Tutorial 6

Q1

X1 = c(1.38,1.26,1.51,1.46,1.61,1.59)
X2 = c(1.36,1.28,1.41,1.39,1.51,1.44)
n1 = c(41,41,33,45,18,23)
n2 = c(40,38,35,46,20,23)

s = c(0.22,0.25,0.31,0.28,0.34,0.46,0.28,0.19,0.27,0.33,0.41,0.30)
n = c(41,41,33,45,18,23,40,38,35,46,20,23)
s2 = sum(s^2 *(n-1))/(sum(n)-12)

d = X1-X2
w = (n1*n2)/(n1+n2)
dbar = sum(w*d)/sum(w)

iss = sum(w*(d-dbar)^2)

F.iss = iss/5/s2

F.tss = dbar^2*sum(w)/s2


Q3 

x1=c(29.33333,28.25,20.4)
x2=c(28,33.5,18.166666667)
x3=c(16.333333333,4.4,8.5)
x4=c(13.6,12.83333333,14.2)
n1=c(6,4,5)
n2=c(5,4,6)
n3=c(3,5,4)
n4=c(5,6,5)
sd1 =c(13.02,5.85,13.37)
sd2 =c(10.98,2.08,12.53)
sd3 =c(14.19,6.91,9)
sd4 =c(10.55,10.34,8.93)
sd = c(sd1,sd2,sd3,sd4)
x=c(x1,x2,x3,x4)
n=c(n1,n2,n3,n4)

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"))
lm.fit=lm(x~S+T, x=T, weights=n)
anova(lm.fit)
V=vcov(lm.fit)

(i)

sd =c(13.02,5.85,13.37,10.98,2.08,12.53,
      14.19,6.91,9,10.55,10.34,8.93)
s2 = sum(sd^2*(n-1))/(sum(n)-12)

(iii)

V.b = V[4:6,4:6]/117.88*s2

b = lm.fit$coef[4:6]

# The three contrasts corresponding to:
# C1:  b[3]
# C2:  b[1]-b[3]
# C3:  b[2]-b[3]

c1 = c(0,0,1)
c2 = c(1,0,-1)
c3 = c(0,1,-1)

L1 = t(c1)%*%b / sqrt(  t(c1)%*%V.b%*%c1 ) 
L2 = t(c2)%*%b / sqrt(  t(c2)%*%V.b%*%c2 ) 
L3 = t(c3)%*%b / sqrt(  t(c3)%*%V.b%*%c3 ) 

c(L1,L2,L3)

#  Dunnett's critical value:  d(3,40,0.05) = 2.44, d(3,60,0.05) = 2.41

Dunnett.d = (46-40)*(2.41-2.44)/(60-40)+2.44