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

Abstract

Suppose a single contrast $y=\sum c_j{u}_j$, where $\sum c_j=0$, is to be tested as a basis for detecting differences among unknown parameters ${\mu }_j$, where $y_j={\mu }_j+{ϵ}_j$, and the ${ϵ}_j$ are independent and normally distributed with mean zero and variance ${\sigma }^2$. Write ${\mu }_j=\alpha +\beta x_j$. Then the problem is to detect $\beta \ne 0$. If $\sum x_j=0$, and $\sum x_j^2=1$, the noncentrality of $y$, referred to its standard deviation, is $(\beta /\sigma )$ times the formal correlation coefficient $r$ between the $c_j$ and the $x_j$. If the $x_j$ are known, the $c_j$ can be chosen to make the correlation unity. If the $x_j$ are wholly unknown, no single contrast can guarantee power in detecting $\beta \ne 0$. Intermediate situations, where we know something but not everything about the $x_j$, occur frequently. If our knowledge can be placed in the form of linear inequalities restricting the ${\mu }_j$ (equivalently the $x_j$) the problem of choosing a contrast $\{c_j\}$ which will give relatively good power against the unknown (latent) configuration $\{x_j\}$ is a relatively manageable one. The problem is to obtain a large value of $r^2$ between $\{c_j\}$ which is at our choice, and $\{x_j\}$, which is only partially known. A conservative approach is to try to select the $\{c_j\}$ so that the minimum value of $r^2$ compatible with the restrictions on $\{x_j\}$ is maximized, or nearly so. The maximization of minimum $r^2$ 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 $$, which we call simple rank order. A formula is found for the maximum contrast which maximizes minimum $$, and its coefficients are given for $$. The "linear-2-4" contrast, constructed from the usual linear contrast by quadrupling $$ and $$, and doubling $$ and $$, is a reasonable approximation to the maximum contrast for small or medium $$, and its minimum $$ remains above 90{\tt\%} of the maximum possible for $$ (Table 2). Knowing only simple rank order for the $$, good practice seems to indicate the use of "maximum" or "linear-2-4" contrasts in careful work. If more information or insight about the $$ is available, some other contrast may be preferable.