Introduction to R for Data Science :: Session 7 [Multiple 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 Serbia
branko.kovac@gmail.com
dr Goran S. Milovanović
Data Scientist at DiploFoundation
Data Science Serbia
goran.s.milovanovic@gmail.com
goranm@diplomacy.edu

MultipleLinear Regression in R
• Dummy coding of categorical predictors
• Multiple regression
• Nested models and Partial
F-test
• Partial and Part Correlation
• Multicolinearity
• {Lattice} plots
• Prediction, Confidence
Intervals, Residuals
• Influential Cases and
the Influence Plot
Intro to R for Data Science
Session 7: Multiple Linear Regression in R

########################################################
# Introduction to R for Data Science
# SESSION 7 :: 9 June, 2016
# Multiple Linear Regression in R
# Data Science Community Serbia + Startit
# :: Goran S. Milovanović and Branko Kovač ::
########################################################
#### read data
library(datasets)
library(broom)
library(ggplot2)
library(lattice)
#### load
data(iris)
str(iris)
MultipleRegression in R
• Problems with simple linear regression: iris dataset

#### simple linearregression:SepalLength vs Petal
Lenth
# Predictorvs 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 = "black"),
method='lm') +
ggtitle("Sepal Length vs Petal Length") +
xlab("Sepal Length") + ylab("Petal Length") +
theme(legend.position = "none")

# And now for something completelly different(but in
R)...
#### Problemswith linearregressionin iris
# Predictorvs Criterion {ggplot2} - group separation
ggplot(data = iris,
aes(x = Sepal.Length,
y = Petal.Length,
color = Species)) +
geom_point(size = 2) +
xlab("Sepal Length") + ylab("Petal Length")

# Predictorvs Criterion {ggplot2} - separate
regression lines
ggplot(data = iris,
aes(x = Sepal.Length,
y = Petal.Length,
colour=Species)) +
geom_smooth(method=lm) +
geom_point(size = 2) +
xlab("Sepal Length") + ylab("Petal Length")

### better... {lattice}
xyplot(Petal.Length ~ Sepal.Length | Species, #
{latice} xyplot
data = iris,
xlab = "Sepal Length", ylab = "Petal Length"
)

# Petal Length and SepalLength:Conditional
Densities
densityplot(~ Petal.Length | Species, # {latice} xyplot
data = iris,
plot.points=FALSE,
xlab = "Petal Length", ylab = "Density",
main = "P(Petal Length|Species)",
col.line = 'red'
)
densityplot(~ Sepal.Length | Species, # {latice} xyplot
data = iris,
plot.points=FALSE,
xlab = "Sepal Length", ylab = "Density",
main = "P(Sepal Length|Species)",
col.line = 'blue'
)
• Problems with simple linear regression:
iris dataset

# Linearregressionin subgroups
species <- unique(iris$Species)
w1 <- which(iris$Species == species[1]) # setosa
reg <- lm(Petal.Length ~ Sepal.Length, data=iris[w1,])
tidy(reg)
w2 <- which(iris$Species == species[2]) # versicolor
tidy(reg)
w3 <- which(iris$Species == species[3]) # virginica
tidy(reg)
• Simple linear regressions in sub-groups

#### Dummy Coding:Species in the iris dataset
is.factor(iris$Species)
levels(iris$Species)
reg <- lm(Petal.Length ~ Species, data=iris)
tidy(reg)
glance(reg)
# Neverforget whatthe regressioncoefficientfor a dummy variablemeans:
# It tells us aboutthe effectof moving from the baselinetowardsthe respectivereferencelevel!
# Here: baseline = setosa (cmp.levels(iris$Species)vs.the outputof tidy(reg))
# NOTE: watch for the order of levels!
levels(iris$Species) # Levels: setosa versicolor virginica
iris$Species <- factor(iris$Species,
levels = c("versicolor",
"virginica",
"setosa"))
# baseline is now:versicolor
reg <- lm(Petal.Length ~ Species, data=iris)
tidy(reg)# The regression coefficents (!): figure out whathas happened!

### anotherway to do dummy coding
rm(iris); data(iris) # ...justto fix the order of Species backto default
contrasts(iris$Species) = contr.treatment(3, base = 1)
contrasts(iris$Species) # this probably whatyou rememberfrom your stats class...
iris$Species <- factor(iris$Species,
levels = c ("virginica","versicolor","setosa"))
contrasts(iris$Species) = contr.treatment(3, base = 1)
# baseline is now:virginica
contrasts(iris$Species) # considercarefully whatyou need to do

### Petal.Length ~ Species(Dummy Coding)+ Sepal.Length
rm(iris); data(iris) # ...just to fix the order of Species backto default
reg <- lm(Petal.Length ~ Species + Sepal.Length, data=iris)
# BTW: since is.factor(iris$Species)==T,R does the dummy coding in lm() for you
regSum <- summary(reg)
regSum$r.squared
regSum$coefficients
# compare w. Simple LinearRegression
reg <- lm(Petal.Length ~ Sepal.Length, data=iris)
regSum <- summary(reg)
regSum$r.squared
regSum$coefficients
• Multiple regression with dummy-coded categorical predictors

