Fitting nested terms in SAS and R:

This lecture has code for a nested experiment. The data is from the 
text "Design of Experiments: Statistical Principles of Research Design 
and Analysis" by Robert O. Kuehl (second edition)

Glucose was measured in a laboratory on three different days, with two separate 
runs per day. Thus runs are nested within days.

SAS

In SAS, to specify a nested term use parentheses. In the code below we 
see that run is nested within day by writing "run(day)"


 proc glm data = table77 ;
  class day run ;
  model glucose = day run(day) ;
  random day run(day) / test ;
  lsmeans day / e = run(day) stderr ;
  run ;

 proc mixed data = table77 ;
  class day run ;
  model glucose = ;
  random day run(day)  ;
 run ;


R 

R has more than one way to specify a nested term, depending on what 
package is used. Listed below are two ways to use with the aov function:

summary(aov(glucose ~ day + Error(day/run), data = table77))

summary(aov(glucose ~ day + run %in% day, data = table77))

Either way calculates the sums of squares for run within day correctly, 
but they do not form the correct F ratio. Also, the two notations above 
are not often recognized by other packages. 

Another way is more reliable, but less transparent. This way is based on 
the identity that:

SSB(A) = SSB + SSA*B

(In other words, the SS for a nested effect equals what SSB + SSA*B would 
have been had the factors been crossed instead of nested) Thus one way to 
calculate nested SS is to pretend the effect is crossed, calculate both SSB 
and SSA*B and add them:

summary(aov(glucose ~ day + run + run:day, data = table77a))

However this is inconvenient and sometimes not possible depending on how the 
nested effect was coded. Instead we can use the fact that if a model is specified
with an interaction but excluding a lower-order term, the software generally 
includes the term automatically. In other words if the software sees:

summary(aov(glucose ~ day + run:day, data = table77a))

It will automatically use SSrun + SSrun:day as the result for SSrun:day. This
approach works more widely than just with the aov function, many of my 
examples use it.


Altered data for this example:

data <- matrix(scan(),ncol=3,byrow=TRUE)
 1 1 42.5  1 1 43.3  1 1 42.9  1 2 42.2  1 2 41.4  1 2 41.8 
 2 1 48.0  2 1 44.6  2 1 43.7  2 2 42.0  2 2 42.8  2 2 42.8 
 3 1 41.7  3 1 43.4  3 1 42.5  3 2 40.6  3 2 41.8  3 2 41.8 

data
table77a  <- data.frame(data)
colnames(table77a)  <- c("day","run","glucose")
rm(data)

table77a$day  <- as.factor(table77a$day)
table77a$run  <- as.factor(table77a$run)


summary(aov(glucose ~ day + run %in% day, data = table77a))

summary(aov(glucose ~ day + run + run:day, data = table77a))

summary(aov(glucose ~ day + run:day, data = table77a))