Univariable Optimal Discriminant Analysis:One-Tailed Hypotheses

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Abstract
For applications involving a single attribute, univariable optimal discriminant analysis (UniODA) is appropriate when one desires to identify a discriminant classifier that explicitly maximizes classification accuracy for a given sample of data. An open-form enumerable solution for the theoretical distribution of optimal values (number of misclassifications) arising from two-category UniODA of continuous random data has recently been discovered, for the case of a two-tailed (nondirectional) hypothesis. This article describes the theoretical distribution of optima arising from two-category UniODA of continuous random data for a one-tailed (directional) hypothesis. Enumeration of the one-tailed distribution and examination of the corresponding figurate triangle reveals a closed-form solution, for which computation time is linear over sample size. Using this result, the exact Type I error rate may also be easily computed for any orthogonal (equal class sizes) two-tailed UniODA design when classification accuracy is at least 75%. Directional two-category UniODA is illustrated using an application investigating the relationship between depression and monamine turnover in the brain.