This is Regression Models Project on the Coursera courses, part of the Data Science specialization. In this project we will perform data analysis in mtcars dataset. We will run numbers of analysis in this project but specifically relationship between gas expenses miles per gallon (MPG) and many variables in this dataset.

Types of variables in the dataset: mpg = Miles/(US) gallon; cyl = Number of cylinders; wt = Weight (lb/1000); vs = V/S; am = Transmission (0 = automatic, 1 = manual);gear = Number of forward gears; carb = Number of carburetors

Dataset- Motor Trend Car Toad Test

library(dplyr);library(ggplot2); library(grid); library(gridExtra); library(corrplot)
## 
## Attaching package: 'dplyr'
## 
## The following object is masked from 'package:stats':
## 
##     filter
## 
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
## Warning: package 'ggplot2' was built under R version 3.1.3
data(mtcars)

Creation of correlation model includes all variables in the initial model includes all variables as predictors of mpg. Creation of correlation on model includes all variables as predictors of mpg. In base our results in all cells for regression models are compatible in linear model. This model will also need to be considered when fitting a linear model later on.

``` Creation of the model that reports fuel consumption on the basis of observed variables in the dataset mtcars.

l1<-qplot(am, mpg, data = mtcars)+ stat_smooth(method="lm", se=FALSE)
coef(lm(mpg ~ am, data = mtcars))
## (Intercept)          am 
##   17.147368    7.244939
l2<-qplot(wt, mpg, data = mtcars)+ geom_abline(intercept = 37, slope = -5)
coef(lm(mpg ~ wt, data = mtcars))               
## (Intercept)          wt 
##   37.285126   -5.344472
l3<-qplot(cyl, mpg, data = mtcars)+ geom_abline(intercept = 37.8, slope = -2.8)
coef(lm(mpg ~ cyl, data = mtcars))
## (Intercept)         cyl 
##    37.88458    -2.87579
l4<-qplot(vs, mpg, data = mtcars)+ geom_abline(intercept = 16.6, slope = 7.9)
coef(lm(mpg ~ vs, data = mtcars))
## (Intercept)          vs 
##   16.616667    7.940476
l5<-qplot(gear, mpg, data = mtcars)+ geom_abline(intercept = 5.6 , slope = 3.9)
coef(lm(mpg ~ gear, data = mtcars))
## (Intercept)        gear 
##    5.623333    3.923333
l6<-qplot(carb, mpg, data = mtcars)+ geom_abline(intercept = 25.8 , slope = -2)
coef(lm(mpg ~ carb, data = mtcars))
## (Intercept)        carb 
##   25.872334   -2.055719
grid.arrange(l1, l2, l3, l4, l5, l6, ncol=3, main ="mtcars")

Multivariate Model Creation of the models that reports fuel consumption on the basis of observed variables in the dataset mtcars. l1 - Value of MPG and type of transmission have linear relationship and from the model, we can infer that transmission mode has an impact on MPG. Model suggests that average MPG for automatic transmission is 17 and average MPG for manual transmission is 24. Manual transmission get 7.2 MPG improvement when compared to automatic. l2-Value of MPG and weight of car mpg have a clear correlation, will decrease by 5 for every 1000 lb increase in wt. l3 - Shows number of cylinders, cyl increases from 4 to 6 and 8, mpg will decrease by a factor of 2.8 respectively. l4- Value of MPG and VS have clear positive correlation by increasing factor 7.9 l5- Value of MPG and number of cylinders have positive correlation, cyl increases from 4 to 6 and 8, mpg will increase by a factor of 3.9 l6- Value of MPG and number of carburetors have clear negative correlation by decreasing factor 2

Inference of output The analysis suggests that selected model has statistically importance for variables in the mtcars dataset. Thi analysis will perform this selection by calling lm repeatedly. It selects the best variables to use in predicting mpg with other variables “cyl”,“hp”, “am”, “disp”, “gear” and “carb”.

