Mathematical incompleteness of the fragility index

Background The Fragility Index (FI) is intended to quantify how many outcome changes would be required to convert a statistically significant two-arm trial result into a non-significant one. For a metric defined on valid 2×2 trial tables, mathematical completeness means that a finite numeric value is obtainable for every valid input in its intended domain. This study evaluated whether the Fragility Index satisfies that property. Methods FI was analyzed as defined: baseline significance required (p < 0.05), one-way movement only, and outcome changes restricted to converting a nonevent to an event in the arm with fewer events while keeping arm size fixed. Mathematical incompleteness was assessed by determining whether valid 2×2 tables exist for which no finite FI can be obtained under these rules. Evidence is provided through formal counterexamples, complete enumeration of all valid nondegenerate 2×2 tables up to total sample size N = 60, and empirical evaluation of published two-arm trials with binary outcomes. Results Valid baseline-significant 2×2 tables exist for which FI is not attainable. A simple counterexample is {3,0,4,11}: baseline Fisher's exact p = 0.04289216, the arm with fewer events is uniquely identified, but that arm has no nonevents available for the required toggle, so no legal FI path exists. Enumeration showed that unattainable cases first appeared at N = 18 and then recurred at every larger sample size through N = 60; by N = 60, 2,390 of 20,774 evaluable baseline-significant tables were unattainable (11.5%). In an empirical dataset of published trials, 2 of 82 baseline-significant evaluable trials (2.4%) were not attainable. Conclusions The Fragility Index is mathematically incomplete. This is a structural property of the FI algorithm, not a data validity issue, confirmed by complete table enumeration and real published trial data.