Analysis of the unbalanced linear model based on the balanced model

Abstract

For a given unbalanced linear model g = X j + E, the computations of the estimator of the parameter j and sums of squares are based on computation of a generalized inverse ( X ' X ) -and the projection matrix P, = X ( X ' X ) -X ' . The design matrix X can be expressed as a product of two matrices Tand X,, namely X = ' IX,, where X, is the design matrix of the corresponding balanced model assuming that the model contains exactly one observation in each cell and Tis the matrix indicating the replications of each cell. In this paper we express P, in terms of Tand Po = X,(X;X,)-X;, the projection matrix of the corresponding balanced model. Using this result and the results from the corresponding balanced model, we can reduce a great amount of the computational storages required to compute the necessary statistics.