Questions:

- Is an automatic or manual transmission better for MPG

- Quantify the MPG difference between automatic and manual transmissions

Begin by loading and reviewing data:

library(ggplot2)
library(dplyr)
## Warning: package 'dplyr' was built under R version 3.2.2
## 
## Attaching package: 'dplyr'
## 
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## 
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(gridExtra)
## Warning: package 'gridExtra' was built under R version 3.2.2
data(mtcars)

cols<- c('mpg','am')
mt.data <- mtcars[cols]

str(mt.data)
## 'data.frame':    32 obs. of  2 variables:
##  $ mpg: num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
##  $ am : num  1 1 1 0 0 0 0 0 0 0 ...

Summarize auto vs man:

by(mt.data$mpg, INDICES = list(mt.data$am), summary)
## : 0
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   10.40   14.95   17.30   17.15   19.20   24.40 
## -------------------------------------------------------- 
## : 1
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   15.00   21.00   22.80   24.39   30.40   33.90

Overall manual is better than automatic in terms of mpg.Figure 1 in the appendix shows a boxplot of the manual vs automatic transmission by mpg. Figure 2 in the appendix shows a more detailed of mgp vs wt by transmission (am)

Quantify mpg difference:

mt.autompg <- mt.data[mt.data$am == 0,]
mt.manmpg <- mt.data[mt.data$am == 1,]
t.test(mt.autompg$mpg, mt.manmpg$mpg)
## 
##  Welch Two Sample t-test
## 
## data:  mt.autompg$mpg and mt.manmpg$mpg
## 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 of x mean of y 
##  17.14737  24.39231

95% CI = [-11.280194, -3.209684] (zero NOT in interval) p-val = 0.001374 according to the test. There is a significant difference between the automatic vs manual cars in terms of mpg, but this is assuming that the cars are the same (ie: same weight, cylinder size, horsepower)

Evaluate other variables using correlation analysis:

sort(cor(mtcars)[1,])
##         wt        cyl       disp         hp       carb       qsec 
## -0.8676594 -0.8521620 -0.8475514 -0.7761684 -0.5509251  0.4186840 
##       gear         am         vs       drat        mpg 
##  0.4802848  0.5998324  0.6640389  0.6811719  1.0000000

Evaluate transmission variable w/ mpg:

mt.fit1 <- lm(mpg~am, data = mtcars)
summary(mt.fit1)
## 
## Call:
## lm(formula = mpg ~ 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 ***
## am             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

On average, manual transmission cars have 7.245 MPGs more than automatic transmission. We see that the r squared value is 0.3598. This means that our model only explains 35.98% of the variance.

I need to understand the impact of transmission with other factors to quantify the mpg difference between automatic and manual transmission.

Evaluate stepwise analysis for most relevant variables:

mt.stepwise = step(lm(data = mtcars, mpg ~ .),direction = "both")
## Start:  AIC=70.9
## mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb
## 
##        Df Sum of Sq    RSS    AIC
## - cyl   1    0.0799 147.57 68.915
## - vs    1    0.1601 147.66 68.932
## - carb  1    0.4067 147.90 68.986
## - gear  1    1.3531 148.85 69.190
## - drat  1    1.6270 149.12 69.249
## - disp  1    3.9167 151.41 69.736
## - hp    1    6.8399 154.33 70.348
## - qsec  1    8.8641 156.36 70.765
## <none>              147.49 70.898
## - am    1   10.5467 158.04 71.108
## - wt    1   27.0144 174.51 74.280
## 
## Step:  AIC=68.92
## mpg ~ disp + hp + drat + wt + qsec + vs + am + gear + carb
## 
##        Df Sum of Sq    RSS    AIC
## - vs    1    0.2685 147.84 66.973
## - carb  1    0.5201 148.09 67.028
## - gear  1    1.8211 149.40 67.308
## - drat  1    1.9826 149.56 67.342
## - disp  1    3.9009 151.47 67.750
## - hp    1    7.3632 154.94 68.473
## <none>              147.57 68.915
## - qsec  1   10.0933 157.67 69.032
## - am    1   11.8359 159.41 69.384
## + cyl   1    0.0799 147.49 70.898
## - wt    1   27.0280 174.60 72.297
## 
## Step:  AIC=66.97
## mpg ~ disp + hp + drat + wt + qsec + am + gear + carb
## 
##        Df Sum of Sq    RSS    AIC
## - carb  1    0.6855 148.53 65.121
## - gear  1    2.1437 149.99 65.434
## - drat  1    2.2139 150.06 65.449
## - disp  1    3.6467 151.49 65.753
## - hp    1    7.1060 154.95 66.475
## <none>              147.84 66.973
## - am    1   11.5694 159.41 67.384
## - qsec  1   15.6830 163.53 68.200
## + vs    1    0.2685 147.57 68.915
## + cyl   1    0.1883 147.66 68.932
## - wt    1   27.3799 175.22 70.410
## 
## Step:  AIC=65.12
## mpg ~ disp + hp + drat + wt + qsec + am + gear
## 
##        Df Sum of Sq    RSS    AIC
## - gear  1     1.565 150.09 63.457
## - drat  1     1.932 150.46 63.535
## <none>              148.53 65.121
## - disp  1    10.110 158.64 65.229
## - am    1    12.323 160.85 65.672
## - hp    1    14.826 163.35 66.166
## + carb  1     0.685 147.84 66.973
## + vs    1     0.434 148.09 67.028
## + cyl   1     0.414 148.11 67.032
## - qsec  1    26.408 174.94 68.358
## - wt    1    69.127 217.66 75.350
## 
## Step:  AIC=63.46
## mpg ~ disp + hp + drat + wt + qsec + am
## 
##        Df Sum of Sq    RSS    AIC
## - drat  1     3.345 153.44 62.162
## - disp  1     8.545 158.64 63.229
## <none>              150.09 63.457
## - hp    1    13.285 163.38 64.171
## + gear  1     1.565 148.53 65.121
## + cyl   1     1.003 149.09 65.242
## + vs    1     0.645 149.45 65.319
## + carb  1     0.107 149.99 65.434
## - am    1    20.036 170.13 65.466
## - qsec  1    25.574 175.67 66.491
## - wt    1    67.572 217.66 73.351
## 
## Step:  AIC=62.16
## mpg ~ disp + hp + wt + qsec + am
## 
##        Df Sum of Sq    RSS    AIC
## - disp  1     6.629 160.07 61.515
## <none>              153.44 62.162
## - hp    1    12.572 166.01 62.682
## + drat  1     3.345 150.09 63.457
## + gear  1     2.977 150.46 63.535
## + cyl   1     2.447 150.99 63.648
## + vs    1     1.121 152.32 63.927
## + carb  1     0.011 153.43 64.160
## - qsec  1    26.470 179.91 65.255
## - am    1    32.198 185.63 66.258
## - wt    1    69.043 222.48 72.051
## 
## Step:  AIC=61.52
## mpg ~ hp + wt + qsec + am
## 
##        Df Sum of Sq    RSS    AIC
## - hp    1     9.219 169.29 61.307
## <none>              160.07 61.515
## + disp  1     6.629 153.44 62.162
## + carb  1     3.227 156.84 62.864
## + drat  1     1.428 158.64 63.229
## - qsec  1    20.225 180.29 63.323
## + cyl   1     0.249 159.82 63.465
## + vs    1     0.249 159.82 63.466
## + gear  1     0.171 159.90 63.481
## - am    1    25.993 186.06 64.331
## - wt    1    78.494 238.56 72.284
## 
## Step:  AIC=61.31
## mpg ~ wt + qsec + am
## 
##        Df Sum of Sq    RSS    AIC
## <none>              169.29 61.307
## + hp    1     9.219 160.07 61.515
## + carb  1     8.036 161.25 61.751
## + disp  1     3.276 166.01 62.682
## + cyl   1     1.501 167.78 63.022
## + drat  1     1.400 167.89 63.042
## + gear  1     0.123 169.16 63.284
## + vs    1     0.000 169.29 63.307
## - am    1    26.178 195.46 63.908
## - qsec  1   109.034 278.32 75.217
## - wt    1   183.347 352.63 82.790
summary(mt.stepwise)
## 
## 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 ***
## am            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-11

