Orthogonalization-Triangularization Methods in Statistical Computations

Procedures for reducing a data matrix to triangular form using orthogonal transformations are presented, e.g., Householder, Givens, and examples these procedures are compard to procedure operating on normal equations. We show how an analysis of variance can be constructed from the triangular reduction of the data matrix. Procedures for calculating sums of squares, degrees of freedom, and expected mean squares are presented. These procedures apply even with mixed models and missing data. It is demonstrated that all statistics needed for inference on linear combinations of parameters of a linear model may be calculated from the triangular reduction of the data matrix. Also included is a test for estimability. We also demonstrate that if the computations are done properly some inference is warranted even when the X matrix is ill-conditioned.