STA114/MTH136: Statistics Lab 10, Apr 5 2001

Today we will continue to analyze the asking prices of used Mercedes. The data was taken from the advertising pages of the Sunday Times a few years, and which represent cars for sale in the UK (mainly in and around London) in pounds. The asking prices are classified according to type/model of car, age of car (in six-month units based on date of registration), recorded mileage, and vendor.

Interest lies in explaining price, distinguishing models according to price, depreciation with age and mileage, possible varying depreciation rates across models of car, collinearity issues for age and mileage, possible vendor differences on asking prices, outliers, influential cases, predicting prices

Download data from here. If you have downloaded the data in lab 9, you don't have download it again this time.

Data columns:

  1. Case number 1,2, .... , n=54
  2. Asking price in pounds (NB: Note the "rounding" up to ..995, etc!)
  3. Type/Model of car (0=model 500, 1=450, 2=380, 3=280, 4=200)
  4. Age of car in six-month units (based on advertised registration date)
  5. Recorded mileage (in thousands of miles)
  6. Vendor (0,1,2,3 are different dealerships, 4 means sale by owner) 

merc <- read.table("mercedes")
i    <- order(merc[,5]); merc <- merc[i,]  # Sortin order of mileage
y    <- log(merc[,2])                      # Why *log* price?  (NB: # not $)
mod  <- merc[,3]                           # Model: 0=500, 1=450, ..., 4=200
age  <- merc[,4]                           # Age in units of 6-mo
mile <- merc[,5]                           # Mileage in thousands
vend <- merc[,6]                           # Four different vendors

Mod <-  as.factor(mod)			# Treat as category, not number

#------------------------------------------------------------------------ 
#First let's plot log(price) against miles for different models
plot( mile, y, xlab="Miles",ylab="Log Price",type='n')
for (j in 0:4) {
    ok <- mod==j;
    points(mile[ok], y[ok], pch=paste(j), col=j+1);
    fit <- lm(y[ok]~mile[ok]);
    abline(fit, col=j+1);
}

# do you think the slopes are different with miles ?
# we fit log(price) on mod + mile
# and then fit log(price) on mod + mile + mod/mile
# if the slopes are statistically different,
# we'd expect the second model is better than the first one
# the F-test would be significant

fit  <-  lm( y~ Mod + mile , x=T)
summary(fit)
newfit  <-  lm( y ~  Mod+mile+Mod/mile)    # Try also:      y ~ Mod/mile
summary(newfit, corr=F)

# Subset F test -- are all "interactions" significant?
# check where we got the numbers from
print(fit); 
print(newfit)
Qb <- 48*0.136^2; 
Qa <- 44*0.131^2; 
ssa <- 0.131^2; 
fobs <- (Qb-Qa)/(4*ssa)
1-pf(fobs,4,44)
# doesn't look statistically significant overall

#-----------------------------------------------------------------------
# Check also age:

 plot( mile, age, xlab="Miles",ylab="Age",type='n')
 for (j in 0:4) { i <- mod==j; points(mile[i],age[i],pch=paste(j),col=j+1) }

# compare models including Age variable rather than or with Miles 
# e.g.,
 fit <- lm(y ~ Mod+age)
 summary(fit)
# now maybe slopes are really not different ...

 pre <- predict.lm(fit,se.fit=T)
 plot( age, y, xlab="Age",ylab="Log Price",type='n')
 for (j in 0:4) {
     ok<-mod==j; points(age[ok],y[ok],pch=paste(j),col=j+1);
     lines(age[ok],pre$fit[ok],col=j+1);
   }

 summary( fit1 <- lm(y ~ Mod+age+Mod/age) , correlation=F)

# subset F test for different slopes:

  Qb  <- 48*0.1019^2; 
  Qa  <- 44*0.105^2; 
  ssa <- 0.105^2;
  fobs <- (Qb-Qa)/(4*ssa)
  1-pf(fobs,4,44)