library(MASS)
## Warning: package 'MASS' was built under R version 4.0.3
#create data
y=c(1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 6, 7, 8)
x=c(7, 7, 8, 3, 2, 4, 4, 6, 6, 7, 5, 3, 3, 5, 8)
#fit linear regression model
model <- lm(y~x)
#find optimal lambda for Box-Cox transformation
bc <- boxcox(y ~ x)
lambda <- bc$x[which.max(bc$y)]
lambda
## [1] -0.4242424
#fit new linear regression model using the Box-Cox transformation
new_model <- lm(((y^lambda-1)/lambda) ~ x)
#define plotting area
op <- par(pty = "s", mfrow = c(1, 2))
#Q-Q plot for the residuals of the original model
qqnorm(model$residuals)
qqline(model$residuals)
#Q-Q plot for Box-Cox transformed model
qqnorm(new_model$residuals)
qqline(new_model$residuals)
#display both Q-Q plots
par(op)
load("EXEXSAL2.Rdata")
names(EXEXSAL2)
## [1] "ID" "Y" "X1" "X2" "X3" "X4" "X5" "X6" "X7" "X8" "X9" "X10"
EXEXSAL2 <-EXEXSAL2[,-1]
names(EXEXSAL2)
## [1] "Y" "X1" "X2" "X3" "X4" "X5" "X6" "X7" "X8" "X9" "X10"
reg_full<-lm(Y~., data=EXEXSAL2)
plot(reg_full)
The dataset describes the attibutes of various cars and how these relate to the dependent variable mpg i.e. how to things like weight, no of cylinders and no of gears affect miles per gallon (mpg). For this example we will use mpg (mpg) vs weight (wt).
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 4.0.3
data("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 1
# Fitting the Regression Line and its Residuals
d <- mtcars
fit <- lm(mpg ~ wt, data = d) # fit the model
d$predicted <- predict(fit) # Save the predicted values
d$residuals <- residuals(fit) # Save the residual values
ggplot(d, aes(x = wt, y = mpg)) +
geom_smooth(method = "lm", se = FALSE, color = "lightgrey") + # regression line
geom_segment(aes(xend = wt, yend = predicted), alpha = .2) + # draw line from point to line
geom_point(aes(color = abs(residuals), size = abs(residuals))) + # size of the points
scale_color_continuous(low = "green", high = "red") + # colour of the points mapped to residual size - green smaller, red larger
guides(color = FALSE, size = FALSE) + # Size legend removed
geom_point(aes(y = predicted), shape = 1) +
theme_bw()
## `geom_smooth()` using formula 'y ~ x'
summary(fit)
##
## Call:
## lm(formula = mpg ~ wt, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.5432 -2.3647 -0.1252 1.4096 6.8727
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 37.2851 1.8776 19.858 < 2e-16 ***
## wt -5.3445 0.5591 -9.559 1.29e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared: 0.7528, Adjusted R-squared: 0.7446
## F-statistic: 91.38 on 1 and 30 DF, p-value: 1.294e-10
# Residuals vs Fitted Plot
plot(fit, which=1, col=c("blue")) # Residuals vs Fitted Plot
plot(fit, which=2, col=c("red")) # Q-Q Plot
plot(fit, which=3, col=c("blue")) # Scale-Location Plot
plot(fit, which=5, col=c("blue")) # Residuals vs Leverage