Splus handout:  Manipulating matrices

Some useful Splus commands:
 +matrix multiplication in Splus:  %*%
 +matrix inversion (and more):  solve()
 +transpose: t()

Recall the boiling point data from Splus handout #3

#X=boiling point (degrees F)
#Y=log(barometric pressure) (inches of mercury)
#17 locations in the Alps and in Scotland
#_________________________________________________________________________
> bp<-read.table("/home/fac/d/jah/DATA/alr/alr.3",header=T)
> bp$pressure<-bp$pressure*100
> 
> X<-cbind(rep(1,17),bp$boil)
> Y<-cbind(bp$pressure)
> n<-length(Y)

#_________________________________________________________________________
#Compute beta.hat

> XtX<-t(X)%*%X
> beta.hat<- solve(XtX)%*%t(X)%*%Y
> beta.hat
            [,1] 
[1,] -42.1308708
[2,]   0.8954625


> #compare to coefficients from lm:
> bp.lm<-lm(pressure~boil,bp)
> coef(bp.lm)
 (Intercept)      boil 
   -42.13087 0.8954625
#_________________________________________________________________________
#Use hat matrix

> H<-X%*%solve(XtX)%*%t(X)
> y.hat<-H%*%Y
> y.hat
          [,1] 
 [1,] 132.0366
 [2,] 131.8575
 [3,] 135.0812
 [4,] 135.5289
 [5,] 136.4244
 [6,] 136.8721
 [7,] 137.7676
 [8,] 137.9466
 [9,] 138.2153
[10,] 138.1257
[11,] 140.1853
[12,] 141.0808
[13,] 145.4685
[14,] 144.6626
[15,] 146.5431
[16,] 147.6176
[17,] 147.8863


> #compare to the predicted values from Splus command:
>predict(bp.lm)
>         1        2        3        4        5        6        7        8 
 132.0366 131.8575 135.0812 135.5289 136.4244 136.8721 137.7676   137.9466
        9       10       11       12       13       14       15  	16 
 138.2153 138.1257 140.1853 141.0808 145.4685 144.6626 146.5431   147.6176
       17 
 147.8863


#_________________________________________________________________________
#Compute SSTO and SSE

> SSTO<-t(Y)%*%Y-1/n*t(Y)%*%J%*%Y
> SSTO
         [,1] 
[1,] 427.7628
> 
> 
> SSE<-t(Y)%*%Y-t(beta.hat)%*%t(X)%*%Y
> SSE
         [,1] 
[1,] 2.153319
> 
> 
> #compare to ANOVA output
> anova(bp.lm)
Analysis of Variance Table

Response: pressure

Terms added sequentially (first to last)
          Df Sum of Sq  Mean Sq  F Value Pr(F) 
boil       1  425.6095 425.6095 2964.792     0
Residuals 15    2.1533   0.1436