## Section 5.8 examples

## Grasses Example:

# These are actually within-block ranks, but for 
# Friedman's test we can treat them as if they are raw data:

grass.data <- matrix(c(4,3,2,1,
4,2,3,1,
3,1.5,1.5,4,
3,1,2,4,
4,2,1,3,
2,2,2,4,
1,3,2,4,
2,4,1,3,
3.5,1,2,3.5,
4,1,3,2,
4,2,3,1,
3.5,1,2,3.5),
nrow=12, ncol=4, byrow=T)

friedman.test(grass.data) 
# this assumes the columns represent the treatments and the rows represent the blocks

# The data could be given in long vector form, with vectors 
# for the treatment-group labels and the block labels:


grass.vec <- c(4,3,2,1,4,2,3,1,3,1.5,1.5,4,3,1,2,4,4,2,1,3,2,2,2,4,1,3,2,4,2,4,1,3,3.5,1,2,3.5,4,1,3,2,4,2,3,1,3.5,1,2,3.5)

grasstype <- rep(1:4,times=12)
homeowner <- rep(1:12, each=4)

friedman.test(grass.vec, groups = grasstype, blocks = homeowner)

## Multiple Comparisons:

alpha <- 0.05
b <- 12
k <- 4
Rj <- colSums(matrix(unlist(tapply(grass.vec,homeowner,rank)), nrow=b, ncol=k, byrow=T))
A1 <- sum(unlist(tapply(grass.vec,homeowner,rank))^2)

abs(outer(Rj, Rj, '-')) > qt(1-alpha/2, df=((b-1)*(k-1)), lower=T)*sqrt((2*(b*A1-sum(Rj^2)))/((b-1)*(k-1)))

## Which pairs of grasses are significantly different?