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).