_extended(cyl+hp+wt+am) is lower than base model. Thus this reject the null hypothesis that confounder variables (cyl+hp+wt) don’t contribute to the accuracy of the model.
Residuals Analysis
Now, plotting the residual plots for the selected model: Extended model (cyl+hp+wt+am)
par(mfrow = c(2, 2))
plot(model_extended)
Observation:
- Residuals vs Fitted: Seems to be randomly scattered and there is no significant pattern identified.
- The Normal Q−Q: Most of the plots are near to the line thus saying that redsiduals are distributed normally.
- Scale−Location: All the points are randomly distributed and scattered thus inidicates a constant variance
- Residuals vs Leverage: The are some insteresting points. Possible some leverage points.
Conclusion
Based the observations from selected model:
- Manual transmission cars is more efficient than automatic transmission.
- MPG decrease slighly with increase of hp
Appendixes
Appendix 1: Boxplot of Miles/gallon(MPG) by Transmission type
boxplot(mpg~am, data=mtcars, main="Boxplot of Miles/gallon(MPG) by Transmission type", ylab="Miles/gallon", xlab="Transmission type")
Appendix 2: Point plot of Miles/gallon(MPG) vs Transmission type
ggplot(mtcars, aes(x = factor(am), y =mpg ))+geom_point()
Appendix 3: Pair graphs
pairs(mtcars, panel=panel.smooth, main="Pair Graph of 1974 Motor Trend")