Running head : REML USING DERIVATIVES Restricted Maximum Likelihood Estimation for Animal Models Using Derivatives of the Likelihood

Restricted Maximum Likelihood estimation using first and second derivatives of the likelihood is described. It relies on the calculation of derivatives without the need for large matrix inversion using an automatic differentiation procedure. In essence, this is an extension of the Cholesky factorisation of a matrix. A reparameterisation is used to transform the constrained optimisation problem imposed in estimating covariance components to an unconstrained problem, thus making the use of Newton-Raphson and related algorithms feasible. A numerical example is given to illustrate calculations. Several modified Newton-Raphson and Method of Scoring algorithms are compared for applications to analyses of beef cattle data, and contrasted to a derivative-free algorithm.