x=c(29.33333,28.25,20.4, 
     28,33.5,18.166666667,
     16.333333333,4.4,8.5, 
     13.6,12.83333333,14.2)
n=c(6,4,5,5,4,6,3,5,4,5,6,5)
S=factor(rep(c(1,2,3),4))
T=factor(c(rep(1,3),rep(2,3),rep(3,3),rep(4,3)))

options(contrasts=c("contr.treatment","contr.poly"))

L1=lm(x~S*T, x=T, weights=n)
anova(L1)

L2=lm(x~S+T, x=T, weights=n)
summary(L2)
anova(L2)
V=vcov(L2)

VT=V[4:6,4:6]

VT/117.88*110.4564


b=L2$coef
bt=b[4:6]
t(bt)%*%solve(Vt)%*%bt*117.88