Matrix oriented least squares or regression algorithms require substantial amounts of computer storage in order to solve analysis of variance problems. However, iterative methods exist which are capable of reducing the storage problem. These employ well-known balanced analysis of variance computations which do not require computer storage. The normal equations corresponding to a linear model with unbalanced data can be expressed in terms of the design matrix X for the cell means model. This fact can be used to construct algorithms which require a balanced analysis of variance problem to be solved in each iteration. A rule for constructing a generalized inverse of X\u27X which is positive definite and lower triangular is given. An iterative algorithm based on the modified conjugate gradient method to obtain the parameter estimates of an analysis of variance problem without storing X or X\u27X is developed using this inverse. This algorithm reduces the number of iterations required as compared to algorithms given previously. Further, the algorithm does not require reparameterization of the X matrix. An iterative method is also developed for calculating the sum of squares for testing a linear hypothesis in the original overparameterized model directly. Programs are implemented which compute the analysis of variance table and parameter estimates for linear models with unbalanced data using the above algorithms.
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