Analysis of the Unbalanced Linear Model Based on the Balanced Model

Abstract

For a given unbalanced linear model y = X beta+epsilon, the computations of the estimator of the parameter beta and sums of squares are based on computation of a generalized inverse (X'X)(-) and the projection matrix P-X = X(X'X)(-)X'. The design matrix X can be expressed as a product of two matrices T and X(0), namely X = TX(0), where X(0) is the design matrix of the corresponding balanced model assuming that the model contains exactly one observation in each cell and T is the matrix indicating the replications of each cell. In this paper we espress P-X in terms of T and P-0 = X(0)(X(0)'X(0))-X(0)', 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.