A Bayes Rule for the Symmetric Multiple Comparisons Problem

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Abstract
A simple LSD (least significant difference) rule is presented for simultaneously testing the differences between n treatments considered in all possible pairs. This rule is a simpler, fully completed, version of the Bayes rule presented for special cases in Duncan [4]. It is based on the same multiple decision theory model except for a modified and extended use of a conjugate chi-square density in the prior. The new rule has the same intuitively appealing dependence on the between-treatment F ratio, varying from a sensitive comparisonwise-α-like rule when F is large or moderate, to a conservative experimentwise-α-like rule when F is small. Tables of t for computing the LSD are presented for three choices of a type-1 to type-2 error-seriousness ratio, k = 50, 100 and 500 (analogous to the usual choices of α = .10, .05 and .01 in testing a single difference), and for full ranges of F and its degree of freedom q and f.