In this report, mtcars data set will be analyzed, to explore the relationship between a the variables and Miles Per Gallon(MPG). 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).
Regression models and exploratory data analyses are used to mainly explore how automatic (am = 0) and manual (am = 1) transmissions features affect the MPG feature. T-test shows that the performance difference between cars with automatic and manual transmission. And it is about 7 MPG more for cars with manual transmission than those with automatic transmission. Then, several linear regression models are fitted with lowest AIC and with lowest deviance values is selected. So, given that weight and 1/4 mile time are held constant.
from the help(mtcars) we get the data Description: 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).
and variables names data frame with 32 observations on 11 (numeric) variables:
- [, 1] mpg Miles/(US) gallon.
- [, 2] cyl Number of cylinders.
- [, 3] disp Displacement (cu.in.).
- [, 4] hp Gross horsepower.
- [, 5] drat Rear axle ratio.
- [, 6] wt Weight (1000 lbs).
- [, 7] qsec 1/4 mile time.
- [, 8] vs Engine (0 = V-shaped, 1 = straight).
- [, 9] am Transmission (0 = automatic, 1 = manual).
- [,10] gear Number of forward gears.
- [,11] carb Number of carburetors.
check categorical variables to factorize them.
lapply(mtcars, unique,axis = 1 )## $mpg
## [1] 21.0 22.8 21.4 18.7 18.1 14.3 24.4 19.2 17.8 16.4 17.3 15.2 10.4 14.7 32.4
## [16] 30.4 33.9 21.5 15.5 13.3 27.3 26.0 15.8 19.7 15.0
##
## $cyl
## [1] 6 4 8
##
## $disp
## [1] 160.0 108.0 258.0 360.0 225.0 146.7 140.8 167.6 275.8 472.0 460.0 440.0
## [13] 78.7 75.7 71.1 120.1 318.0 304.0 350.0 400.0 79.0 120.3 95.1 351.0
## [25] 145.0 301.0 121.0
##
## $hp
## [1] 110 93 175 105 245 62 95 123 180 205 215 230 66 52 65 97 150 91 113
## [20] 264 335 109
##
## $drat
## [1] 3.90 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.07 2.93 3.00 3.23 4.08 4.93 4.22
## [16] 3.70 3.73 4.43 3.77 3.62 3.54 4.11
##
## $wt
## [1] 2.620 2.875 2.320 3.215 3.440 3.460 3.570 3.190 3.150 4.070 3.730 3.780
## [13] 5.250 5.424 5.345 2.200 1.615 1.835 2.465 3.520 3.435 3.840 3.845 1.935
## [25] 2.140 1.513 3.170 2.770 2.780
##
## $qsec
## [1] 16.46 17.02 18.61 19.44 20.22 15.84 20.00 22.90 18.30 18.90 17.40 17.60
## [13] 18.00 17.98 17.82 17.42 19.47 18.52 19.90 20.01 16.87 17.30 15.41 17.05
## [25] 16.70 16.90 14.50 15.50 14.60 18.60
##
## $vs
## [1] 0 1
##
## $am
## [1] 1 0
##
## $gear
## [1] 4 3 5
##
## $carb
## [1] 4 1 2 3 6 8
Form unique values we can know which variables are the categorical variables.
mtcars$cyl <-as.factor( mtcars$cyl)
mtcars$vs <-as.factor( mtcars$vs)
mtcars$carb <-as.factor( mtcars$carb)
mtcars$am <-as.factor( mtcars$am)
mtcars$gear <-as.factor( mtcars$gear)check the summary for changes
summary(mtcars)## mpg cyl disp hp drat
## Min. :10.40 4:11 Min. : 71.1 Min. : 52.0 Min. :2.760
## 1st Qu.:15.43 6: 7 1st Qu.:120.8 1st Qu.: 96.5 1st Qu.:3.080
## Median :19.20 8:14 Median :196.3 Median :123.0 Median :3.695
## Mean :20.09 Mean :230.7 Mean :146.7 Mean :3.597
## 3rd Qu.:22.80 3rd Qu.:326.0 3rd Qu.:180.0 3rd Qu.:3.920
## Max. :33.90 Max. :472.0 Max. :335.0 Max. :4.930
## wt qsec vs am gear carb
## Min. :1.513 Min. :14.50 0:18 0:19 3:15 1: 7
## 1st Qu.:2.581 1st Qu.:16.89 1:14 1:13 4:12 2:10
## Median :3.325 Median :17.71 5: 5 3: 3
## Mean :3.217 Mean :17.85 4:10
## 3rd Qu.:3.610 3rd Qu.:18.90 6: 1
## Max. :5.424 Max. :22.90 8: 1
boxplot(mpg ~ am,data = mtcars)
title(main ="miles per gallon for each transmission type")
legend("topleft", inset=.02, title="Transmission Type",c("0 automatic","1 manual"), fill=topo.colors(0), horiz=TRUE, cex=0.8)
The plot shows the mpg mean of the manual transmission is higher then automatic transmission ,and can use two tail t test to verify it.
