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The method of least squares, fitting constants, is commonly used as an approximate solution for the analysis of variance when the model contains more than one random component. Techniques are available for obtaining the expected mean squares for balanced designs and for completely nested (hierarchal) experiments. For unbalanced designs containing crossed (factorial) variates, the expected mean squares are generally difficult to obtain. The expected mean squares for only a few relatively simple crossed or mixed (crossed and nested) experiments are readily available. This paper provides a single unified procedure for obtaining the expected mean squares from the forward solution of the Abbreviated Doolittle or Square Root Methods. This procedure encompasses any experimental design, balanced or unbalanced, for crossed, nested, or mixed classifications containing any type of effect (fixed, finite, or random).
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