analysis <- aov(mpg ~ ., data = mtcars); summary(analysis)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## cyl          1  817.7   817.7 116.425 5.03e-10 ***
## disp         1   37.6    37.6   5.353  0.03091 *  
## hp           1    9.4     9.4   1.334  0.26103    
## drat         1   16.5    16.5   2.345  0.14064    
## wt           1   77.5    77.5  11.031  0.00324 ** 
## qsec         1    3.9     3.9   0.562  0.46166    
## vs           1    0.1     0.1   0.018  0.89317    
## am           1   14.5    14.5   2.061  0.16586    
## gear         1    1.0     1.0   0.138  0.71365    
## carb         1    0.4     0.4   0.058  0.81218    
## Residuals   21  147.5     7.0                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
lm<- lm(mpg ~ ., data = mtcars); summary(lm)
## 
## Call:
## lm(formula = mpg ~ ., data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4506 -1.6044 -0.1196  1.2193  4.6271 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) 12.30337   18.71788   0.657   0.5181  
## cyl         -0.11144    1.04502  -0.107   0.9161  
## disp         0.01334    0.01786   0.747   0.4635  
## hp          -0.02148    0.02177  -0.987   0.3350  
## drat         0.78711    1.63537   0.481   0.6353  
## wt          -3.71530    1.89441  -1.961   0.0633 .
## qsec         0.82104    0.73084   1.123   0.2739  
## vs           0.31776    2.10451   0.151   0.8814  
## am           2.52023    2.05665   1.225   0.2340  
## gear         0.65541    1.49326   0.439   0.6652  
## carb        -0.19942    0.82875  -0.241   0.8122  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.65 on 21 degrees of freedom
## Multiple R-squared:  0.869,  Adjusted R-squared:  0.8066 
## F-statistic: 13.93 on 10 and 21 DF,  p-value: 3.793e-07
anova(lm)
## Analysis of Variance Table
## 
## Response: mpg
##           Df Sum Sq Mean Sq  F value    Pr(>F)    
## cyl        1 817.71  817.71 116.4245 5.034e-10 ***
## disp       1  37.59   37.59   5.3526  0.030911 *  
## hp         1   9.37    9.37   1.3342  0.261031    
## drat       1  16.47   16.47   2.3446  0.140644    
## wt         1  77.48   77.48  11.0309  0.003244 ** 
## qsec       1   3.95    3.95   0.5623  0.461656    
## vs         1   0.13    0.13   0.0185  0.893173    
## am         1  14.47   14.47   2.0608  0.165858    
## gear       1   0.97    0.97   0.1384  0.713653    
## carb       1   0.41    0.41   0.0579  0.812179    
## Residuals 21 147.49    7.02                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The adjusted R-squared value of 0.80 tells us that 80% of the variability is explained by this model. From these results we can conclude that more than 80% of the variability is explained by the above model. The p- value obtained with anova test is significant (p<0.05) and we reject the null hypothesis in the variables wt and disp.

Regression through the origin This analysis and graphic using the origin as a pivot point picking the line that minimizes the sum of the squared vertical distances of the points to the line for MPG and mode of transmission( in two levels “Automatic” and “Manual”). Subtract the means so that the origin is the mean of the MPG and mode of transmission as factor. (Fig.3)

data(mtcars)
mtcars<-data.frame(mtcars)
a<-factor(mtcars$mpg)
x <-mtcars$mpg - mean(mtcars$mpg)
y<-factor(mtcars$am,labels=c("Automatic","Manual"))
car <- as.data.frame(table(a,x, y))
names(car) <- c("MPG","mpg", "am", "freq")

g <- ggplot(filter(car, freq > -1), aes(am, MPG), + facet_grid(.~am))
g <- g + xlab("Type of car transmission")
g <- g + ylab("Miles per gallon")
g <- g  + scale_size(range = c(4, 20), guide = "none" )
g <- g + geom_point(colour="grey90",  aes(size = freq+ 0.2, show_guide = FALSE))
g <- g + geom_point(aes(colour=freq, size = freq))
g <- g + geom_point(colour="grey80")
g <- g + scale_colour_gradient(low = "yellow", high="red") 
g <- g + geom_smooth(aes(group=freq), method="lm", fullrange=TRUE)
g