t.test(mtcars$mpg ~ mtcars$am, alternative = "two.sided", var.equal = FALSE)##
## Welch Two Sample t-test
##
## data: mtcars$mpg by mtcars$am
## t = -3.7671, df = 18.332, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -11.280194 -3.209684
## sample estimates:
## mean in group 0 mean in group 1
## 17.14737 24.39231
The mean of the manual transmission is more then The mean of automatic transmission with approximate 7 miles per gallons but if we take that finding without any further investigations will be too hasty and there another variables in the data has effect on miles per gallons so fitting models will help explore the relation between variable
using the AIC to chose best model fit :
model1 <- lm(mpg ~ am+ disp+ wt+ cyl+ drat +gear +qsec +vs +carb + hp, data = mtcars)
modl1 <- step(model1 ,trace = FALSE , k = log(nrow(mtcars) ))using trace = false to hide step output as it very long and store the lowest AIC model in modl1.
using model$anova to preview model selection process and which variables has effect on AIC we excluded.
modl1$anova## Step Df Deviance Resid. Df Resid. Dev AIC
## 1 NA NA 15 120.4027 101.32090
## 2 - carb 5 13.5988573 20 134.0015 87.41652
## 3 - gear 2 5.0215145 22 139.0230 81.66228
## 4 - cyl 2 10.4247254 24 149.4478 77.04464
## 5 - vs 1 0.6454862 25 150.0933 73.71682
## 6 - drat 1 3.3445512 26 153.4378 70.95632
## 7 - disp 1 6.6286537 27 160.0665 68.84398
## 8 - hp 1 9.2194693 28 169.2859 67.17025
The - in each row mean eliminating associated variable in second row -carb means eliminating variable carb from the model this lower the AIC to 87.4 so we the best AIC 67.1 after eliminate hp.
The deviance also increase and that bad sign, so will fit two model one with lowest deviance and one with lowest AIC and use coefficients and diagnostics plots to reach to best fit.
Examining the model model1 with lowest deviance:
variables and coefficients of the model.
model1 <- lm(mpg ~ ., data = mtcars)
summary(model1)$call; summary(model1)$coef## lm(formula = mpg ~ ., data = mtcars)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 23.87913244 20.06582026 1.19004018 0.25252548
## cyl6 -2.64869528 3.04089041 -0.87102622 0.39746642
## cyl8 -0.33616298 7.15953951 -0.04695316 0.96317000
## disp 0.03554632 0.03189920 1.11433290 0.28267339
## hp -0.07050683 0.03942556 -1.78835344 0.09393155
## drat 1.18283018 2.48348458 0.47627845 0.64073922
## wt -4.52977584 2.53874584 -1.78425732 0.09461859
## qsec 0.36784482 0.93539569 0.39325050 0.69966720
## vs1 1.93085054 2.87125777 0.67247551 0.51150791
## am1 1.21211570 3.21354514 0.37718957 0.71131573
## gear4 1.11435494 3.79951726 0.29328856 0.77332027
## gear5 2.52839599 3.73635801 0.67670068 0.50889747
## carb2 -0.97935432 2.31797446 -0.42250436 0.67865093
## carb3 2.99963875 4.29354611 0.69863900 0.49546781
## carb4 1.09142288 4.44961992 0.24528452 0.80956031
## carb6 4.47756921 6.38406242 0.70136677 0.49381268
## carb8 7.25041126 8.36056638 0.86721532 0.39948495
deviance(model1)## [1] 147.4944
interpret some variable when holding the other variables as constants:
- the Intercept is am0 ,cyl4,vs0,gear3* coefficient with value
23.88means the associated coefficient has23.88Miles per gallon (MPG). - am1 coefficient with value
1.21means the manual Transmission has25.091Miles per gallon (MPG). - wt for each 1000 lbs increase in vehicle weight the (MPG) decreases with
4.52. - qsec for 1/4 mile time increases by unit time (second) the (MPG) increases with
1.2 - carb 2,3,4,6,8 has huge impact on MPG first if car with
carb2will lower theMPGbut with each increase incarbincrease MPG rapidly.
