Tyrha FINAL.R

Tyrha — Dec 15, 2013, 11:42 PM

setwd("~/final")

Error: cannot change working directory

attach(final)

Error: object 'final' not found

summary(final)

Error: object 'final' not found

log_Price=log(Price)

Error: object 'Price' not found

log_Beds=log(Beds)

Error: object 'Beds' not found

log_Price[is.infinite(log_Price)==T] <- NA

Error: object 'log_Price' not found

log_Beds[is.infinite(log_Beds)==T] <- NA

Error: object 'log_Beds' not found

## Q1: non-constant variance, non-normality ##
g <- lm(log_Beds~log_Price + Baths + Year + Size)

Error: object 'log_Beds' not found

# NON-CONSTANT VARIANCE
# 1. Use graphs
plot(fitted(g), abs(residuals(g)), xlab="Fitted", ylab="|Residuals|")

Error: object 'g' not found

# 2. Use tests
summary(lm(abs(residuals(g)) ~ fitted(g))) 

Error: object 'g' not found

# Checking if nonconstant variance is related to a predictor Beds

par(mfrow=c(2,3))
plot(log_Beds, log_Beds, xlab="Beds", ylab="Residuals")

Error: object 'log_Beds' not found

plot(log_Beds, Price, xlab="Price", ylab="Residuals")

Error: object 'log_Beds' not found

plot(log_Beds, Baths, xlab="Baths", ylab="Residuals")

Error: object 'log_Beds' not found

plot(log_Beds, Size, xlab="Size", ylab="Residuals")

Error: object 'log_Beds' not found

plot(log_Beds, Price_Sqft, xlab="Price_Sqft", ylab="Residuals")

Error: object 'log_Beds' not found

plot(log_Beds, Year, xlab="Year", ylab="Residuals")

Error: object 'log_Beds' not found

# NON-NORMALITY #
# 1. QQ-plots for detecting non-normality 
qqnorm(residuals(g), ylab="Residuals")

Error: object 'g' not found

qqline(residuals(g)) 

Error: object 'g' not found

# The histogram is not suitable for detecting nonnormality
hist(residuals(g)) 

Error: object 'g' not found

par(mfrow=c(1,1))

# 2. Test of non-normal errors
shapiro.test(residuals(g))  

Error: object 'g' not found

# 3. Makes adjustments (improve the non-constant variance)
gg <- lm(log_Beds~log_Price + Baths + Size + Price_Sqft + Year)

Error: object 'log_Beds' not found

gs <- lm(sqrt(log_Beds)~log_Price + Baths + Size + Price_Sqft + Year)

Error: object 'log_Beds' not found

par(mfrow=c(2,1))
plot(fitted(gg), residuals(gg), xlab="Fitted", ylab="Residuals")

Error: object 'gg' not found

plot(fitted(gs), residuals(gs), xlab="Fitted", ylab="Residuals")

Error: object 'gs' not found

## Q2: Unusual observations ##
# 1. Use graphs
# 2. Leverage, Cook's distance, and added variable plots:
# Cook's Distance for detecting influential outliers #
cook <- cooks.distance(g)

Error: object 'g' not found

# Half normal plot of Cook's Distance with labels of three largest values #
halfnorm(cook,3,labs=Beds$Price,ylab="Cook's distance")

Error: could not find function "halfnorm"

# Model fit excluding observation with largest Cook's Distance #
g1 <- lm(log_Beds~log_Price + Baths + Size + Price_Sqft + Year, subset=(cook < max(cook)))

Error: object 'log_Beds' not found

# Comparison of model fits with and without influential observation #
coef(g1); coef(g); summary(g1); summary(g) 

Error: object 'g1' not found

# Added variable plot #
prplot(g,1) # The most important predictor is Beds

Error: could not find function "prplot"

halfnorm(lm.influence(g)$hat, labs=Beds$Price, ylab="Leverages")

Error: could not find function "halfnorm"

cook <- cooks.distance(g)

Error: object 'g' not found

## Q3: Generalized Least Squares(GLS) ##(GLS) is a statistical tool used for estimating
##the unknown variables in a linear regression model.

## Q4: Weighted Least Squares ## (WLS)
# WLS is another way of dealing with non-constant error variance is that
# when errors are uncorrelated but have unequal variance where the form of 
# the inequality is known, we can use weighted least squares to handle
# 2. Use a model for error variance as a function of a predictor
g <- lm(Beds~Price + Baths + Size + Price_Sqft + Year)

Error: object 'Beds' not found

g1 <- lm(log(residuals(g)^2)~Price + Baths + Size + Price_Sqft + Year)

Error: object 'g' not found

summary(g1)

Error: object 'g1' not found

# This model is marginally significant; reading from the results, it means that 
# the log of squared residuals is a function of Price_Sqft

# 3. Fit linear model using WLS
# Size is the variable that is going to be used as weight
Beds$Size[Beds$Size == "1450"] <- 1

Error: object 'Beds' not found

Beds$Size[Beds$Size == "1600"] <- 2

Error: object 'Beds' not found

Beds$Size[Beds$Size == "2100"] <- 3

Error: object 'Beds' not found

Beds$Size[Beds$Size == "2400"] <- 4

Error: object 'Beds' not found

Beds$Size[Beds$Size == "2200"] <- 5

Error: object 'Beds' not found

Beds$Size[Beds$Size == "1900"] <- 6

Error: object 'Beds' not found

Beds$Size[Beds$Size == "1800"] <- 7

Error: object 'Beds' not found

Beds$Size[Beds$Size == "2605"] <- 8

Error: object 'Beds' not found

# This syntax output is the results of the WLS regression
g <- lm(log_Beds~ Size + log_Price + Baths + Price_Sqft + Year,weight = 1/Size)