### Comparingnestedmodels
reg1 <- lm(Petal.Length ~ Sepal.Length, data=iris)
reg2 <- lm(Petal.Length ~ Species + Sepal.Length, data=iris) # reg1 is nested under reg2
# terminology:reg2 is a "full model"
# this terminology will be used quite often in Logistic Regression
# NOTE: Nested models
# There is a set of coefficientsfor the nested model(reg1)such thatit
# can be expressedin terms of the full model(reg2); in our case it is simple
# HOME: - figure it out.
anova(reg1, reg2) # partial F-test; Speciescertainly has an effect beyond Sepal.Length
# NOTE: for partial F-test, see:
# http://pages.stern.nyu.edu/~gsimon/B902301Page/CLASS02_24FEB10/PartialFtest.pdf
• Comparison of nested models

#### Multiple Regression - by the book
# Following: http://www.r-tutor.com/elementary-statistics/multiple-linear-regression
# (that's from yourreading list, to remind you...)
data(stackloss)
str(stackloss)
# Data set description
# URL: https://stat.ethz.ch/R-manual/R-devel/library/datasets/html/stackloss.html
stacklossModel = lm(stack.loss ~ Air.Flow + Water.Temp + Acid.Conc.,
data=stackloss)
# let's see:
summary(stacklossModel)
glance(stacklossModel) # {broom}
tidy(stacklossModel) # {broom}
# predictnew data
obs = data.frame(Air.Flow=72, Water.Temp=20, Acid.Conc.=85)
predict(stacklossModel, obs)
• By the book: two or three continuous predictors…

# confidence intervals
confint(stacklossModel, level=.95) #
95% CI
confint(stacklossModel, level=.99) #
99% CI
# 95% CI for Acid.Conc.only
confint(stacklossModel, "Acid.Conc.",
level=.95)
# defaultregressionplots in R
plot(stacklossModel)
• By the book: two or three continuous predictors…

# multicolinearity
library(car) # John Fox's carpackage
VIF <- vif(stacklossModel)
VIF
sqrt(VIF)
# Variance Inflation Factor(VIF)
# The increasein the ***variance***of an regression ceoff.due to colinearity
# NOTE: sqrt(VIF)= how much larger the ***SE*** of a reg.coeff.vs. whatit would be
# if there were no correlationswith the other predictors in the model
# NOTE: lower_bound(VIF)= 1; no upperbound;VIF > 2 --> (Concerned== TRUE)
Tolerance <- 1/VIF # obviously,tolerance and VIF are redundant
Tolerance
# NOTE: you can inspectmulticolinearity in the multiple regressionmode
# by conductinga PrincipalComponentAnalysis overthe predictors;
# when the time is right.
• Assumptions: multicolinearity

#### R for partial and part (semi-partial)correlations
library(ppcor) # a good one;there are many ways to do this in R
#### partialcorrelation in R
dataSet <- iris
str(dataSet)
dataSet$Species <- NULL
irisPCor <- pcor(dataSet, method="pearson")
irisPCor$estimate # partialcorrelations
irisPCor$p.value # results of significancetests
irisPCor$statistic # t-test on n-2-k degrees offreedom ;k = num. of variablesconditioned
# partial correlation between x and y while controlling forz
partialCor <- pcor.test(dataSet$Sepal.Length, dataSet$Petal.Length,
dataSet$Sepal.Width,
method = "pearson")
partialCor$estimate
partialCor$p.value
partialCor$statistic
• Partial Correlation in R

#### semi-partialcorrelation in R
# NOTE: ... Semi-partialcorrelation is the correlation of two variables
# with variation from a third or more othervariables removedonly
# from the ***second variable***
# NOTE: The first variable <- rows, the secondvariable <-columns
# cf. ppcor:An R Packagefor a FastCalculationto Semi-partialCorrelation Coefficients(2015)
# SeonghoKim, BiostatisticsCore,Karmanos CancerInstitute,Wayne State University
# URL: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4681537/
irisSPCor <- spcor(dataSet, method = "pearson")
irisSPCor$estimate
irisSPCor$p.value
irisSPCor$statistic
partCor <- spcor.test(dataSet$Sepal.Length, dataSet$Petal.Length,
dataSet$Sepal.Width,
method = "pearson")
# NOTE: this is a correlation of dataSet$Sepal.Length w. dataSet$Petal.Length
# when the variance ofdataSet$Petal.Length(2nd variable)due to dataSet$Sepal.Width
# is removed!
partCor$estimate
partCor$p.value
• Part (semi-partial) Correlation in R

# NOTE: In multiple regression,this is the semi-partial(or part) correlation
# that you need to inspect:
# assume a modelwith X1, X2, X3 as predictors,and Y as a criterion
# You need a semi-partialof X1 and Y following the removalof X2 and X3 from Y
# It goes like this: in Step 1, you perform a multiple regression Y ~ X2 + X3;
# In Step 2, you take the residualsof Y, call them RY; in Step 3, you regress (correlate)
# RY ~ X1: the correlation coefficientthat you get from Step 3 is the part correlation
# that you're looking for.
• NOTE on semi-partial (part) correlation in multiple regression…

Introduction to R for Data Science :: Session 7 [Multiple Linear Regression in R]

Introduction to R for Data Science :: Session 7 [Multiple Linear Regression in R]

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