Sparse Matrix Representations of Linear Mixed Models

We describe a representation of linear mixed-effects models using a sparse semidefinite matrix. This representation provides for efficient evaluation of the profiled log-likelihood or profiled restricted loglikelihood of the model, given the relative precision parameters for the random effects. The evaluation is based upon the LDLT form of the Cholesky decomposition of the augmented sparse representation. Additionally, we can use information from this representation to evaluate ECME updates and the gradient of the criterion being optimized. The sparse matrix methods that we employ have both a symbolic phase, in which the number and the positions of nonzero elements in the result are determined, and a numeric phase, in which the actual numeric values are determined. The symbolic phase need only be done once and it can be accomplished knowing only the grouping factors with which the random effects are associated. An important part of the symbolic phase is determination of a fill-minimizing permutation of the rows and columns of the sparse semidefinite matrix. This matrix has a special structure in the linear mixed-effects problem and we provide a new fill-minimizing algorithm tuned to this structure.