Efficient Utilization of Non-Numerical Information in Quantitative Analysis General Theory and the Case of Simple Order

This is the authors' abstract. We don't add key points for this paper.

Suppose a single contrast y = ∑ cⱼ uⱼ, where ∑ cⱼ = 0, is to be tested as a basis for detecting differences among unknown parameters μⱼ, where yⱼ = μⱼ + εⱼ, and the εⱼ are independent and normally distributed with mean zero and variance σ². Write μⱼ = α + β xⱼ. Then the problem is to detect β ≠ 0. If ∑ xⱼ = 0, and ∑ x²ⱼ = 1, the noncentrality of y, referred to its standard deviation, is (β/σ) times the formal correlation coefficient r between the cⱼ and the xⱼ. If the xⱼ are known, the cⱼ can be chosen to make the correlation unity. If the xⱼ are wholly unknown, no single contrast can guarantee power in detecting β ≠ 0. Intermediate situations, where we know something but not everything about the xⱼ, occur frequently. If our knowledge can be placed in the form of linear inequalities restricting the μⱼ (equivalently the xⱼ) the problem of choosing a contrast ⱼ\ which will give relatively good power against the unknown (latent) configuration ⱼ\ is a relatively manageable one. The problem is to obtain a large value of r² between ⱼ\ which is at our choice, and ⱼ\, which is only partially known. A conservative approach is to try to select the ⱼ\ so that the minimum value of r² compatible with the restrictions on ⱼ\ is maximized, or nearly so. The maximization of minimum r² when response patterns are constrained by linear homogeneous inequalities leads to the mathematical problem of finding the geometric direction whose maximum angle with a given set of directions is least. The solution to this problem is characterized and proven unique (Sections 8, 17-20). No useful algorithm which is absolutely certain to reach the solution in a few steps appears to exist. However, procedures are discussed (Sections 10 and 11) which reach a solution relatively rapidly in the instances we have considered. The procedures are illustrated on selected examples (Sections 15-16). The general theory is applied (Sections 13-14) to the latent configuration defined by x₁ x₂ x₃ ⋯ xₙ, which we call simple rank order. A formula is found for the maximum contrast which maximizes minimum r², and its coefficients are given for n 20. The "linear-2-4" contrast, constructed from the usual linear contrast by quadrupling c₁ and cₙ, and doubling c₂ and cₙ₋₁, is a reasonable approximation to the maximum contrast for small or medium n, and its minimum r² remains above 90{%} of the maximum possible for n 50 (Table 2). Knowing only simple rank order for the μⱼ, good practice seems to indicate the use of "maximum" or "linear-2-4" contrasts in careful work. If more information or insight about the xⱼ is available, some other contrast may be preferable.