Homework 3
Christopher Salinas
2024-10-08
Data Exploration
- Load Data
mtcarsand review its codebook along with its visualizations and summary statistics.
data(mtcars)
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. :8.000#load package for visualization
library(ggplot2)Visualizations code
ggplot(mtcars, aes(x = hp, y = mpg)) +
geom_point() +
labs(title = "MPG vs Horsepower",
x = "Horsepower (hp)",
y = "Miles per Gallon (mpg)") 
ggplot(mtcars, aes(x = mpg)) +
geom_histogram(binwidth = 5, fill = "blue", alpha = 0.7) +
ggtitle("Distribution of MPG") +
xlab ("Miles per Gallon (mpg)") +
ylab ("Frequency") 
Patterns/correlations
Noticed that mpg and hp have a negative relationship in which the higher the hp the lower mpg. For the distribution of mpg it is skewed to the left a little bit with most mpg values falling in the 15-25 range.
corr_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 1.00000000It seems that wt, cyl, and
disp have strong negative correlations with
mpg.
Data Reprocessing
- Check for missing values
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#No missing values2.Check for inconsistent/invalid data
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. :8.000Does not seem like the data has any invalid data all the numbers make sense when considering the variable they’re under.
Linear Regression using lm
- Create linear regression model to predict mpg based on other variables.
mtcars_lm <- lm(mpg ~ hp + wt, data = mtcars)
summary(mtcars_lm)##
## Call:
## lm(formula = mpg ~ hp + wt, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.941 -1.600 -0.182 1.050 5.854
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 37.22727 1.59879 23.285 < 2e-16 ***
## hp -0.03177 0.00903 -3.519 0.00145 **
## wt -3.87783 0.63273 -6.129 1.12e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.593 on 29 degrees of freedom
## Multiple R-squared: 0.8268, Adjusted R-squared: 0.8148
## F-statistic: 69.21 on 2 and 29 DF, p-value: 9.109e-12Interpret coefficients The p-values for both
hpandwtare statistically significant. The coefficients for both variables are negative meaning a negative relationship between horsepower to mpg and weight to mpg. This means as horsepower or weight go up the mpg goes down.Assumptions made in linear regression include linearity, homoscedasticity, normality of residuals, no multicollinearity, and independence of errors
par(mfrow = c(2, 2))
plot(mtcars_lm)
For the residuals vs fitted there is some pattern but overall I’d say
linearity holds. Q-Q residuals are all mostly following a straight line
indicating normality. Scale-location plot has constant variance
confirming homoscedasticity. Residuals vs leverage has some outliers but
are mostly in the middle of the 0.5 so they are not too problematic.
car::vif(mtcars_lm)## hp wt
## 1.766625 1.766625no multicollinearity or not enough to be significant.
- Evaluate model using MSE
lm_mse <- mean((mtcars_lm$fitted.values - mtcars$mpg)^2)
print(paste("Mean Squared Error for Linear Model:", round(lm_mse, 2)))## [1] "Mean Squared Error for Linear Model: 6.1"Since there is more than just the 2 variables I included the mse is
high, making this model not bad but definitely not the best model if we
wanted to accurately predict mpg.
- Add interaction terms to the model.
mtcars_lm_it <- lm(mpg ~ hp + wt + hp*wt, data = mtcars)
summary(mtcars_lm_it)##
## Call:
## lm(formula = mpg ~ hp + wt + hp * wt, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.0632 -1.6491 -0.7362 1.4211 4.5513
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 49.80842 3.60516 13.816 5.01e-14 ***
## hp -0.12010 0.02470 -4.863 4.04e-05 ***
## wt -8.21662 1.26971 -6.471 5.20e-07 ***
## hp:wt 0.02785 0.00742 3.753 0.000811 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.153 on 28 degrees of freedom
## Multiple R-squared: 0.8848, Adjusted R-squared: 0.8724
## F-statistic: 71.66 on 3 and 28 DF, p-value: 2.981e-13mtcars_sum_lm = summary(mtcars_lm_it)
mtcars_sum_lm$adj.r.squared## [1] 0.872417After creating the linear regression model with an interaction
between hp and wt the adj. r squared has
increased from .8148 to .8724 making the new model better. Also the
interactions p-value is significant.
6.Any outliers? If so apply winsorization.
boxplot(mtcars, las=2, cex.axis=0.6)
Really no outliers for this data set only one value that is extreme in
hp.
- Does improved r squared really improve model predictability? Higher r squared values mean that the model explains more variance of the data but it does not mean that the model is better for predictability. Adj. r squared and MSE would probably fare better at deciding whether the model is good or not.