By Sahai H., Ojeda M.M.
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Extra resources for Analysis of variance for random models, vol.2: Unbalanced data
5), the desired estimate of µ is µˆ = G(Y ) = y¯. Thus, in this case, the estimate coincides with the usual sample mean. 7 MAXIMUM LIKELIHOOD ESTIMATION Maximum likelihood (ML) equations for estimating variance components from unbalanced data cannot be solved explicitly. Thus, for unbalanced designs, explicit expressions for the ML estimators of variance components cannot be found in general and solutions have to obtained using some iterative procedures. The application of maximum likelihood estimation to the variance components problem in a general mixed model has been considered by Hartley and Rao (1967) and Miller (1977, 1979), among others.
The mean squares of these analyses (weighted or unweighted) can then be used for estimating variance components in random as well as mixed models. Estimators of the variance components are obtained in the usual manner of equating the mean squares to their expected values and solving the resulting equations for the variance components. The estimators, thus obtained, are unbiased. This is, of course, only an approximate procedure, with the degree of approximation depending on the extent to which the unbalanced data are not balanced.
1) In Method III, the reductions in sums of squares are calculated for a variety of submodels of the model under consideration, which may be either a random or a mixed model. Then the variance components are estimated by equating each computed reduction in sum of squares to its expected value under the full model, and solving the resultant equations for the variance components. 4). 2) where β = (β1 , β2 ), without any consideration as to whether they represent ﬁxed or random effects. 2). 2). 1, we have E(Y QY ) = E(Y )QE(Y ) + tr[Q Var(Y )].
Analysis of variance for random models, vol.2: Unbalanced data by Sahai H., Ojeda M.M.