##### Demonstration Script #6a
##### MATH322
#####
##### A quick ANOVA example in R
##### 




### Illustrate multicollinearity 
#   and its issue

X = matrix( c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
			 1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,
			 0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,
			 0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,
			 0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1), ncol=5 )

det( t(X)%*%X )

solve( t(X)%*%X )






### Doing ANOVA in R

y = c(6,2,9,8,2,3,7,3,8,7,7,9,6,4,6,3)
x = c("A","A","A","A", "B","B","B","B", "C","C","C","C", "D","D","D","D" )

# one can also do
x = c( rep("A",4), rep("B",4), rep("C",4), rep("D",4) )



# To get effects model
mod = aov(y~x)
summary(mod)      # F-test for yield variable effect
summary.lm(mod)   # t-tests for variety effects as compared to tau A


# To get means model
mod2 = aov(y~x+0)
summary(mod2)     # Actually meaningless F-test for mean(data)=0
summary.lm(mod2)  # What do these estimates estimate?