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# Remove most objects from the working environment
rm(list = ls())
options(stringsAsFactors = F)
# 8.2.1. Fitting regression models with lm()
myfit <- lm(formula, data)
# 8.2.2. Simple linear regression
# code listing Simple linear regression
fit <- lm(weight ~ height, data = women)
colnames(women)
summary(fit)
# Call:
# lm(formula = weight ~ height, data = women)
#
# Residuals:
# Min 1Q Median 3Q Max
# -1.7333 -1.1333 -0.3833 0.7417 3.1167
#
# Coefficients:
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) -87.51667 5.93694 -14.74 1.71e-09 ***
# height 3.45000 0.09114 37.85 1.09e-14 ***
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Residual standard error: 1.525 on 13 degrees of freedom
# Multiple R-squared: 0.991, Adjusted R-squared: 0.9903
# F-statistic: 1433 on 1 and 13 DF, p-value: 1.091e-14
fitted(fit)
women$weight
residuals(fit)
# Figure 8.1.
plot(women$height, women$weight,
xlab = "Height (in inches)",
ylab = "Weight (in pounds)")
abline(fit)
# 8.2.3. Polynomial regression
fit2 <- lm(weight ~ height + I(height^2), data=women)
# code listing 8.2. Polynomial regression
fit2 <- lm(weight ~ height + I(height^2), data = women)
summary(fit2)
# Call:
# lm(formula = weight ~ height + I(height^2), data = women)
#
# Residuals:
# Min 1Q Median 3Q Max
# -0.50941 -0.29611 -0.00941 0.28615 0.59706
#
# Coefficients:
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) 261.87818 25.19677 10.393 2.36e-07 ***
# height -7.34832 0.77769 -9.449 6.58e-07 ***
# I(height^2) 0.08306 0.00598 13.891 9.32e-09 ***
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Residual standard error: 0.3841 on 12 degrees of freedom
# Multiple R-squared: 0.9995, Adjusted R-squared: 0.9994
# F-statistic: 1.139e+04 on 2 and 12 DF, p-value: < 2.2e-16
# Figure 8.2
plot(women$height, women$weight,
xlab = "Height (in inches)",
ylab = "Weight (in pounds)")
lines(women$height, women$weight)
# with library(car) in the "car" package
# Figure 8.3.
library(car)
scatterplot(weight ~ height,
data=women,
spread=FALSE, smoother=list(lty=2),
pch=19,
main="Women Age 30-39",
xlab="Height (inches)",
ylab="Weight (lbs.)")
# 8.2.4. Multiple linear regression
# code listing 8.3 Examining bivariate relationships
states <- as.data.frame(state.x77[,c("Murder", "Population",
"Illiteracy", "Income", "Frost")])
cor(states)
library(car)
# Figure 8.4.
scatterplotMatrix(states, spread=F, lty.smooth=2,
main = "Scatter Plot Matrix")
# code listing 8.4 Multiple linear regression
states <- as.data.frame(state.x77[,c("Murder", "Population",
"Illiteracy", "Income", "Frost")])
fit <- lm(Murder ~ Population + Illiteracy + Income + Frost,
data = states)
summary(fit)
# Call:
# lm(formula = Murder ~ Population + Illiteracy + Income + Frost,
# data = states)
#
# Residuals:
# Min 1Q Median 3Q Max
# -4.7960 -1.6495 -0.0811 1.4815 7.6210
#
# Coefficients:
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) 1.235e+00 3.866e+00 0.319 0.7510
# Population 2.237e-04 9.052e-05 2.471 0.0173 *
# Illiteracy 4.143e+00 8.744e-01 4.738 2.19e-05 ***
# Income 6.442e-05 6.837e-04 0.094 0.9253
# Frost 5.813e-04 1.005e-02 0.058 0.9541
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Residual standard error: 2.535 on 45 degrees of freedom
# Multiple R-squared: 0.567, Adjusted R-squared: 0.5285
# F-statistic: 14.73 on 4 and 45 DF, p-value: 9.133e-08
#===================================================================
# 8.2.5. Multiple linear regression with interactions
# code listing 8.5 Multiple linear regression with a significant interaction term
fit <- lm(mpg ~ hp + wt + hp:wt, data=mtcars)
summary(fit)
# 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-13
#====================================================================
# install.packages("effects")
library(effects)
