🤖 AI Summary
Conventional p-values suffer from unintuitive interpretation, lack of comparability across test statistics, and difficulty in combining independent evidence. Method: We propose quantile-based standardized measures—s-values (measuring tail distance in semi-tail units, where each unit halves the tail probability) and ζ-values (a two-sided symmetric extension)—establishing the semi-tail unit as a unified scale of extremity. Our approach integrates quantile standardization, logarithmic transformation, Bahadur slope analysis, and additive evidence synthesis, enabling linear critical-value inference and direct summation of s-values for multi-source evidence integration. Contribution/Results: We introduce an interpretable logarithmic-scale s-value; construct the two-sided compatible ζ-value; and propose a novel asymptotic efficiency metric based on slope differences. Empirical validation across diverse settings—including standardized testing and poker hand distributions—demonstrates the naturalness and broad applicability of the framework.
📝 Abstract
We introduce $ζ$- and $s$-values as quantile-based standardizations that are particularly suited for hypothesis testing. Unlike p-values, which express tail probabilities, $s$-values measure the number of semi-tail units into a distribution's tail, where each unit represents a halving of the tail area. This logarithmic scale provides intuitive interpretation: $s=3.3$ corresponds to the 10th percentile, $s=4.3$ to the 5th percentile, and $s=5.3$ to the 2.5th percentile. For two-tailed tests, $ζ$-values extend this concept symmetrically around the median.
We demonstrate how these measures unify the interpretation of all test statistics on a common scale, eliminating the need for distribution-specific tables. The approach offers practical advantages: critical values follow simple arithmetic progressions, combining evidence from independent studies reduces to the addition of $s$-values, and semi-tail units provide the natural scale for expressing Bahadur slopes. This leads to a new asymptotic efficiency measure based on differences rather than ratios of slopes, where a difference of 0.15 semi-tail units means that the more efficient test moves samples 10% farther into the tail. Through examples ranging from standardized test scores to poker hand rankings, we show how semi-tail units provide a natural and interpretable scale for quantifying extremeness in any ordered distribution.