Weighted Least Squares (WLS)
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Data set:  from Graybill and Iyer (1994), page 576

Goal: investigate the relationship between carbon monoxide in the air
and number of automobiles for 13 cities of population > 50,000 that do
not have clean air programs.

Y=carbon monoxide in the air (units: parts per million)
X=number of automobiles (in thousands)

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Splus commands:

No new Splus commands, just a new option in lsfit and lm:

lsfit(x, y, wt=...)

lm(formula, data=, weights=...)
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carb<-read.table("/home/fac/d/jah/DATA/grayiyer/carbmon.dat",header=T)

>carb
      Y    X 
 1 5817  873
 2 1063  109
 3 2616  398
 4 2018  353
 5 3147  506
 6 7210 1026
 7 4339  862
 8 5153  742
 9 4450  786
10 5591  896
11 2747  377
12 3712  720
13 2354  655


#OLS model
carb.lsfit<-lsfit(carb[,2],carb[,1])
ls.print(carb.lsfit)

Residual Standard Error = 742.6138,  Multiple R-Square = 0.8365
N = 13,  F-statistic = 56.2719 on 1 and 11 df, p-value = 0

             coef  std.err t.stat p.value 
Intercept 40.5045 549.5987 0.0737  0.9426
        X  5.9846   0.7978 7.5015  0.0000


#RESIDUAL PLOTS FOR OLS MODEL
e.star<-ls.diag(carb.lsfit)$stud.res

par(mfrow=c(2,3))
plot(carb$X,carb$Y,xlab="# of automobiles",ylab="CO levels")
abline(carb.lsfit)
title("Fig 1: Scatter plot with fitted OLS reg. line")
plot(carb$X,e.star)
abline(h=0)
title("Fig 2: Residual plot for OLS")
qqnorm(e.star)
title("Fig 3: QQ plot for OLS")


#WLS model

#Note:  it is suspected that the standard deviation of carbon monoxide
#levels is related to the number of automobiles.  Researchers have
#suggested using weights equal to 1/x.

carb.lsfit2<-lsfit(carb[,2],carb[,1],wt=1/carb[,2])
ls.print(carb.lsfit2)

Residual Standard Error = 27.7958,  Multiple R-Square = 0.8975
N = 13,  F-statistic = 96.356 on 1 and 11 df, p-value = 0

              coef  std.err t.stat p.value 
Intercept 371.6209 297.5527 1.2489  0.2376
        X   5.4662   0.5569 9.8161  0.0000




#Residual plots for WLS model
e.star2<-ls.diag(carb.lsfit2)$stud.res

plot(carb$X,carb$Y,xlab="# of automobiles",ylab="CO levels")
abline(carb.lsfit2)
title("Fig 4: Scatter plot with fitted WLS reg. line")
plot(carb$X,e.star2)
abline(h=0)
title("Fig 5: Residual plot for WLS")
qqnorm(e.star2)
title("Fig 6: QQ plot for WLS")


#THE WLS USING THE LM COMMAND
carb.lm<-lm(Y~X,data=carb,weights=1/X)
summary(carb.lm)

Coefficients:
               Value Std. Error  t value Pr(>|t|) 
(Intercept) 371.6209 297.5527     1.2489   0.2376
          X   5.4662   0.5569     9.8161   0.0000

Residual standard error: 27.8 on 11 degrees of freedom
Multiple R-Squared: 0.8975 
F-statistic: 96.36 on 1 and 11 degrees of freedom, the p-value is 8.897e-07 

**SEE CLASS DISCUSSION REGARDING THE RESIDUAL PLOTS