Restricted Maximum Likelihood to estimate variance components for mixed models with two random factors

A Restricted Maximum Likelihood procedure is described to estimate variance components for a univariate mixed model with two random factors. An EM-type algorithm is presented with a reparameterisation to speed up the rate of convergence. Computing strategies are outlined for models common to the analysis of animal breeding data, allowing for both a nested and a crossclassified design of the 2 random factors. Two special cases are considered: firstly, the total number of levels of fixed effects is small compared to the number of levels of both random factors " secondly, one fixed effect with a large number of levels is to be fitted in addition to other fixed effects with few levels. A small numerical example is given to illustrate details.