# Figure 8.5.
plot(effect("hp:wt", fit,
vcov. = vcov,
list(wt=c(2.2,3.2,4.2))),
multiline=TRUE)
# 8.3. Regression diagnostics
fit <- lm(Murder ~ Population + Illiteracy + Income + Frost, data=states)
confint(fit)
# 2.5 % 97.5 %
# (Intercept) -6.552191e+00 9.0213182149
# Population 4.136397e-05 0.0004059867
# Illiteracy 2.381799e+00 5.9038743192
# Income -1.312611e-03 0.0014414600
# Frost -1.966781e-02 0.0208304170
# 8.3.1. A typical approach
fit <- lm(weight ~ height, data=women)
par(mfrow=c(2,2))
# Figure 8.6.
plot(fit)
fit2 <- lm(weight ~ height + I(height^2), data=women)
par(mfrow=c(2,2))
# Figure 8.7.
plot(fit2)
fit <- lm(Murder ~ Population + Illiteracy + Income + Frost, data=states)
par(mfrow=c(2,2))
# Figure 8.8.
plot(fit)
# 8.3.2. An enhanced approach
# Normality
library(car)
fit <- lm(Murder ~ Population + Illiteracy + Income + Frost, data=states)
# Figure 8.9.
qqPlot(fit, labels=row.names(states), id.method="identify",
simulate=TRUE, main="Q-Q Plot")
states["Nevada",]
# Murder Population Illiteracy Income Frost
# Nevada 11.5 590 0.5 5149 188
fitted(fit)["Nevada"]
# Nevada
# 3.878958
residuals(fit)["Nevada"]
# Nevada
# 7.621042
rstudent(fit)["Nevada"]
# Nevada
# 3.542929
# Normality
# code listing 8.6. Function for plotting studentized residuals
residplot <- function(fit, nbreaks=10) {
z <- rstudent(fit)
hist(z, breaks=nbreaks, freq=FALSE,
xlab="Studentized Residual",
main="Distribution of Errors")
rug(jitter(z), col="brown")
curve(dnorm(x, mean=mean(z), sd=sd(z)),
add=TRUE, col="blue", lwd=2)
lines(density(z)$x, density(z)$y,
col="red", lwd=2, lty=2)
legend("topright",
legend = c( "Normal Curve", "Kernel Density Curve"),
lty=1:2, col=c("blue","red"), cex=.7)
}
# Figure 8.10.
residplot(fit)
# Independence of Errors
durbinWatsonTest(fit)
# lag Autocorrelation D-W Statistic p-value
# 1 -0.2006929 2.317691 0.306
# Alternative hypothesis: rho != 0
# Linearity
library(car)
# Figure 8.11.
crPlots(fit)
# Homoscedasticity
# code listing 8.7. Assessing homoscedasticity
library(car)
ncvTest(fit)
# Non-constant Variance Score Test
# Variance formula: ~ fitted.values
# Chisquare = 1.746514, Df = 1, p = 0.18632
# Figure 8.12.
spreadLevelPlot(fit)
# Suggested power transformation: 1.209626
# 8.3.3. Global validation of linear model assumption
# code listing 8.8. Global test of linear model assumptions
# install.packages(gvlma)
library(gvlma)
gvmodel <- gvlma(fit)
summary(gvmodel)
# Call:
# lm(formula = Murder ~ Population + Illiteracy + Income + Frost,
# data = states)
#
# Residuals:
# Min 1Q Median 3Q Max
# -4.7960 -1.6495 -0.0811 1.4815 7.6210
#
# Coefficients:
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) 1.235e+00 3.866e+00 0.319 0.7510
# Population 2.237e-04 9.052e-05 2.471 0.0173 *
# Illiteracy 4.143e+00 8.744e-01 4.738 2.19e-05 ***
# Income 6.442e-05 6.837e-04 0.094 0.9253
# Frost 5.813e-04 1.005e-02 0.058 0.9541
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Residual standard error: 2.535 on 45 degrees of freedom
