library(Stat2Data)
library(ggplot2) 
#------------------------------------------------------------------------

#Example 1: Can age and height predict sales price in horses?

#Get data
data(HorsePrices)

#Test Assumptions

#Independence
#depends on sample design and taken to be valid here

#Normality
hist(HorsePrices$Price, col="brown") #not great
hist(log(HorsePrices$Price), col="brown") #not much better
hist(sqrt(HorsePrices$Price), col="brown") #good enough
HorsePrices$sPrice <-sqrt(HorsePrices$Price)

#Multicollinearity
round(cor(na.omit(HorsePrices[,3:4])),2) #not correlated

#Linear Relationship and Homeoscedasticity
plot(sPrice~Age, data=HorsePrices) #Non-linearity present
plot(sPrice~Height, data=HorsePrices) #Heteroscedascicity present
lm1 <- lm(sPrice~Age+Height, data=HorsePrices)
plot(fitted(lm1), residuals(lm1)); abline(h=0, lty=2) #problematic

#Multiple Linear Regression
lm1 <- lm(sPrice~Age+Height, data=HorsePrices)
summary(lm1)
# Variable    Beta    t-value   p-value
# Age         -6.0    -4.6      <0.0001
# Height      43.4     8.8      <0.0001
  #do not trust because assumptions are not met
#------------------------------------------------------------------------


#Example 2: Can skull length, bill depth, and bill length predict overall body mass in blue jays?

#Get data
data(BlueJays)

#Test Assumptions

#Independence
#depends on sample design and taken to be valid here

#Normality
hist(BlueJays$Mass, col='blue') #good

#Multicollinearity
Jay_Xvars <-data.frame(BlueJays$Skull, BlueJays$BillDepth, BlueJays$BillLength)
round(cor(Jay_Xvars),2) #not highly correlated

#Linear Relationship and Homeoscedasticity
plot(Mass~Skull, data=BlueJays)      #looks good
plot(Mass~BillDepth, data=BlueJays)  #looks good
plot(Mass~BillLength, data=BlueJays) #looks good
lm2 <- lm(Mass~Skull + BillDepth + BillLength, data=BlueJays)
plot(fitted(lm2), residuals(lm2)); abline(h=0, lty=2) #looks good

#Multiple Linear Regression
lm2 <- lm(Mass~Skull + BillDepth + BillLength, data=BlueJays)
summary(lm2)
# Variable      Beta    t-value   p-value
# Skull         2.4     6.8      <0.0001
# BillDepth     2.8     2.8       0.0063
# BillLength    0.9     2.9       0.0051
#------------------------------------------------------------------------


#Example 3: Can SAT score, ACT score, adjusted GPA, and HS class size predict math placement exam scores in college students?

#Get data
data(MathPlacement)

#Test Assumptions

#Independence
#depends on sample design and taken to be valid here

#Normality
hist(MathPlacement$PlcmtScore, col="green") #looks good

#Multicollinearity
Math_Xvars <-data.frame(MathPlacement$SATM, MathPlacement$ACTM, MathPlacement$GPAadj, MathPlacement$Size)
round(cor(na.omit(Math_Xvars)),2) #SAT and ACT highly correlated
Math_vars2 <-data.frame(MathPlacement$PlcmtScore ,MathPlacement$SATM, MathPlacement$ACTM)
round(cor(na.omit(Math_vars2)),2) #ACT more highly correlated with Placement Score
  #Keep ACT
  #Drop SAT

#Linear Relationship and Homeoscedasticity
plot(PlcmtScore~ACTM, data=MathPlacement) #good
plot(PlcmtScore~GPAadj, data=MathPlacement) #good enough
plot(PlcmtScore~Size, data=MathPlacement) #good

lm3 <- lm(PlcmtScore~ACTM + GPAadj + Size, data=MathPlacement)
plot(fitted(lm3), residuals(lm3)); abline(h=0, lty=2) #some Heteroskedastisticity

#Multiple Linear Regression
lm3 <- lm(PlcmtScore~ACTM + GPAadj + Size, data=MathPlacement)
summary(lm3)
# Variable    Beta    t-value   p-value
# ACT         1.6     67.7      <0.0001
# GPA         1.1     56.1      <0.0001
# Size       <0.1     <0.1       0.9950
  #grain of salt trust because assumptions tilted