Statistical matrices, D. Bates, Univ. Wisconsin
There will be no fill-in regardless of the ordering and it is even the case
that the inverse has the same pattern of nonzeros as the original L factor.
To explain why this happens it may be best if I explain what "nested grouping
factors" means. In the Chem97 example in that writeup I sent the matrix Zt is
generated according to the school and the lea (local education authority - the
British equivalent of a school district) that the student was in when they took
the exam. The rows of Zt correspond to schools and to lea's. The rows
correspond to students. The "grouping factors" are school and lea. We say
that school is nested within lea in the sense that each school occurs in one
and only one lea. This is in contrast to the other example using the "star"
data where the grouping factors are student, teacher and school. Multiple
observations on the same student are often with different teachers so student
is not nested within teacher.
Returning to the Chem97 example, the Zt matrix for this model is an indicator
matrix of the school for each student vertically concatenated with the
indicator matrix of the lea for each student. The structure of Zt (Zt)' has
a diagonal block corresponding to schools, a diagonal block corresponding to
lea's and the off-diagonal block. The point is that there is exactly one
nonzero in each column of the off-diagonal block in the lower-left.
The way I would reorder these columns and rows to show the structure would be
to put all the schools associated with the first lea followed by the first lea
followed by all the schools associated with the second lea followed by the
second lea ... Then the L matrix will be block diagonal with each block
corresponding to an lea. Within each block the structure is a diagonal matrix
in the first k-1 rows and columns and a dense row underneath it.
By the way, don't try to decompose ZtZ as it is only positive semidefinite.
The matrix that is decomposed is ZtZ + Omega where Omega is diagonal with the
diagonal consisting of 2410 elements of 4.419696 (corresponding to schools) and
131 elements of 349.0238 (corresponding to lea's).