This shows that in addition to transmission, weight of the vehicle and accelaration/speed have the highest relation to explaining the variation in mpg. The adjusted r squared value is 85% which means that the model explains 85% of the variation in mpg. This is a reasonably robust and predictive model.

Best fit model using variables from stepwise (r squared = 85%) vs fit model:

mt.bestfit <- lm(mpg~am + wt + qsec, data = mtcars)
anova(mt.fit1, mt.bestfit)
## Analysis of Variance Table
## 
## Model 1: mpg ~ am
## Model 2: mpg ~ am + wt + qsec
##   Res.Df    RSS Df Sum of Sq      F   Pr(>F)    
## 1     30 720.90                                 
## 2     28 169.29  2    551.61 45.618 1.55e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Model shows 85% of the variability in mpg. p-val = 1.55e-09, we reject the null hypothesis and claim that the stepwise model is significantly different from the simple/reduced linear regression model.

Figure 3 in the appendix shows the checking of residuals for signs of non-normality & residuals vs fitted values plot for signs of heteroskedasticity. Residual diagnostics show normality & no evidence of heteroskedasticity.

###Results:

summary(mt.bestfit)
## 
## Call:
## lm(formula = mpg ~ am + wt + qsec, 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    
## am            2.9358     1.4109   2.081 0.046716 *  
## wt           -3.9165     0.7112  -5.507 6.95e-06 ***
## qsec          1.2259     0.2887   4.247 0.000216 ***
## ---
## 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-11

The model explains 83.36% of the variance in mpg. Results of wt(-3.9165) and qsec(1.2259) did confound the relationship between am and mpg. In conclusion, manual transmission cars have 2.94 mpg more than automatic transmission cars, on average. But the effect is not as high as the original simple/reduced model since weight and qsec were not adjusted for.

## Appendix:

Figure 1:

par(mar=c(5.5, 5.5,5.5,5.5))
boxplot(mpg~am, data = mt.data,
        col = c("blue", "pink"),
        xlab = "Transmission Type",
        ylab = "Miles per Gallon",
        main = "MPG by Transmission Type")

Figure 2:

print(qplot(x=wt, y=mpg, 
            facets=.~am, 
            data=mtcars,
            main="mgp vs. wt by transmission"))

Figure 3:

par(mfrow = c(2,2))
plot(mt.bestfit)

Correlations:

pairs(mpg ~ ., data = mtcars)

corr <- select(mtcars, mpg,cyl,qsec,am)
pairs(corr)

plot1 <- qplot(am, mpg, data= mtcars, xlab = c("automatic","Manual"))
plot2 <- qplot(wt, mpg, data = mtcars, geom = c("point","smooth"))
plot3 <- qplot(qsec, mpg, data = mtcars)
grid.arrange(plot1, plot2, plot3, ncol=2)
## geom_smooth: method="auto" and size of largest group is <1000, so using loess. Use 'method = x' to change the smoothing method.