A.1 Ordering strategies

An essential set-up set in REML analyses is to find a permutation of the rows and columns in the coefficient matrix of the mixed model equations that makes the ‘fill-in’ arising during factorisation small and thus minimises the computational efforts per likelihood evaluation and REML iterate. This needs to be done only once per analysis (WOMBAT saves the results from this step for re-use in any subsequent steps). As it can have a dramatic impact on the time and memory required per analysis, it is well worth considerable effort to find the ‘best’ order. Especially for analyses involving large data sets or multiple random effects, the time spend trying several, or even numerous alternatives is readily recouped within the first few iterates [26]. WOMBAT selects a default ordering strategy based on the number of equations in the analysis.

Three different stratgies are implemented :

The multiple minimum degree procedure [21] as implemented in the widely used public domain subroutine genmmd. This is the strategy which has been used in DfReml. For WOMBAT, it is the default for small analyses involving up to 25000  equations.
The approximate minimum degree ordering of Amestoy et al. [1]. This tends to produce orderings of similar quality to the multiple minimum degree procedure, but is considerably faster. Implementation is through the public domain subroutine amd (version 1.1) available at
www.cise.ufl.edu/research/sparse/amd. This is the default for analyses involving more than 25000  and up to 50000  equations.
A multilevel nested dissection procedure, as implemented in subroutine
metis_nodend from the MeTiS package (public domain) of Karypis and Kumar [18], available at at www.cs.umn.edu/∼ karypis/metis. This is the default for large analyses.