## Computational Method

The building blocks of the analysis are the
sums of squares for the dependent variables for each
classification variable within the factors that precede
it in the model, corrected for the factors that follow it.
For example, for a two-factor nested design,
PROC NESTED computes the following sums of squares:

where *y*_{ijr} is the *r*th replication, *n*_{ij} is the
number of replications at level *i* of the first factor
and level *j* of the second, and a dot as a subscript
indicates summation over the corresponding index.
If there is more than one dependent variable, PROC NESTED also
computes the corresponding sums of crossproducts for each pair.
The expected value of the sum of squares for a
given classification factor is a linear combination
of the variance components corresponding to this
factor and to the factors that are nested within it.
For each factor, the coefficients of
this linear combination are computed.
(The efficiency of PROC NESTED is partly due to the fact
that these various sums can be accumulated with just
one pass through the data, assuming that the data
have been sorted by the classification variables.)
Finally, estimates of the variance components are derived
as the solution to the set of linear equations that arise
from equating the mean squares to their expected values.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.