MPG Regression Analysis
Michael Garcia
5/17/2018
Executive Summary
Motor Trend provides information, opinions, and tips about cars to its readers. A topic of interest is energy efficiency of vehicles, specifically autmatic and manual transimission and miles per gallon. The analysis will provide insight into the methods used and the results for answering the question is miles per gallon for automatic vehicles greater than, less than, or equal to vehicles with manual transmission. * “Is an automatic or manual transmission better for MPG” * “Quantify the MPG difference between automatic and manual transmissions”
The analysis includes to sets of stepped models. Those with “Model#” are Generalized Linear Model using binomial distribution and logiscit regression; “LModel#” are the models using Linear Models.
The question involves a response of binary data: is it MPG automatic or manual. The models tested involve linear models and generalized linear model with binomial family function. The coefficient of determination is highest for LModel5. However, the predictors have p-values indiciating they are not significant to the model. The question is whether there probability of a vehicle being automatic with higher miles per gallon. So the a logistic GLM is the better approach for binomial outcome.
The logistic generalized model with the best fit is Model1 or AM = 0.307 MPG - 6.604.
Exploratory Analysis
s
data(cars)
summary(mtcars)## mpg cyl disp hp
## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.20 Median :6.000 Median :196.3 Median :123.0
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000mtcarsdf <- as.data.frame(mtcars)Exploratory Data Analysis- Distribution
The distributions for the mpg for the total dataset are reflected
mtcarsdf$mpg_rnd <- round(mtcarsdf$mpg,0)
par(mfrow = c(1, 2), mar = c(4, 4, 2, 1))
hist(mtcarsdf$mpg,col = "blue", freq = TRUE, xlab = "MPG")
rug(mtcarsdf$mpg, col = "red")
abline(v = mean(mtcarsdf$mpg), col = "red", lwd = 4)
boxplot(mpg ~ am, data = mtcarsdf, col = "green", xlab = "Manual vs Automatic", ylab = "MPG")
You can also embed plots, for example:
Nested Fitting - Generalized Linear Model
Model1 <- glm(am ~ mpg , data = mtcars, family = "binomial")
Model2 <- glm(am ~ mpg + wt, data = mtcars, family = "binomial")
Model3 <- glm(am ~ mpg + wt + hp , data = mtcars, family = "binomial")
Model4 <- glm(am ~ mpg + wt + hp+ disp, data = mtcars, family = "binomial")
#Model5 <- glm(am ~ ., data = mtcars, family = "binomial")ANOVA GLM
anova(Model1, Model2, Model3, Model4)## Analysis of Deviance Table
##
## Model 1: am ~ mpg
## Model 2: am ~ mpg + wt
## Model 3: am ~ mpg + wt + hp
## Model 4: am ~ mpg + wt + hp + disp
## Resid. Df Resid. Dev Df Deviance
## 1 30 29.6752
## 2 29 17.1843 1 12.4909
## 3 28 8.7661 1 8.4181
## 4 27 8.1620 1 0.6041Nested Fitting - Linear Model
LModel1 <- lm(mpg ~ am , data = mtcars)
LModel2 <- lm(mpg ~ am + wt, data = mtcars)
LModel3 <- lm(mpg ~ am + wt + hp , data = mtcars)
LModel4 <- lm(mpg ~ am + wt + hp+ disp, data = mtcars)
LModel5 <- lm(mpg ~ ., data = mtcars)ANOVA LM
anova(LModel1, LModel2, LModel3, LModel4, LModel5)## Analysis of Variance Table
##
## Model 1: mpg ~ am
## Model 2: mpg ~ am + wt
## Model 3: mpg ~ am + wt + hp
## Model 4: mpg ~ am + wt + hp + disp
## Model 5: mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 720.90
## 2 29 278.32 1 442.58 63.0133 9.325e-08 ***
## 3 28 180.29 1 98.03 13.9571 0.001219 **
## 4 27 179.91 1 0.38 0.0546 0.817510
## 5 21 147.49 6 32.41 0.7692 0.602559
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1LModelSum1 <- summary(LModel1)
LModelSum2 <- summary(LModel2)
LModelSum3 <- summary(LModel3)
LModelSum4 <- summary(LModel4)
LModelSum5 <- summary(LModel5)
LModelSum1$r.squared## [1] 0.3597989LModelSum2$r.squared## [1] 0.7528348LModelSum3$r.squared## [1] 0.8398903LModelSum4$r.squared## [1] 0.8402309LModelSum5$r.squared## [1] 0.8690158GLM
The model supports that the MPG increases for vehicles that are automatic or not automatic. We use binomial general linear model given that 1 of 2 outcomes is possible for mileage per gallon.
