The Neutrality Boundary Framework: Quantifying Statistical Robustness Geometrically

TL;DR

The Neutrality Boundary Framework introduces geometric metrics to quantify statistical robustness and fragility, providing a threshold-free, sample-size invariant measure that complements traditional effect size and p-value assessments.

Contribution

It proposes a novel geometric framework for quantifying statistical robustness, deriving a general formula, and providing domain-specific implementations for various statistical measures.

Findings

Provides a threshold-free robustness measure

Proves properties like boundedness and monotonicity

Offers domain-specific implementations for common statistics

Abstract

We introduce the Neutrality Boundary Framework (NBF), a set of geometric metrics for quantifying statistical robustness and fragility as the normalized distance from the neutrality boundary, the manifold where the effect equals zero. The neutrality boundary value nb in [0,1) provides a threshold-free, sample-size invariant measure of stability that complements traditional effect sizes and p-values. We derive the general form nb = |Delta - Delta_0| / (|Delta - Delta_0| + S), where S>0 is a scale parameter for normalization; we prove boundedness and monotonicity, and provide domain-specific implementations: Risk Quotient (binary outcomes), partial eta^2 (ANOVA), and Fisher z-based measures (correlation). Unlike threshold-dependent fragility indices, NBF quantifies robustness geometrically across arbitrary significance levels and statistical contexts.