The Method of Least Squares

This note calls attention to a variant formulation of the least-squares adjustment procedure [1] that should prove increasingly attractive as electronic dataprocessing equipment becomes ever more widely available. It indicates that the statistician need not concern himself with the two rituals featured in traditional textbook presentations—namely, the construction and solution of the conventional "normal equations." By simply arranging the original data (together with l's and O's) in a particular pattern and without engaging in any arithmetic operations, he (or an aide) may instantly form the essential matric system to be solved. This system contains no processed elements—no sums or products—yet is algedracalliy equivalent to the usual normal equations. Thus, model design, data compilation, and explicit statement of the simultaneous equations to be solved are entiiely divorcible, in linear regression analysis, from computation, which may be deferred and delegated to specialists as a unified task. To illustrate the variant approach, we consider the classical linear regression problem in which n observed values of only one varíate are deemed subject to (random) error. This linear model has the matrix form