A General Purpose Approximation to Restricted Maximum Likelihood: The Tilde-Hat Approach

Restricted maximum likelihood estimates of variances and covariances are often preferred by animal breeders but can be expensive or impossible to compute for large data sets. Less expensive, approximate restricted maximum likelihood estimates can be obtained by using quadratic forms that resemble the restricted maximum likelihood quadratics but have expectations easier to compute. Quadratic forms for the tilde-hat approach resemble the restricted maximum likelihood quadratics more closely than previous approximations. The strategy is not difficult computationally even for models containing additive genetic relationships. Of three approximate strategies tested, tilde-hat gave estimates closest to restricted maximum likelihood in an actual data set. All three approximations were strongly biased downward by selection in simulated data. The restricted maximum likelihood quadratic seems to account for selection well, whereas substitute quadratics account for selection poorly.