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Extra resources for Analysis of Variance for Random Models [Vol II - Unbalanced Data]
The analysis was originally proposed by Yates (1934) and provides a simple and efﬁcient method of analyzing data from experimental situations having unbalanced design structure with no empty cells. 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.
Rao (1973, p. 52). 3) with respect to each element of the ﬁxed effects and with respect to each of the variance components. Thus the ML estimators for the variance components do not take into account the loss in degrees of freedom resulting from estimating the ﬁxed effects and may produce biased estimates. 1) with p = 0, Y = Xα + e, and V = σe2 IN , the ML estimator for the single variance component σe2 is σˆ e2 = 1 ˆ (Y − Xα), ˆ (Y − Xα) N where αˆ = X(X X)−1 X Y . 34 Chapter 10. Making Inferences about Variance Components Clearly, σˆ e2 is a biased estimator since E(σˆ e2 ) = σˆ e2 (N − q)/N .
We have seen that for ﬁxed effects, having normal equations X Xβ = X Y , the reduction in sum of squares due to β, denoted by R(β), is R(β) = Y X(X X)− X Y . 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.
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