Homework3
Garrett Smith
2024-10-03
data(mtcars)
attach(mtcars)
head(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
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1summary(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. :8.000library(corrplot)## Warning: package 'corrplot' was built under R version 4.3.3## corrplot 0.94 loadedcorr_matrix <- cor(mtcars)
print(corr_matrix)## mpg cyl disp hp drat wt
## mpg 1.0000000 -0.8521620 -0.8475514 -0.7761684 0.68117191 -0.8676594
## cyl -0.8521620 1.0000000 0.9020329 0.8324475 -0.69993811 0.7824958
## disp -0.8475514 0.9020329 1.0000000 0.7909486 -0.71021393 0.8879799
## hp -0.7761684 0.8324475 0.7909486 1.0000000 -0.44875912 0.6587479
## drat 0.6811719 -0.6999381 -0.7102139 -0.4487591 1.00000000 -0.7124406
## wt -0.8676594 0.7824958 0.8879799 0.6587479 -0.71244065 1.0000000
## qsec 0.4186840 -0.5912421 -0.4336979 -0.7082234 0.09120476 -0.1747159
## vs 0.6640389 -0.8108118 -0.7104159 -0.7230967 0.44027846 -0.5549157
## am 0.5998324 -0.5226070 -0.5912270 -0.2432043 0.71271113 -0.6924953
## gear 0.4802848 -0.4926866 -0.5555692 -0.1257043 0.69961013 -0.5832870
## carb -0.5509251 0.5269883 0.3949769 0.7498125 -0.09078980 0.4276059
## qsec vs am gear carb
## mpg 0.41868403 0.6640389 0.59983243 0.4802848 -0.55092507
## cyl -0.59124207 -0.8108118 -0.52260705 -0.4926866 0.52698829
## disp -0.43369788 -0.7104159 -0.59122704 -0.5555692 0.39497686
## hp -0.70822339 -0.7230967 -0.24320426 -0.1257043 0.74981247
## drat 0.09120476 0.4402785 0.71271113 0.6996101 -0.09078980
## wt -0.17471588 -0.5549157 -0.69249526 -0.5832870 0.42760594
## qsec 1.00000000 0.7445354 -0.22986086 -0.2126822 -0.65624923
## vs 0.74453544 1.0000000 0.16834512 0.2060233 -0.56960714
## am -0.22986086 0.1683451 1.00000000 0.7940588 0.05753435
## gear -0.21268223 0.2060233 0.79405876 1.0000000 0.27407284
## carb -0.65624923 -0.5696071 0.05753435 0.2740728 1.00000000corrplot(corr_matrix, method="circle", type="upper", order="hclust",
tl.col="black", tl.srt=45)
The variables most strongly associated with mpg are wt, cyl, and disp.
Now let’s look at these variables graphically.
library(ggplot2)## Warning: package 'ggplot2' was built under R version 4.3.3##
## Attaching package: 'ggplot2'## The following object is masked from 'mtcars':
##
## mpgggplot(mtcars, aes(x = wt, y = mpg)) +
geom_point() +
labs(title = "Weight vs Miles per Gallon", x = "Weight (1000 lbs)", y = "Miles per Gallon")
ggplot(mtcars, aes(x=factor(cyl), y=mpg)) +
geom_boxplot(aes(fill=factor(cyl))) +
ggtitle("Miles per Gallon by Number of Cylinders") +
xlab("Number of Cylinders") +
ylab("Miles per Gallon")
ggplot(mtcars, aes(x=disp, y=mpg)) +
geom_point(aes(color=disp)) +
ggtitle("Displacement vs. Miles per Gallon") +
xlab("Displacement (cubic inches)") +
ylab("Miles per Gallon") +
scale_color_continuous(name="Index")
An increase in weight, number of cylinders, and displacement correlates to a decrease in miles per gallon.
