Genetics Selection Evolution 29 : 97-116
A Quasi-Newton Restricted Maximum Likelihood algorithm which approximates the Hessian matrix with the average of observed and expected information is described for the estimation of covariance components or covariance functions under a linear mixed model. The computing strategy outlined relies on sparse matrix tools and automatic differentiation of a matrix, and does not require inversion of large, sparse matrices.
For the special case of a model with only one random factor and equal design matrices for all traits, calculations to evaluate the likelihood, first and `average' second derivatives can be carried out trait by trait, collapsing computational requirements of a multivariate analysis to those of a series of univariate analyses. This is facilitated by a canonical decomposition of the covariance matrices and corresponding transformation of the data to new, uncorrelated traits.
The rank of the estimated genetic covariance is determined by the number of non-zero eigenvalues of the canonical decomposition, and thus can be reduced by fixing a number of eigenvalues at zero. This limits the number of univariate analyses needed to the required rank. It is particularly useful for the estimation of covariance function when a potentially large number of highly correlated traits can be described by a low order polynomial.
Key words : REML, average information, covariance components, reduced rank, covariance function, equal design matrices