The model is given by: probautomatic = .307MPG - 6.6035. So for every increase in distance of .307MPG theres a higher probabilty that the vehicle is automatic.
logCars <- glm(mtcars$am~ mtcars$mpg, family = "binomial")
summary(logCars)##
## Call:
## glm(formula = mtcars$am ~ mtcars$mpg, family = "binomial")
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5701 -0.7531 -0.4245 0.5866 2.0617
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -6.6035 2.3514 -2.808 0.00498 **
## mtcars$mpg 0.3070 0.1148 2.673 0.00751 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 43.230 on 31 degrees of freedom
## Residual deviance: 29.675 on 30 degrees of freedom
## AIC: 33.675
##
## Number of Fisher Scoring iterations: 5logCars$fitted## 1 2 3 4 5 6
## 0.46109512 0.46109512 0.59789839 0.49171990 0.29690087 0.25993307
## 7 8 9 10 11 12
## 0.09858705 0.70846924 0.59789839 0.32991148 0.24260966 0.17246396
## 13 14 15 16 17 18
## 0.21552479 0.12601104 0.03197098 0.03197098 0.11005178 0.96591395
## 19 20 21 22 23 24
## 0.93878132 0.97821971 0.49939484 0.13650937 0.12601104 0.07446438
## 25 26 27 28 29 30
## 0.32991148 0.85549212 0.79886349 0.93878132 0.14773451 0.36468861
## 31 32
## 0.11940215 0.49171990logCars$coefficients## (Intercept) mtcars$mpg
## -6.6035267 0.3070282GLM Summary
The models for the GLM are summarized here. The Akaike Information Criterion (aic) is proper for the model as we are looking at the likelihood of the vehicle being either automatic or manual. The aic measures the dispersion of data points for models of likelihood. The AM = .307 - 6.60 has the largest AIC compared to the rest of the models
summary(Model1)##
## Call:
## glm(formula = am ~ mpg, family = "binomial", data = mtcars)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5701 -0.7531 -0.4245 0.5866 2.0617
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -6.6035 2.3514 -2.808 0.00498 **
## mpg 0.3070 0.1148 2.673 0.00751 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 43.230 on 31 degrees of freedom
## Residual deviance: 29.675 on 30 degrees of freedom
## AIC: 33.675
##
## Number of Fisher Scoring iterations: 5summary(Model2)##
## Call:
## glm(formula = am ~ mpg + wt, family = "binomial", data = mtcars)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.50806 -0.45191 -0.04684 0.24664 2.01168
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 25.8866 12.1935 2.123 0.0338 *
## mpg -0.3242 0.2395 -1.354 0.1759
## wt -6.4162 2.5466 -2.519 0.0118 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 43.230 on 31 degrees of freedom
## Residual deviance: 17.184 on 29 degrees of freedom
## AIC: 23.184
##
## Number of Fisher Scoring iterations: 7summary(Model3)##
## Call:
## glm(formula = am ~ mpg + wt + hp, family = "binomial", data = mtcars)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.93381 -0.09191 -0.00913 0.01139 1.47331
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -15.72137 40.00281 -0.393 0.6943