colSums(is.na(mtcars))## mpg cyl disp hp drat wt qsec vs am gear carb
## 0 0 0 0 0 0 0 0 0 0 0#There is no missing data.
boxplot(mtcars, las=2, cex.axis=0.6)
mtcars_lm <- lm(mpg ~ wt + disp + cyl + hp, data = mtcars)
summary(mtcars_lm)##
## Call:
## lm(formula = mpg ~ wt + disp + cyl + hp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.0562 -1.4636 -0.4281 1.2854 5.8269
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 40.82854 2.75747 14.807 1.76e-14 ***
## wt -3.85390 1.01547 -3.795 0.000759 ***
## disp 0.01160 0.01173 0.989 0.331386
## cyl -1.29332 0.65588 -1.972 0.058947 .
## hp -0.02054 0.01215 -1.691 0.102379
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.513 on 27 degrees of freedom
## Multiple R-squared: 0.8486, Adjusted R-squared: 0.8262
## F-statistic: 37.84 on 4 and 27 DF, p-value: 1.061e-10#There is a significant correlation between mpg and wt and mpg and cyl, but not mpg and disp.
mean(mtcars_lm$residuals^2)## [1] 5.326386#The Means Squared Error is 5.326386.Looking through the data, the displacement variable has the largest range. Let’s winsorize this variable and also truncate hp to remove the large outlier.
mt_cars_sorted <- mtcars[mtcars$hp <= 300, ]
lower_bound_disp <- quantile(mt_cars_sorted$disp, 0.05, na.rm = TRUE)
upper_bound_disp <- quantile(mt_cars_sorted$disp, 0.95, na.rm = TRUE)
mt_cars_sorted$disp[mt_cars_sorted$tax < lower_bound_disp] <- lower_bound_disp
mt_cars_sorted$disp[mt_cars_sorted$tax > upper_bound_disp] <- upper_bound_dispNow lets run the same linear regression tests as before.
mtcars_lm2 <- lm(mpg ~ wt + disp + cyl + hp, data = mt_cars_sorted)
summary(mtcars_lm2)##
## Call:
## lm(formula = mpg ~ wt + disp + cyl + hp, data = mt_cars_sorted)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.9483 -1.2877 -0.3379 0.7952 5.7201
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 41.13342 2.78171 14.787 3.6e-14 ***
## wt -3.91774 1.01973 -3.842 0.000705 ***
## disp 0.01502 0.01229 1.221 0.232863
## cyl -1.20061 0.66446 -1.807 0.082366 .
## hp -0.03123 0.01661 -1.879 0.071432 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.518 on 26 degrees of freedom
## Multiple R-squared: 0.8501, Adjusted R-squared: 0.827
## F-statistic: 36.86 on 4 and 26 DF, p-value: 2.325e-10mean(mtcars_lm2$residuals^2)## [1] 5.315611#The means squared error is 5.315611.Lastly, lets run these two regressions with an interaction term of cyl * hp and see if there is a correlation between these two variables.
mtcars_lm <- lm(mpg ~ wt + disp + cyl + hp + cyl * hp, data = mtcars)
summary(mtcars_lm)##
## Call:
## lm(formula = mpg ~ wt + disp + cyl + hp + cyl * hp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.395 -1.426 -0.612 1.219 4.312
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 52.164411 5.031802 10.367 9.96e-11 ***
## wt -3.302628 0.946204 -3.490 0.00174 **
## disp 0.003063 0.011144 0.275 0.78561
## cyl -2.780502 0.826175 -3.366 0.00238 **
## hp -0.159849 0.054760 -2.919 0.00716 **
## cyl:hp 0.018380 0.007077 2.597 0.01527 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.281 on 26 degrees of freedom
## Multiple R-squared: 0.8798, Adjusted R-squared: 0.8567
## F-statistic: 38.07 on 5 and 26 DF, p-value: 3.697e-11mtcars_lm2 <- lm(mpg ~ wt + disp + cyl + hp + cyl * hp, data = mt_cars_sorted)
summary(mtcars_lm2)##
## Call:
## lm(formula = mpg ~ wt + disp + cyl + hp + cyl * hp, data = mt_cars_sorted)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.3926 -1.4547 -0.7343 1.2200 4.3282
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 51.959186 5.294022 9.815 4.67e-10 ***
## wt -3.325420 0.975411 -3.409 0.00222 **
## disp 0.003811 0.012326 0.309 0.75972
## cyl -2.731965 0.897450 -3.044 0.00543 **
## hp -0.158458 0.056522 -2.803 0.00963 **
## cyl:hp 0.017967 0.007682 2.339 0.02764 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.326 on 25 degrees of freedom
## Multiple R-squared: 0.877, Adjusted R-squared: 0.8524
## F-statistic: 35.65 on 5 and 25 DF, p-value: 1.343e-10There is a relatively significant relationship between the number of cylinders and horsepower.