# One cannot conclude from the variances of pre/post values on the variance of the pre-post difference
# without knowing/making assumptions about the correkation.

# Example: pre-values x, post values y
x <- c(0,0.5,1)
y <- c(2,1.5,1)
mean(x)
var(x)
sd(x)  
mean(y)
var(y)  
sd(y)  
# Changes d.yx
d.yx <- y-x
mean(d.yx) 
var(d.yx)  
sd(d.yx)   
# Permutation of the post values provides other post values with the same mean and variance:
z <- c(1,1.5,2)
mean(z)
var(z) 
sd(z) 
# New changes d.zx have the same mean, but different variance:
d.zx <- z-x
mean(d.zx)
var(d.zx)
sd(d.zx)

# The reason is the that the correlation differs:
cor(x,y)   # negative correlation
cor(x,z)   # positive correlation

opar <- par(mfrow=c(1,2),pty="s")
plot(c(1,2),c(x[1],y[1]),type="n",
     main="Pre- and post values",
     xlab=" ",ylab=" ",ylim=c(0,2))
for (i in 1:3){
  lines(c(1,2),c(x[i],y[i]))
}
legend(1,2,bty="n",expression(paste("Negative Correlation: ",rho," = -0.25")))

plot(c(1,2),c(x[1],z[1]),type="n",
     main="Pre- and post values",
     xlab=" ",ylab=" ",ylim=c(0,2))
for (i in 1:3){
  lines(c(1,2),c(x[i],z[i]))
}
legend(1,2,bty="n",expression(paste("Perfect positive Correlation: ",rho," = 1")))

par(opar)