Examining the model modl1 with lowest AIC:
variables and coefficients of the model.
summary(modl1)$call; summary(modl1)$coef## lm(formula = mpg ~ am + wt + qsec, data = mtcars)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.617781 6.9595930 1.381946 1.779152e-01
## am1 2.935837 1.4109045 2.080819 4.671551e-02
## wt -3.916504 0.7112016 -5.506882 6.952711e-06
## qsec 1.225886 0.2886696 4.246676 2.161737e-04
deviance(modl1)## [1] 169.2859
the model with 3 variables am + wt + qsec , interpret each variable when holding the other variables as constants:
- the Intercept is am0 coefficient with value
9.6means the automatic Transmission has mean9.6Miles per gallon (MPG). - am1 coefficient with value
2.9means the manual Transmission has mean12.55Miles per gallon (MPG). - wt for each 1000 lbs increase in vehicle weight the (MPG) decreases with
3.9. - qsec for 1/4 mile time increases by unit time (second) the (MPG) increases with
1.2
Since regression based on certain assumptions. Regression diagnostics used to evaluate the model assumptions and investigate whether holds or not.
Residuals vs fitted values plot: to check linearity assumption
plot(modl1,which = 1)
The linearity assumption dose not hold as the red line is not approximate to the dashed line in the graph, means the mean of residues is varies.
The normal QQ and standardized shape of residuals to check the residuals normality.
par(mfrow = c(1, 2))
plot(modl1,which = 2)
resdulpdf<-density(modl1$residuals /summary.lm(modl1)$sigma)
plot(resdulpdf,main='standardized Residual pdf shape',xlab='Residuals')
abline(v = 0)
the residuals is approximate to normality but the curve shifted as the peek between zero and one.
The Scale-location plot to check the Homoscedasticity.
par(mfrow = c(1, 1))
plot(modl1,which = 3)
variance around the line The spread of standardized residuals around the red line doesn’t vary with respect to the fitted values,Homoscedasticity holds.
we can also verify our finding by using bptest from lmtest library,the bptest function test the hypothesis of homoskedasticity.
bptest(modl1)##
## studentized Breusch-Pagan test
##
## data: modl1
## BP = 6.1871, df = 3, p-value = 0.1029
the p-value is bigger then .05 so the assumption of homoskedasticity is holding as we expected.
The cook's distance and residuals vs leverage to spot the outliers
par(mfrow = c(1, 2))
plot(modl1,which = c(4,5))
from plots the data dose not contain any outliers.
so diagnostic conclusion for this model we need model contains interactions between variables or polynomial terms but first lets do same diagnostics for the model with lowest deviance.
Residuals vs fitted values plot: to check linearity assumption:
plot(model1,which = 1)
The linearity assumption holds as the red line is approximate to the dashed line in the graph, but we can see some point of residual far away from the others.
The normal QQ and standardized shape of residuals to check the residuals normality:
par(mfrow = c(1, 2))
plot(model1,which = 2)## Warning: not plotting observations with leverage one:
## 30, 31
resdulpdf<-density(model1$residuals /summary.lm(model1)$sigma)
plot(resdulpdf,main='standardized Residual pdf shape',xlab='Residuals')
abline(v = 0)
The residuals is approximate to normality but the curve between 2 standard deviations that violate the standardized residuals as approximately 99.9% between 3 standard deviations.
The Scale-location plot to check the Homoscedasticity.
par(mfrow = c(1, 1))
plot(model1,which = 3)## Warning: not plotting observations with leverage one:
## 30, 31
variance is constant around the line,Homoscedasticity is holding.
bptest(model1)##
## studentized Breusch-Pagan test
##
## data: model1
## BP = 18.294, df = 16, p-value = 0.307
the p-value is less then .05 so the assumption of homoskedasticity is holding as we expected.
The cook's distance and residuals vs leverage to spot the outliers but the model assumptions are violated so the outliers analysis dose not matter as the model is poorly fit.
so diagnostic conclusion for this model the model has low deviance but the Regression model assumptions violated.
based from the finding from the two models and mean test the manual transmissions has higher miles per gallon so the manual transmissions and is better for saving fuel.