An 'average information' restricted maximum likelihood algorithm for estimating reduced rank genetic covariance matrices or covariance functions for animal models with equal design matrices

A quasi-Newton restricted maximum likelihood algorithm that 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.