A Method with Low Computing Cost to Approximate Increments of Determinants Needed for Derivative-Free Restricted Maximum Likelihood

¹ Departamento de Producción Animal, Escuela Técnica Superior de lngenieros Agrónomos, Universidad Politécnica de Madrid, E28040 Madrid, Spain
² National Institute of Animal Science, Dept. of Research in Pigs and Horses, PO Box 39, DK-8830 Tjele, Denmark

Estimation of variance components by REML via derivative-free algorithms requires the determinant of the coefficient matrix of mixed model equations. A Monte Carlo-based method is proposed to approximate the increment of the logarithm of the determinant of this coefficient matrix that corresponds to increments of the ratio between residual and additive genetic variance. The computing cost of this method is linear with the order of the coefficient matrix. Results of approximate and exact methods were compared. A bias, detected when the difference of the ratio between residual and additive genetic variance is large, vanishes as convergence is reached. The proposed procedure is accurate enough to estimate the maximum of the likelihood function. The REML estimates of variance components using the Monte Carlo approximation are also presented.

Key Words: derivative-free algorithm • restricted maximum likelihood • Monte Carlo methods

Submitted on May 24, 1994
Accepted on September 26, 1994