Error: object 'log_Beds' not found

coef(g)

Error: object 'g' not found

## Q5: Test of Lack of Fit ##
g <- lm(log_Beds~Size)

Error: object 'log_Beds' not found

summary(g)

Error: object 'g' not found

plot(log_Beds~Size)

Error: object 'log_Beds' not found

# Therefore, the regression of Beds on Size is below and found model fits well (Multiple R-squared:  0.1537; p-value: 0.02648)

ga <- lm(log_Beds~factor(Size))

Error: object 'log_Beds' not found

summary(ga)

Error: object 'ga' not found

# Fitting the data with a group means model to see if there is any improved fit
# Results reveal that there is indeed an improved fit (Multiple R-squared:  0.9738; p-value: .0539)

# Testing the difference between model(g) and model(ga)
anova(g, ga)

Error: object 'g' not found

#Therefore, Model 2 is a better fitting model than Model 1.

## Q6: Robust Regression ##
# Variables used: Beds, Price, Baths, Size, Price_Sqft, Year
g <- lm(log_Beds~log_Price + Baths + Size + Price_Sqft + Year)

Error: object 'log_Beds' not found

summary(g)$coef

Error: object 'g' not found

shapiro.test(residuals(g))

Error: object 'g' not found

library(MASS)
gr <- rlm(log_Beds~log_Price + Baths + Size + Price_Sqft + Year)

Error: object 'log_Beds' not found

summary(gr)

Error: object 'gr' not found

# Huber method of regression: 
gr <- rq(log_Beds~log_Price + Baths + Size + Price_Sqft + Year)

Error: could not find function "rq"

summary(gr)$coef

Error: object 'gr' not found

#LAD method regression: Shows some improvement
ltsreg(log_Beds~log_Price + Baths + Size + Price_Sqft + Year)$coef

Error: object 'log_Beds' not found

# Comparison of Huber, LAD, LTS
plot(log_Beds ~ log_Price)

Error: object 'log_Beds' not found

plot(log_Beds ~ log_Price)

Error: object 'log_Beds' not found

abline(lm(log_Beds ~ log_Price)$coef) # LS

Error: object 'log_Beds' not found

abline(rlm(log_Beds ~ log_Price)$coef, lty=2) # Huber

Error: object 'log_Beds' not found

abline(reg(log_Beds ~ log_Price)$coef, lty=5) # LAD

Error: could not find function "reg"

abline(ltsreg(log_Beds ~ log_Price)$coef, lty=7) # LTS

Error: object 'log_Beds' not found

## Q7: Automated Variable Selection ##
# Variables used:Beds, Price, Baths, Size, Price_Sqft, Year
# 1. Role of ANOVA in Big vs. Small models
g1 <- lm(log_Beds~log_Price + Baths + Size + Price_Sqft + Year)

Error: object 'log_Beds' not found

summary(g1)

Error: object 'g1' not found

# Baths is least sigificant, so we remove it in a smaller model
g2 <- lm(log_Beds~log_Price + Size + Price_Sqft + Year)

Error: object 'log_Beds' not found

summary(g2)

Error: object 'g2' not found

#p value is slightly marginally significant when you remove "Baths"

g3 <- lm(log_Beds~log_Price + Size + Price_Sqft)

Error: object 'log_Beds' not found

summary(g3)

Error: object 'g3' not found

# Removing Year; makes the p value of the model marginally significant

# 2. Backward variable selection
g <- lm(log_Beds~log_Price + Baths + Size + Price_Sqft + Year)

Error: object 'log_Beds' not found

summary(g)

Error: object 'g' not found

#Remove Baths
g <- lm(log_Beds~log_Price + Size + Price_Sqft + Year)

Error: object 'log_Beds' not found

summary(g)

Error: object 'g' not found

#Remove Year
g <- lm(log_Beds~log_Price + Size + Price_Sqft)

Error: object 'log_Beds' not found

summary(g)

Error: object 'g' not found

#Therefore, when you remove Baths and Year, the model is marginally significant
#Remove Size
g <- lm(log_Beds~log_Price + Price_Sqft)

Error: object 'log_Beds' not found

summary(g)

Error: object 'g' not found

#Overall, once you remove Baths, Year and Size, the model becomes significant with a p=.02726 (p<.05)

# 3. Stepwise regression
g <- lm(log_Beds~log_Price + Baths + Price_Sqft + Size + Year)

Error: object 'log_Beds' not found

step(g)

Error: object 'g' not found

# 'AIC" is a criterion-based procedure and stands for Akaike Information Criterion: 
# Selected variables are listed in the last step when AIC=-99.72: Price
# Essentially, Stepwise regression had the same results that backward variable selection had

## Q8: Cross - Validation ##
# Variables used: Beds, Price, Baths, Price_Sqft, Size, Year
# Cross-Validation- the goal of cross-validation is to predict how well the model will perform based on test or practice datasets
# 3. 3-fold cross-validation for this dataset
g <- lm(MV~Beds + Price + Baths + Price_Sqft + Size + Year)

Error: object 'MV' not found

g.step <- step(g)

Error: object 'g' not found

summary(g.step)

Error: object 'g.step' not found

#Cross Validation is significant with the p=.008 (p<.05)

par(mfrow=c(1,1))