Introduction to R for Data Science :: Session 6 [Linear Regression in R]

Introduction to R for Data Science

Lecturers

dipl. ing Branko Kovač

Data Analyst at CUBE/Data Science Mentor

at Springboard

Data Science zajednica Srbije

[email protected]

dr Goran S. Milovanović

Data Scientist at DiploFoundation

Data Science zajednica Srbije

[email protected]

[email protected]

mailto:[email protected]



Linear Regression in R

• Exploratory Data Analysis

• Assumptions of the Linear Model

• Correlation

• Normality Tests

• Linear Regression

• Prediction, Confidence

Intervals, Residuals

• Influential Cases and

the Influence Plot

Intro to R for Data Science

Session 6: Linear Regression in R

# Introduction to R for Data Science # SESSION 6 :: 02 June, 2016

# clear rm(list=ls())

#### read data

library(datasets) data(iris)

### iris data set description: # https://stat.ethz.ch/R-manual/R-devel/library/iriss/html/iris.html

### Exploratory Data Analysis (EDA) str(iris)

summary(iris)


• Before modeling: Assumptions and Exploratory Data Analysis (EDA)




### EDA plots # plot layout: 2 x 2

par(mfcol = c(2,2)) # boxplot iris$Sepal.Length

boxplot(iris$Sepal.Length, horizontal = TRUE, xlab="Sepal Length")

# histogram: iris$Sepal.Length hist(iris$Sepal.Length,

main="", xlab="Sepal.Length", prob=T)

# overlay iris$Sepal.Length density function over the empirical distribution lines(density(iris$Sepal.Length),

lty="dashed", lwd=2.5, col="red")

EDA




• EDA






## Pearson correlation in R {base} cor1 <- cor(iris$Sepal.Length, iris$Petal.Length,

method="pearson") cor1

par(mfcol = c(1,1)) plot(iris$Sepal.Length, iris$Petal.Length, main = "Sepal Length vs Petal Length",

xlab = "Sepal Length", ylab = "Petal Length")

## Correlation matrix and treatment of missing data dSet <- iris # Remove one discrete variable

dSet$Species <- NULL # introduce NA in dSet$Sepal.Length[5]

dSet$Sepal.Length[5] <- NA # Pairwise and Listwise Deletion:

cor1a <- cor(dSet,use="complete.obs") # listwise deletion cor1a <- cor(dSet,use="pairwise.complete.obs") # pairwise deletion

cor1a <- cor(dSet,use="all.obs") # all observations - error

Correlation




library(Hmisc) cor2 <- rcorr(iris$Sepal.Length,

iris$Petal.Length, type="pearson")

cor2$r # correlations cor2$r[1,2] # that's what you need, right cor2$P # significant at

cor2$n # num. observations # NOTE: rcorr uses Pairwise deletion!

# Correlation matrix cor2a <- rcorr(as.matrix(dSet),

type="pearson") # NOTE: as.matrix # select significant at alpha == .05

w <- which(!(cor2a$P<.05),arr.ind = T) cor2a$r[w] <- NA cor2a$P # compare w.

cor2a$r

Correlation {Hmisc}




# Linear Regression: lm() # Predicting: Petal Length from Sepal Length

reg <- lm(Petal.Length ~ Sepal.Length, data=iris) class(reg)

summary(reg) coefsReg <- coefficients(reg) coefsReg

slopeReg <- coefsReg[2] interceptReg <- coefsReg[1]

# Prediction from this model newSLength <- data.frame(Sepal.Length = runif(100,

min(iris$Sepal.Length), max(iris$Sepal.Length))

) # watch the variable names in the new data.frame! predictPLength <- predict(reg, newSLength) predictPLength

Linear Regression with lm()




# Standardized regression coefficients {QuantPsych} library(QuantPsyc)

lm.beta(reg)

# Reminder: standardized regression coefficients are... # What you would obtain upon performing linear regression over standardized variables # z-score in R

zSLength <- scale(iris$Sepal.Length, center = T, scale = T) # computes z-score zPLength <- scale(iris$Petal.Length, center = T, scale = T) # again; ?scale

# new dSet w. standardized variables dSet <- data.frame(Sepal.Length <- zSLength, Petal.Length <- zPLength)

# Linear Regression w. lm() over standardized variables reg1 <- lm(Petal.Length ~ Sepal.Length, data=dSet)

summary(reg1) # compare coefficients(reg1)[2] # beta from reg1

lm.beta(reg) # standardized beta w. QuantPscy lm.beta from reg

Standardized Regression Coefficients




# plots w. {base} and {ggplot2} library(ggplot2)

# Predictor vs Criterion {base} plot(iris$Sepal.Length, iris$Petal.Length,

main = "Petal Length vs Sepal Length", xlab = "Sepal Length", ylab = "Petal Length"

) abline(reg,col="red")

# Predictor vs Criterion {ggplot2} ggplot(data = iris,

aes(x = Sepal.Length, y = Petal.Length)) + geom_point(size = 2, colour = "black") +

geom_point(size = 1, colour = "white") + geom_smooth(aes(colour = "red"), method='lm') +

ggtitle("Sepal Length vs Petal Length") + xlab("Sepal Length") + ylab("Petal Length") +

theme(legend.position = "none")

Plots {base} vs {ggplot2}



Plots {base} vs {ggplot2}




# Predicted vs. residuals {ggplot2} predReg <- predict(reg) # get predictions from reg

resReg <- residuals(reg) # get residuals from reg # resStReg <- rstandard(reg) # get residuals from reg

plotFrame <- data.frame(predicted = predReg, residual = resReg); ggplot(data = plotFrame,

aes(x = predicted, y = residual)) + geom_point(size = 2, colour = "black") +

geom_point(size = 1, colour = "white") + geom_smooth(aes(colour = "blue"), method='lm',

se=F) + ggtitle("Predicted vs Residual Lengths") +

xlab("Predicted Lengths") + ylab("Residual") + theme(legend.position = "none")

Predicted vs Residuals



Predicted vs Residuals




## Detect influential cases infReg <- as.data.frame(influence.measures(reg)$infmat)

# Cook's Distance: Cook and Weisberg (1982): # values greater than 1 are troublesome

wCook <- which(infReg$cook.d>1) # we're fine here # Average Leverage = (k+1)/n, k - num. of predictors, n - num. observations # Also termed: hat values, range: 0 - 1

# see: https://en.wikipedia.org/wiki/Leverage_%28statistics%29 # Various criteria (twice the leverage, three times the average...)

wLev <- which(infReg$hat>2*(2/length(iris$price))) # we seem to be fine here too... ## Influence plot infReg <- as.data.frame(influence.measures(reg)$infmat)

plotFrame <- data.frame(residual = resStReg, leverage = infReg$hat,

cookD = infReg$cook.d)

Infulential Cases + Infulence Plot




ggplot(plotFrame, aes(y = residual,

x = leverage)) + geom_point(size = plotFrame$cookD*100, shape = 1) +

ggtitle("Influence Plot\nSize of the circle corresponds to Cook's distance") + theme(plot.title = element_text(size=8, face="bold")) + ylab("Standardized Residual") + xlab("Leverage")

Infulence Plot



Infulence Plot



Education

Introduction to R for Data Science :: Session 6 [Linear Regression in R]