## mpg 1.22930 1.58109 0.778 0.4369
## wt -6.95492 3.35297 -2.074 0.0381 *
## hp 0.08389 0.08228 1.020 0.3079
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 43.2297 on 31 degrees of freedom
## Residual deviance: 8.7661 on 28 degrees of freedom
## AIC: 16.766
##
## Number of Fisher Scoring iterations: 10summary(Model4)##
## Call:
## glm(formula = am ~ mpg + wt + hp + disp, family = "binomial",
## data = mtcars)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.84992 -0.15966 -0.00615 0.01257 1.46081
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -18.48207 40.90451 -0.452 0.651
## mpg 1.13503 1.55720 0.729 0.466
## wt -4.80560 3.97978 -1.208 0.227
## hp 0.10871 0.09837 1.105 0.269
## disp -0.02588 0.04087 -0.633 0.527
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 43.230 on 31 degrees of freedom
## Residual deviance: 8.162 on 27 degrees of freedom
## AIC: 18.162
##
## Number of Fisher Scoring iterations: 9GLM Step and Best Model Selection
base_model <- glm(am ~ ., data = mtcars)
optimal_model <- step(base_model, direction = "both")## Start: AIC=17.93
## am ~ mpg + cyl + disp + hp + drat + wt + qsec + vs + gear + carb
##
## Df Deviance AIC
## - wt 1 1.5502 15.936
## - carb 1 1.5530 15.995
## - disp 1 1.5567 16.071
## - hp 1 1.5651 16.243
## - drat 1 1.5736 16.416
## - vs 1 1.6219 17.384
## - cyl 1 1.6494 17.922
## <none> 1.5497 17.926
## - mpg 1 1.6605 18.136
## - gear 1 1.6785 18.482
## - qsec 1 1.7247 19.350
##
## Step: AIC=15.94
## am ~ mpg + cyl + disp + hp + drat + qsec + vs + gear + carb
##
## Df Deviance AIC
## - carb 1 1.5533 14.001
## - disp 1 1.5609 14.156
## - hp 1 1.5653 14.248
## - drat 1 1.5736 14.417
## - vs 1 1.6236 15.417
## <none> 1.5502 15.936
## - cyl 1 1.6531 15.993
## - mpg 1 1.6754 16.422
## - gear 1 1.6791 16.493
## + wt 1 1.5497 17.926
## - qsec 1 1.7941 18.613
##
## Step: AIC=14
## am ~ mpg + cyl + disp + hp + drat + qsec + vs + gear
##
## Df Deviance AIC
## - disp 1 1.5613 12.164
## - hp 1 1.5654 12.248
## - drat 1 1.5745 12.434
## - vs 1 1.6238 13.420
## <none> 1.5533 14.001
## - cyl 1 1.6698 14.316
## - gear 1 1.6933 14.762
## - mpg 1 1.7304 15.456
## + carb 1 1.5502 15.936
## + wt 1 1.5530 15.995
## - qsec 1 1.8134 16.954
##
## Step: AIC=12.16
## am ~ mpg + cyl + hp + drat + qsec + vs + gear
##
## Df Deviance AIC
## - hp 1 1.5677 10.296
## - drat 1 1.5842 10.630
## - vs 1 1.6255 11.454
## <none> 1.5613 12.164
## - cyl 1 1.7031 12.947
## - gear 1 1.7371 13.580
## - mpg 1 1.7512 13.839
## + disp 1 1.5533 14.001
## + wt 1 1.5567 14.071
## + carb 1 1.5609 14.156
## - qsec 1 1.8506 15.605
##
## Step: AIC=10.3
## am ~ mpg + cyl + drat + qsec + vs + gear
##
## Df Deviance AIC
## - drat 1 1.5916 8.7807
## - vs 1 1.6264 9.4724
## <none> 1.5677 10.2955
## - cyl 1 1.7185 11.2350
## - mpg 1 