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An ``average information'' Restricted Maximum Likelihood algorithm for
estimating reduced rank genetic covariance matrices or covariance
functions for animal models with equal design matrices.

K. Meyer

*Genetics Selection Evolution* **29** : 97-116

### Abstract

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

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K.Meyer, June 22, 2000