Executive Summary
Data Description
Exploratory Data Analysis
Multivariable Regression Model
Conclusions
Appendix
- A.1 - Data Processing
- A.2 - Multivariable Regression Model and inferences.

Executive Summary

Motor Trend, an automobile trend magazine is interested in exploring the relationship between a set of variables and miles per gallon (MPG) outcome. In this project, we will analyze the mtcars dataset from the 1974 Motor Trend US magazine to answer the following questions:

Is an automatic or manual transmission better for miles per gallon (MPG)?
How different is the MPG between automatic and manual transmissions?

Data Description

We analyze the mtcars data set through Regression Modelling and exploratory analysis to show how automatic (am = 0) and manual (am = 1) transmissions features affect the MPG feature.

The data was extracted from the 1974 Motor Trend US magazine, which comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973-74 models).

The data set consists of a data frame with 32 observations (nrow) and 11 variables (ncol).

mpg: Miles per US gallon
cyl: Number of cylinders
disp: Displacement (cubic inches)
hp: Gross horsepower
drat: Rear axle ratio
wt: Weight (1000 lb)
qsec: 1/4 mile time
vs: Engine (0 = V-shaped, 1 = straight)
am: Transmission (0 = automatic, 1 = manual)
gear: Number of forward gears
carb: Number of carburetors

Exploratory Data Analysis

First we must determine whether or not there really is a difference between automatic and manual transmissions in terms of mpg. For this purpose, a review of the data is made (see Appendix A.1)

In the Figure1 in Appendix A.1 the boxplot with the information of the mileage vs the transmission is shown. In this diagram you can clearly see that there is a difference in the mileage (mpg) depending on the type of transmission (automatic or manual). Where the median mileage (mpg) for an automatic transmission is 17.3 and for a manual transmission is 22.8.

Multivariable Regression Model

For the analysis we will consider 3 models (see Appendix A.2).

fit_base: Regression model between mpg andam.
fit_best: Regression model between mpg the variables determined by thestepAIC function.
fit_full: Regression model between mpg and all the data set variables.

Once the fit_best model has been obtained (mpg ~ cyl + hp + wt + am) we will use theanova function to determine which of these three models best fits the data with the least possible complexity.

According to the analysis of variance (ver Apendix A.2) the p-value is significant, hence we reject the null hypothesis that the variables cyl, hp and wt do not contribute to the accuracy of the model. We double-check the residuals for non-normality (Apendix A.2 - Figure 2) and can see they are all normally distributed and homoskedastic.

d<-data.frame()
d[1,1] <- round(summary(fit_base)$r.squared,4); d[1,2] <- round(summary(fit_best)$r.squared,4)
colnames(d) <- c("fit_base", "fit_best"); rownames(d) <- "R^2^"
knitr::kable(d, align = "c", pading = 3, caption = "Comparacion R^2^ entre modelos")

Comparacion R² entre modelos
	fit_base	fit_best
R²	0.3598	0.8497

The model fit_best explains 84.96% of the variance and as a result, cyl, hp, wt did affect the correlation between mpg and am. Thus, we can say the difference between automatic and manual transmissions is 1.81 MPG.

Conclusions

Based on the observations of our best fit model, we can conclude:

Considering the summaries of the models, it is observed that the model fit_best has a significant difference in the parameters with fit_base. For fit_base, cars with manual transmission get more miles per gallon compared to automobiles with automatic transmission. (7.25 mpg set by hp, cyl, and wt), in the case of fit_best is 1.81 in the same conditions.

The mpg will decrease by 2.5 (adjusted by hp, cyl and am) for every 1000 pounds of weight. mpg decreases negligibly with the increase of hp.

If the number of cylinders, cil increases from 4 to 6 and 8, mpg will decrease by a factor of 3 and 2.2 respectively (set by hp, wt and am).