1.7643 12.0771
## + hp 1 1.5613 12.1642
## + disp 1 1.5654 12.2479
## + wt 1 1.5659 12.2584
## + carb 1 1.5676 12.2950
## - gear 1 1.8101 12.8961
## - qsec 1 1.8965 14.3893
##
## Step: AIC=8.78
## am ~ mpg + cyl + qsec + vs + gear
##
## Df Deviance AIC
## - vs 1 1.6505 7.9429
## <none> 1.5916 8.7807
## + drat 1 1.5677 10.2955
## + hp 1 1.5842 10.6305
## + disp 1 1.5887 10.7212
## + wt 1 1.5897 10.7418
## + carb 1 1.5904 10.7570
## - cyl 1 1.8074 10.8484
## - mpg 1 1.8148 10.9804
## - gear 1 1.9097 12.6110
## - qsec 1 1.9780 13.7359
##
## Step: AIC=7.94
## am ~ mpg + cyl + qsec + gear
##
## Df Deviance AIC
## <none> 1.6505 7.9429
## + vs 1 1.5916 8.7807
## - cyl 1 1.8138 8.9619
## + drat 1 1.6264 9.4724
## - mpg 1 1.8666 9.8803
## + carb 1 1.6489 9.9113
## + hp 1 1.6492 9.9170
## + disp 1 1.6494 9.9219
## + wt 1 1.6505 9.9429
## - gear 1 1.9579 11.4076
## - qsec 1 2.3181 16.8127summary(optimal_model)##
## Call:
## glm(formula = am ~ mpg + cyl + qsec + gear, data = mtcars)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.51557 -0.15860 -0.00793 0.19350 0.35820
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.36836 1.52965 1.548 0.13319
## mpg 0.02703 0.01438 1.880 0.07091 .
## cyl -0.11052 0.06762 -1.634 0.11378
## qsec -0.14810 0.04481 -3.305 0.00269 **
## gear 0.22288 0.09940 2.242 0.03335 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.06112933)
##
## Null deviance: 7.7188 on 31 degrees of freedom
## Residual deviance: 1.6505 on 27 degrees of freedom
## AIC: 7.9429
##
## Number of Fisher Scoring iterations: 2Evaluating the AIC
1-pchisq(Model1$aic,Model1$df.residual)## [1] 0.29400461-pchisq(Model2$aic,Model2$df.residual)## [1] 0.7680441-pchisq(Model3$aic,Model3$df.residual)## [1] 0.9530671-pchisq(Model4$aic,Model4$df.residual)## [1] 0.8984913exp(logCars$coefficients)## (Intercept) mtcars$mpg
## 0.001355579 1.359379288exp(confint(logCars))## Waiting for profiling to be done...## 2.5 % 97.5 %
## (Intercept) 4.425443e-06 0.06255158
## mtcars$mpg 1.129764e+00 1.79946863anova(logCars, test = "Chisq")## Analysis of Deviance Table
##
## Model: binomial, link: logit
##
## Response: mtcars$am
##
## Terms added sequentially (first to last)
##
##
## Df Deviance Resid. Df Resid. Dev Pr(>Chi)
## NULL 31 43.230
## mtcars$mpg 1 13.555 30 29.675 0.0002317 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Appendix
Probability Plot Automatic Transmission
plot(mtcars$mpg,logCars$fitted,pch=19,col="blue",xlab="MPG",ylab="Probability Automatic")
rug(mtcars$mpg, lwd = 2, col = "red")
Residuals and Fit Plots - Logistic GLM Model
par(mfrow = c(2,2))
plot(logCars)
#### Residuals and Fit Plots - Linear Model- Best Model based on Significance and Coefficient of Determination
par(mfrow = c(2,2))
plot(LModel2)
#### Scatterplot Matrix The plot displays the points for pairs of variables.
pairs(mpg ~ ., data = mtcars, col = "black")