The Invariance and Calculation of Method 2 for Estimating Variance Components

Three methods of estimating variance components from unbalanced data are given in Henderson [1953]. Method 2, designed to circumvent biasedness that results from using MIethod 1 on mixed models, is characterized more generally in Searle [1968] as a special way of executing what is described there as a simplified form of a Generalized Method 2. In that description 2 limitations are discussed: (i) It is proven that the simplified form of the Generalized M[ethod 2 can be used only when there are no interactions between fixed and random factors. (ii) It is suggested that the Generalized Method 2 is not uniquely specified. There is no doubt about limitation (i): Henderson's Method 2 can be used only when the model has no interactions between fixed and random effects. However, we show here for a wide class of -models that Henderson's Method 2, despite being one of the methods collectively called Generalized Method 2, does not suffer from limitation (ii). Indeed, it is a generalization of limitation (i) that nullifies (ii); i.e., for a wide class of models, Method 2 is well defined. One comment on limitation (i) is appropriate. Models that include any nesting of fixed and random factors within each other are also excluded from Method 2, because such models are tantamount to having interactions between fixed aind random factors.