Computing Gaussian Likelihoods and Their Derivatives for General Linear Mixed Models

Algorithms are described for computing the Gaussian likelihood or restricted likelihood corresponding to a general linear mixed model. Included are arbitrary covariance structures for both the random effects and errors. Formulas are also given for the first and second derivatives of the likelihoods, thus enabling a Newton–Raphson implementation. The algorithms make heavy use of the Cholesky decomposition, the sweep operator, and the W-transformation. Also described are the modifications needed for variance profiling, Fisher scoring, and MIVQUE(0), as well as the computational order of the procedures.