# Multiple R-squared: 0.567, Adjusted R-squared: 0.5285
# F-statistic: 14.73 on 4 and 45 DF, p-value: 9.133e-08
#
#
# ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
# USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
# Level of Significance = 0.05
#
# Call:
# gvlma(x = fit)
#
# Value p-value Decision
# Global Stat 2.7728 0.5965 Assumptions acceptable.
# Skewness 1.5374 0.2150 Assumptions acceptable.
# Kurtosis 0.6376 0.4246 Assumptions acceptable.
# Link Function 0.1154 0.7341 Assumptions acceptable.
# Heteroscedasticity 0.4824 0.4873 Assumptions acceptable.
#============================================================
# 8.3.4. Multicollinearity
# code listing 8.9. Evaluating multicollinearity
library(car)
vif(fit)
# Population Illiteracy Income Frost
# 1.245282 2.165848 1.345822 2.082547
sqrt(vif(fit)) > 2
# Population Illiteracy Income Frost
# FALSE FALSE FALSE FALSE
#============================================================
# 8.4. Unusual observations
# 8.4.1. Outliers
library(car)
outlierTest(fit)
# 8.4.2. High leverage points
# Figure 8.13.
hat.plot <- function(fit) {
p <- length(coefficients(fit))
n <- length(fitted(fit))
plot(hatvalues(fit), main="Index Plot of Hat Values")
abline(h=c(2,3)*p/n, col="red", lty=2)
identify(1:n, hatvalues(fit), names(hatvalues(fit)))
}
hat.plot(fit)
# 8.4.3. Influential observations
# Figure 8.14.
cutoff <- 4/(nrow(states)-length(fit$coefficients)-2)
plot(fit, which=4, cook.levels=cutoff)
abline(h=cutoff, lty=2, col="red")
# Figure 8.15.
library(car)
avPlots(fit, ask=FALSE, onepage=TRUE, id.method="identify")
# Figure 8.16. Influence plot.
library(car)
influencePlot(fit, id.method="identify", main="Influence Plot",
sub="Circle size is proportional to Cook's distance")
# 8.5.2. Transforming variables
# code listing 8.10. Box–Cox transformation to normality
library(car)
summary(powerTransform(states$Murder))
# bcPower Transformation to Normality
# Est Power Rounded Pwr Wald Lwr Bnd Wald Upr Bnd
# states$Murder 0.6055 1 0.0884 1.1227
#
# Likelihood ratio test that transformation parameter is equal to 0
# (log transformation)
# LRT df pval
# LR test, lambda = (0) 5.665991 1 0.017297
#
# Likelihood ratio test that no transformation is needed
# LRT df pval
# LR test, lambda = (1) 2.122763 1 0.14512
boxTidwell(Murder~Population+Illiteracy,data=states)
# MLE of lambda Score Statistic (z) Pr(>|z|)
# Population 0.86939 -0.3228 0.7468
# Illiteracy 1.35812 0.6194 0.5357
#
# iterations = 19
# 8.6.1. Comparing models
# code listing 8.11. Comparing nested models using the anova() function
fit1 <- lm(Murder ~ Population + Illiteracy + Income + Frost,
data=states)
fit2 <- lm(Murder ~ Population + Illiteracy, data=states)
anova(fit1,fit2)
# Analysis of Variance Table
#
# Model 1: Murder ~ Population + Illiteracy + Income + Frost
# Model 2: Murder ~ Population + Illiteracy
# Res.Df RSS Df Sum of Sq F Pr(>F)
# 1 45 289.17
# 2 47 289.25 -2 -0.078505 0.0061 0.9939
# code listing 8.12. Comparing models with the AIC
fit1 <- lm(Murder ~ Population + Illiteracy + Income + Frost,
data=states)
fit2 <- lm(Murder ~ Population + Illiteracy, data=states)
AIC(fit1, fit2)
# df AIC
# fit1 6 241.6429
# fit2 4 237.6565
# 8.6.2. Variable selection
# code listing 8.13. Backward stepwise selection
library(MASS)
fit1 <- lm(Murder ~ Population + Illiteracy + Income + Frost,
data=states)
stepAIC(fit, direction = "backward")
