Another Look at Henderson's Methods of Estimating Variance Components

Three methods of estimating variance components given by Henderson [1953] are critically reviewed and reformulated in matrix theory. Some modifications, as well as a fourth method, are also considered. Some conclusions are: 1. Method 1 is the easiest method to compute but it is inappropriate for mixed models. 2. The Generalized Method 2 contains elements of arbitrariness and is not uniquely defined: the Simplified Generalized Method 2 cannot be used if there are interactions between fixed and random effects, neither when they are considered fixed nor random. Henderson's Method 2 is but one form of the Simplified Generalized Method 2. 3. Method 3 is the most suitable method for mixed models, and for models involving covariances between sets of random effects. However, it can involve matrices of very large order. 4. Method 4 is a variant of Method 2, involving somewhat simpler calculations but still restricted by the lack of uniqueness in Method 2.