Least Squares and Maximal Contrasts: An Elementary General Approach and Its Application to Simultaneous Inference in Contingency Tables

Abstract

1. Introduction and Summary An elementary proof of some basic relationships in analysis of variance is presented in this paper. This approach is also shown to yield fundamental results in testing and simultaneous inference. The common textbook solution of linear least-squares methods derives the requisite normal equations either by partial differentiation or by appeals to vector spaces and their orthogonal bases (Scheff6, 1959, section 1.3). In either case it is necessary first to formulate a linear model for the expectations. The presentation of simultaneous confidence intervals usually involves a geometric argument which yields the relation between the statistics for individual functions and the sums of for linear sets of functions (Scheff6, 1953 - see also Scheff6, 1959, Appendix III). It is shown here how both these results can be derived simultaneously by an application of the Cauchy-Schwarz inequality - the Theorem of section 2. This approach hinges on the equality of the maximum of the for single functions and the minimum sum of squares for a set of such functions. The equalities and inequalities used in simultaneous inference follow naturally and do not require separate treatment. The present derivation is simpler mathematically than the common ones. Moreover, it does not require postulation of a model on variables and expectations but stresses the algebraic structure of the statistics. This may be useful in yielding data-analytic insights. For these reasons, the present approach is likely to be didactically preferable to the common one, at least at the introductory level. (For more advanced students the geometric approach no doubt