# Start: AIC=97.75
# Murder ~ Population + Illiteracy + Income + Frost
#
# Df Sum of Sq RSS AIC
# - Frost 1 0.021 289.19 95.753
# - Income 1 0.057 289.22 95.759
# <none> 289.17 97.749
# - Population 1 39.238 328.41 102.111
# - Illiteracy 1 144.264 433.43 115.986
#
# Step: AIC=95.75
# Murder ~ Population + Illiteracy + Income
#
# Df Sum of Sq RSS AIC
# - Income 1 0.057 289.25 93.763
# <none> 289.19 95.753
# - Population 1 43.658 332.85 100.783
# - Illiteracy 1 236.196 525.38 123.605
#
# Step: AIC=93.76
# Murder ~ Population + Illiteracy
#
# Df Sum of Sq RSS AIC
# <none> 289.25 93.763
# - Population 1 48.517 337.76 99.516
# - Illiteracy 1 299.646 588.89 127.311
#
# Call:
# lm(formula = Murder ~ Population + Illiteracy, data = states)
#
# Coefficients:
# (Intercept) Population Illiteracy
# 1.6515497 0.0002242 4.0807366
# code listing 8.14. All subsets regression
# install.packages("leaps")
library(leaps)
leaps <-regsubsets(Murder ~ Population + Illiteracy + Income +
Frost, data=states, nbest=4)
plot(leaps, scale="adjr2") # Figure 8.17
library(car)
subsets(leaps, statistic="cp",
main="Cp Plot for All Subsets Regression")
abline(1,1,lty=2,col="red")
# 8.7.1. Cross-validation
# code listing 8.15. Function for k-fold cross-validated R-square
shrinkage <- function(fit, k=10){
require(bootstrap)
theta.fit <- function(x,y){lsfit(x,y)}
theta.predict <- function(fit,x){cbind(1,x)%*%fit$coef}
x <- fit$model[,2:ncol(fit$model)]
y <- fit$model[,1]
results <- crossval(x, y, theta.fit, theta.predict, ngroup=k)
r2 <- cor(y, fit$fitted.values)^2
r2cv <- cor(y, results$cv.fit)^2
cat("Original R-square =", r2, "\n")
cat(k, "Fold Cross-Validated R-square =", r2cv, "\n")
cat("Change =", r2-r2cv, "\n")
}
fit <- lm(Murder ~ Population + Income + Illiteracy + Frost, data=states)
shrinkage(fit)
# Original R-square = 0.5669502
# 10 Fold Cross-Validated R-square = 0.4240783
# Change = 0.1428719
fit2 <- lm(Murder ~ Population + Illiteracy, data=states)
shrinkage(fit2)
# Original R-square = 0.5668327
# 10 Fold Cross-Validated R-square = 0.5332059
# Change = 0.03362679
# 8.7.2. Relative importance
zstates <- as.data.frame(scale(states))
zfit <- lm(Murder~Population + Income + Illiteracy + Frost, data=zstates)
coef(zfit)
# (Intercept) Population Income Illiteracy Frost
# -2.054026e-16 2.705095e-01 1.072372e-02 6.840496e-01 8.185407e-03
# code listing 8.16. relweights() function
# for calculating relative importance of predictors
relweights <- function(fit,...){
R <- cor(fit$model)
nvar <- ncol(R)
rxx <- R[2:nvar, 2:nvar]
rxy <- R[2:nvar, 1]
svd <- eigen(rxx)
evec <- svd$vectors
ev <- svd$values
delta <- diag(sqrt(ev))
lambda <- evec %*% delta %*% t(evec)
lambdasq <- lambda ^ 2
beta <- solve(lambda) %*% rxy
rsquare <- colSums(beta ^ 2)
rawwgt <- lambdasq %*% beta ^ 2
import <- (rawwgt / rsquare) * 100
lbls <- names(fit$model[2:nvar])
rownames(import) <- lbls
colnames(import) <- "Weights"
barplot(t(import),names.arg=lbls,
ylab="% of R-Square",
xlab="Predictor Variables",
main="Relative Importance of Predictor Variables",
sub=paste("R-Square=", round(rsquare, digits=3)),
...)
return(import)
}
relweights2 <- function(fit,...){
R <- cor(fit$model)
nvar <- ncol(R)
rxx <- R[2:nvar, 2:nvar]
rxy <- R[2:nvar, 1]
svd <- eigen(rxx)
evec <- svd$vectors
ev <- svd$values
delta <- diag(sqrt(ev))
lambda <- evec %*% delta %*% t(evec)
lambdasq <- lambda ^ 2
beta <- solve(lambda) %*% rxy
rsquare <- colSums(beta ^ 2)
rawwgt <- lambdasq %*% beta ^ 2
import <- (rawwgt / rsquare) * 100
lbls <- names(fit$model[2:nvar])
rownames(import) <- lbls
colnames(import) <- "Weights"
dotplot(t(import),names.arg=lbls,
ylab="% of R-Square",
xlab="Predictor Variables",
main="Relative Importance of Predictor Variables",
sub=paste("R-Square=", round(rsquare, digits=3)),
...)
return(import)
}
# code listing 8.17. Applying the relweights() function
fit <- lm(Murder ~ Population + Illiteracy + Income + Frost, data=states)
relweights(fit, col="lightgrey")
# Weights
# Population 14.723401
# Illiteracy 59.000195
# Income 5.488962
# Frost 20.787442
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