Motor Trend Car Road Analysis

Transcription

1 Motor Trend Car Road Analysis Zakia Sultana February 28, 2016 Executive Summary 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: 1."Is an automatic or manual transmission better for MPG" 2. "Quantify the MPG difference between automatic and manual transmissions" Data processing library(datasets) data(mtcars) names(mtcars) [1] "mpg" "cyl" "disp" "hp" "drat" "wt" "qsec" "vs" "am" "gear" [11] "carb" str(mtcars) There are 11 variables, since we are interested in the relationshp between mpg and other variables, we first check the correlation between mpg and other variables by using the cor() function cor(mtcars$mpg,mtcars[, 1]) cyl disp hp drat wt qsec [1,] vs am gear carb [1,] From the correlation data, we could see cyl, hp, wt and carb are negatively correlated with mpg. Explorary Data analysis We begin the explorary data analysis by looking at the pairwise scatter plot between all variables.

2 With the distribution of the dependent variable: mpg, and see if it mets the assumptions of regression. From both the histogram and the kernel density, it is approximately normal. pairs(mtcars) par(mfrow=c(2,1)) hist(mtcars$mpg, breaks=10, xlab="mpg", main="mpg histogram") plot(density(mtcars$mpg), main="kernel density", xlab="mpg")

3 Is an automatic or manual transmission better for MPG? For automatic: summary(mtcars[mtcars$am==0,]) For manual: summary(mtcars[mtcars$am==1,]) We plot a boxplot of MPG by transmission types. boxplot(mpg~am, data = mtcars,col="green",xlab = "Transmission",ylab = "Miles per Gallo n",main = "MPG by Transmission Type", names = c("automatic", "Manual"))

4 Hence, from this simple plot, It seems that manual transmission is better in MPG than automatic transmission as the mean of mpg is greater for manual (at 24.4) than automatic (at 17.1). Hypothesis Testing We then perform a t test to confirm this hypothesis i.e., whether manual transmission is better than automatic transmission. t.test(mtcars$mpg~mtcars$am,conf.level=0.95) Welch Two Sample t test data: mtcars$mpg by mtcars$am t = , df = , p value = alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: sample estimates: mean in group 0 mean in group

5 The p value is , we may think it is ok to reject the null hypothesis and conclude automatic has low mpg compared with manual cars however this assumption is based on all other characteristics of auto cars and manual cars are same (e.g: auto cars and manual cars have same weight distribution) which needs to be further explored in the multiple linear regression analysis. Quantify the MPG difference between automatic and manual transmissions In this section we aim to quantify the MPG different between transmission types, and find if there are other variables that account for the MPG differences. First, we try to do a basic linear regression model considering only one variable am: basic = lm( mpg ~ am,mtcars) summary(basic) Call: lm(formula = mpg ~ am, data = mtcars) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e 15 *** am *** Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 30 degrees of freedom Multiple R squared: , Adjusted R squared: F statistic: on 1 and 30 DF, p value: Then, we try to build an initial a multivariate linear regression model with all variables as a predictors: mlr = lm(data = mtcars, mpg ~.) summary(mlr)

6 Call: lm(formula = mpg ~., data = mtcars) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) cyl disp hp drat wt qsec vs am gear carb Signif. codes: 0 '***' '**' 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: F statistic: on 10 and 21 DF, p value: 3.793e 07 From the coefficients, it looks like wt,am changes significantly with mpg. However, including all variables will possibly result overfitting, and so we will perform stepwise model selection to select significant predictors for the best model by using automatic model choosing function in R to choose the best linear regression model. bestmodel < step(mlr,trace=0) summary(bestmodel)

7 Call: lm(formula = mpg ~ wt + qsec + am, data = mtcars) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) wt e 06 *** qsec *** am * Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 28 degrees of freedom Multiple R squared: , Adjusted R squared: F statistic: on 3 and 28 DF, p value: 1.21e 11 It looks like the best model is the one that includes wt, qsec and am, which means besides transmission types, weight and accelearation also needs to be considered. Weight negatively changes with mpg, and qsec and am positively changes. Every lb/1000 weight increase will cause a decrease of roughly 4 mpg, every increase of 1/4 mile time will cause an increase of 1.2 mpg, and on average, manual transmission is 2.9 mpg better than automatic transmission. The model is able to explain 85% of variance. The anova comparison shows a very low f value, so it makes sense to use the fit model instead of the basic. The adjusted R squared is also much better (0.85 fit vs 0.34 basic). anova(basic,bestmodel) Analysis of Variance Table Model 1: mpg ~ am Model 2: mpg ~ wt + qsec + am Res.Df RSS Df Sum of Sq F Pr(>F) e 09 *** Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Therefore given the above analysis, the question of auto car and manual car is not anwsered and have to be considered in the context of weight and accelaration speed. Analysis of the Residuals and Diagnostics Finally, we plot the residue and diagnostic plot for this best model.

8 par(mfrow = c(2,2)) plot(bestmodel) From these plots we can conclude the following: The Residuals vs Fitted plot shows random points on the plot that verifies the indepe ndence condition. In the Normal Q Q plot the points mostly fall on the line indicating that the residua ls are normally distributed. In the Scale Location plot the points are in a constant band pattern, indicating cons tant variance. Finally, the Residuals vs Leverage plot shows that there are a number of outliers (le verage points) in the dataset, specially Toyota Corolla, Fiat 128 and Chrysler Imperial. Now we will compute some regression diagnostics of our model to find out these interesting leverage points. We compute top four points in each case of influence measures. leverage_points < hatvalues(bestmodel) tail(sort(leverage_points), 4)

9 Cadillac Fleetwood Chrysler Imperial Lincoln Continental Merc influence_measure < dfbetas(bestmodel) tail(sort(influence_measure[, 4]), 4) Toyota Corolla Toyota Corona Fiat 128 Chrysler Imperial Looking at this result we see that they the same cars shown in the residual plots. Conclusion On average, manual transmission is better than automatic transmission by 2.9 mpg. However, transmission type is not the only factor accounting for MPG, weight, and acceleration (1/4 mile time) also needs to be considered.