Appendix

A.1 - Data Processing

First we load the data and the libraries.

library(ggplot2); library(dplyr); library(knitr); library(MASS); data(mtcars)
data <- mtcars

Then we start with the analysis, review the data taking a sample and see how it is structured (table 2, 3).

knitr::kable(head(data,5), align = "c", caption = "Sample of data set mtcars");

Sample of data set 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

a<-list();for (i in 1:dim(data)[2]){a[i]<-class(data[,i])}
knitr::kable(data.frame(t(a)), align = "c", pading = 3, 
             caption = "Class of the data set mtcars")

Class of the data set mtcars
X1	X2	X3	X4	X5	X6	X7	X8	X9	X10	X11
numeric	numeric	numeric	numeric	numeric	numeric	numeric	numeric	numeric	numeric	numeric

Observing the data, let’s adjust the variables cyl, hp, am, gear and carb which are numeric but should be factor type.

mtcars$cyl <- factor(mtcars$cyl)
mtcars$am <- factor(mtcars$am,labels=c("Automatic","Manual"))
mtcars$vs <- factor(mtcars$vs,labels=c("V-shaped","straight"))
mtcars$gear <- factor(mtcars$gear); mtcars$carb <- factor(mtcars$carb)

With this we can compare how the type of transmission (automatic - manual) affects the mileage (mpg).

p <- ggplot(mtcars, aes(x=factor(am,labels=c("Automatic","Manual")), y=mpg, 
      fill = factor(am,labels=c("Automatic","Manual")))) + geom_boxplot() +
      geom_dotplot(binaxis='y', stackdir='center', dotsize=1, fill ="black") +
      scale_fill_discrete(name = "Transmission") + xlab("Transmission") + 
      ylab("Miles per US gallon") + labs(caption = 
      "Figure 1: Miles per US gallon v/s Transmission (Automatic/Manual)");p

A.2 - Multivariable Regression Model and inferences.

Next, 3 linear models are generated:

fit_base <- lm(mpg ~ am, data = mtcars)
fit_full <- lm(mpg ~ ., data = mtcars)
fit_best <- stepAIC(fit_full, direction = "both")

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
+ carb  5   13.5989 120.40 76.403

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
+ gear  2    5.0215 134.00 69.828
- wt    1   31.1896 170.21 71.482
+ carb  5   14.6475 124.38 73.442

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
+ drat  1    0.9672 139.02 67.005
- hp    1   17.7382 157.73 67.044
- am    1   19.4660 159.46 67.393
+ gear  2    4.8033 135.19 68.110
- wt    1   30.7151 170.71 69.574
+ carb  5   13.0509 126.94 72.095

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
+ disp  1     1.247 139.99 65.227
+ drat  1     0.666 140.57 65.359
- hp    1    18.184 159.42 65.386
- am    1    18.885 160.12 65.527
+ gear  2     4.684 136.55 66.431
- wt    1    39.645 180.88 69.428
+ carb  5     2.331 138.91 72.978

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
+ qsec  1     2.442 141.24 63.511
- am    1    16.443 160.12 63.527
+ disp  1     0.589 143.09 63.928
+ drat  1     0.330 143.35 63.986
+ gear  2     3.437 140.24 65.284
- hp    1    36.344 180.02 67.275
- wt    1    41.088 184.77 68.108
+ carb  5     3.480 140.20 71.275

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
+ vs    1     7.346 143.68 62.059
+ qsec  1     7.044 143.98 62.126
- cyl   2    29.265 180.29 63.323
+ disp  1     0.617 150.41 63.524
+ drat  1     0.220 150.81 63.608
+ gear  2     1.361 149.66 65.365
- hp    1    31.943 182.97 65.794
- wt    1    46.173 197.20 68.191
+ carb  5     5.633 145.39 70.438

Analysis of variance of the models.

anova(fit_base, fit_best)

Res.Df	RSS	Df	Sum of Sq	F	Pr(>F)
30	720.8966	NA	NA	NA	NA
26	151.0256	4	569.871	24.52671	0

Summary fit_base and fit_best.

summary(fit_base)


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 ***
amManual       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

summary(fit_best)


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 ** 
amManual     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-10

Check residuals

par(mfrow=c(2,2), oma=c(2,0,0,0))
plot(fit_best)
mtext(side=1,line=0,cex=1, outer=T,"Figure 2: Check residuals")

Regression Models Course Project

crsd

December 08, 2018